Exact Solutions to the Beam Equations

Exact Solutions to the Beam Equations

Chapter 2 Exact Solutions to the Beam Equations Contents 2.1. Introduction 2.2. Some Facts from the Theory of Continuous Groups 2.2.1. Definitions ...
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Chapter

2 Exact Solutions to the Beam Equations

Contents

2.1. Introduction 2.2. Some Facts from the Theory of Continuous Groups 2.2.1. Definitions and Examples of Groups 2.2.2. Infinitely Small Transformation and Infinitesimal Operator 2.2.3. The Functions Invariant with Respect to a Given Group and Invariant Manifolds 2.2.4. Extended Group 2.2.5. Main Group and Defining Equations 2.3. Calculation of the Main Group for the Equations Describing a One-Dimensional Nonstationary Beam 2.3.1. Beam Equations and Coordinates of the Operater of the Extended Group 2.3.2. Splitting the Equations 2.3.3. Intermediate Conclusion 2.3.4. Solving the Defining Equations 2.3.5. Main Group 2.4. Group Properties of the Beam Equations 2.4.1. Coordinates of Infinitesimal Operators in Different Systems 2.4.2. Preliminary Considerations 2.4.3. Non-Relativistic Beam 2.4.4. The Relativistic Beam 2.5. Invariant Solutions 2.5.1. The Concept of Invariant Solutions 2.5.2. An Example: The H-Solution 2.5.3. Substantially Different Invariant Solutions

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Advances in Imaging and Electron Physics, Volume 166, ISSN 1076-5670, DOI: 10.1016/B978-0-12-381310-7.00002-8. Copyright # 2011 Elsevier Inc. All rights reserved.

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Exact Solutions to the Beam Equations

2.6. Optimal Systems of the Subgroups for a Three-Dimensional Stationary Beam 2.6.1. The Adjoint Group 2.6.2. Optimal System of One-Parametric Subgroups 2.6.3. Some Comments 2.6.4. Invariant Solutions of the Rank 2 2.6.5. Two-Parametric Subgroups and Invariant Solutions of the Rank 1 2.7. Results of Constructing the Invariant Solutions 2.7.1. System of Three-Parametric Subgroups 2.7.2. Stationary Flows 2.7.3. Electrostatic Beams 2.7.4. Beams in a Magnetic Field 2.7.5. Relativistic Flows 2.7.6. Nonstationary Flows 2.8. Solutions Invariant with Respect to the Transformations with Arbitrary Functions of Time 2.8.1. H-Solutions of the Rank 1 2.8.2. H-Solutions of the Rank 3 2.8.3. z-Solenoidal Flows 2.8.4. z-Potential Flows 2.8.5. Laminated Flows 2.8.6. H-Solutions Inessentially Different with Respect to Infinite Subgroups 2.9. Invariant Solutions of the Geometrized Beam Equations 2.9.1. On the Exact Solutions of the Geometrized Beam Equations 2.9.2. Group Properties and Invariant Solutions 2.10. The Exact Solutions, Whose Relation to the Group Properties Is Yet Unknown 2.10.1. Terminology 2.10.2. Planar Solenoidal Flows 2.10.3. Degenerate Flows 2.10.4. Generalized Brillouin Flows 2.10.5. Reducing the Problem to a Linear Partial Differential Equation

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2.1. INTRODUCTION The purpose of this Chapter is to derive the most complete set of exact solutions to the beam equations. The exact solutions satisfy a system of ordinary differential equations and describe the principal regularities peculiar to such complicated nonlinear media as the dense beam (the 3/2 law). These solutions can be exceedingly useful in designing real electron guns

Exact Solutions to the Beam Equations

47

(e.g., the magnetron-injection guns used in gyrotrons; in this case, the separation of variables in spherical coordinates is available). The solutions also may serve as zero approximations to the relevant asymptotic series; they describe the equilibrium state when investigating the flow stability; and eventually, they can be used as reliable testing models in developing approximate and numerical techniques. In this last case, factors such as the fulfillment of the thermoemission condition on the curvilinear thermoemitter with a nonhomogeneous current density, taking into consideration the self-magnetic field of the beam and the nonhomogeneous external magnetic field, may be of profound importance. Because there is no exact solution that simultaneously addresses all these peculiarities, a set of various models should be used instead. This implies that any exact solution is of interest and the task of deriving such solutions is extremely urgent. The most effective approach to constructing exact solutions to a system of partial differential equations consists of studying its group properties—in other words, revealing all transformations of independent and dependent variables that leave the system of equations unchanged. Thus, we come to the notion of invariant solution (Ovsyannikov, 1958, 1959, 1962, 1978; Ovsyannikov and Ibragimov, 1975). Elementary facts from the theory of continuous groups (Ince, 1956), which are required to understand the essence of the approach, are provided in Section 2.2. Later in this section we demonstrate how the main group can be calculated, outline the preliminary considerations, which before the use of the general approach, allow deriving essential information on the group, illustrate the most important notions of the method, and discuss the results of studying the group properties of the equations describing the nonstationary spatial flows. The majority of the exact solutions known from the literature appear to be invariant. Among them are such ‘‘exotic’’ specimens as the stationary solutions being invariant with respect to the infinite groups with arbitrary functions of time; such groups ‘‘preserve’’ the equations of a nonstationary beam. We analyze below those few exact solutions whose relation to the group properties of a beam is yet unknown. The content of this section is based primarily on works by Syrovoy (1962, 1963, 1964a, 1965a,c,d, 1985a,b, 2003a,b, 2004b, 2005a, 2008b, 2009), Vashkovskii and Syrovoy (1983, 1991, 1992), and Vashkovskii et al. (1996).

2.2. SOME FACTS FROM THE THEORY OF CONTINUOUS GROUPS 2.2.1. Definitions and Examples of Groups Let us consider a set of transformations of the points (x, y) belonging to a 2D space, defined as

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Exact Solutions to the Beam Equations

 x ¼ jðx; y; aÞ;

 y ¼ cðx; y; aÞ;

(2.1)

where a is a parameter that continuously varies within a definite interval. If (1) the given set of transformations contains the identical transformation, (2) each of the transformations of the set possesses its own inverse transformation, and (3) two consecutive transformations (2.1) with the parameters a1 and a2 are equivalent to a third transformation with the parameter a3, so that jðjðx; y; a 1 Þ; cðjðx; y; a 1 Þ;

cðx; y; a1 Þ; a2 Þ ¼ jðx; y; a3 Þ; cðx; y; a2 Þ; a2 Þ ¼ cðx; y; a3 Þ;

(2.2)

we say that the transformations (2.1) comprise a finite continuous group. Let us consider some relevant examples.  The group of shifts (translations) parallel to the x-axis:

 x ¼ x þ a;

 y ¼ y:

(2.3)

 The group of rotations around the origin of coordinates:

 x ¼ x cosa  y sina;

 y ¼ x sina þ y cosa:

(2.4)

 y ¼ ay:

(2.5)

 The scaling group:

 x ¼ ax;

Obviously, the transformations (2.3)–(2.5) satisfy the group properties formulated above.

2.2.2. Infinitely Small Transformation and Infinitesimal Operator Let us assume that a ¼ 0 in Eq. (2.1) corresponds to an identical transformation and consider a function f ð x;  yÞ expanded into a power series 1 00 f ð x;  yÞ ¼ f0 þ f00 a þ f0 a2 þ . . . ; 2   x;a þ f;y y;a f0 ¼ f ðx; yÞ; f00 ¼ f;x 

(2.6) a¼0

with respect to a in the vicinity of the point a ¼ 0. An infinitely small increment of the function f ð x;  yÞ is determined by the linear part of this expansion:

Exact Solutions to the Beam Equations

  df ¼ f ð x; yÞ f ðx; yÞ ¼ xðx; yÞf;x þ Zðx; yÞf;y a; xðx; yÞ ¼ j;a a¼0 ; Zðx; yÞ ¼ c;a a¼0 ; f;x ¼ f;x x;x þ f;y y;x :

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(2.7)

;x ! 1; y;x ! 0; f;x ! f;x at a ! 0. We have taken into account that x The expression for df may be considered as a result of applying the operator X ¼ xð@=@xÞ þ Zð@=@yÞ;

df ¼ Xf a

(2.8)

;  to the original function f ðx yÞ. The operator X is called the infinitesimal operator and the function x,Z its coordinates. Inasmuch as the second derivative with respect to a in (2.6) signifies, by definition, the repeated applying of the operator d/da, this expansion can be represented as     (2.9) f ð x;  yÞ ¼ f ðx; yÞ þ ða=1!ÞXf þ a2 =2! X2 f þ a3 =3! X3 f þ . . . With the transition from (2.6) to (2.9), one should consider the relations     X2 f ¼ X xf;x þ X Z f;y ;   X xf;x ¼ x2 f;xx þ xx;x f;x þ xZ f;xy þ Zx;y f;x ; (2.10)   X Z f;y ¼ xZ f;xy þ xZ;x f;y þ Z2 f;yy þ ZZ;y f;y : Replacing f in (2.8) by x, y with regard to the fact that a ¼ da in the vicinity of a ¼ 0, we obtain the coordinate increments for an infinitely small transformation in the following form: dx ¼ xðx; yÞda;

dy ¼ Zðx; yÞda:

(2.11)

Thus, if we know the infinitesimal operator of a group, the finite equations of such a group can be found by means of integrating the differential equations d x=xð x;  yÞ ¼ d  y=Zð x;  yÞ ¼ da;

a ¼ 0 : x ¼ x;

y ¼ y:

(2.12)

The finite equations of a group also follow from the expansion (2.9) in the form    x ¼ x þ ða=1!ÞXx þ a2 =2! X2 x þ . . . ;    (2.13) y ¼ y þ ða=1!ÞXy þ a2 =2! X2 y þ . . . Let us construct the infinitesimal operators for the groups (2.3)–(2.5). By comparing (2.3) and (2.11), for the group of translations along the x-axis we have x ¼ 1;

Z ¼ 0;

X ¼ @=@x:

(2.14)

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Exact Solutions to the Beam Equations

Similar consideration for the group of rotations (2.4) in the vicinity of a ¼ 0 gives   x ¼ x  ay; y ¼ y þ ax; X ¼ yð@=@xÞ þ xð@=@yÞ:

x ¼ y;

Z ¼ x;

(2.15)

For the scaling group, the identical transformation corresponds to a ¼ 1, da ¼ a  1; therefore,   x ¼ x þ xda; y ¼ y þ yda; X ¼ xð@=@xÞ þ yð@=@yÞ:

x ¼ x;

Z ¼ y;

(2.16)

Let us demonstrate how the finite equations (2.4) can be derived using Eq. (2.13) when the operator (2.15) is known: X ¼ yð@=@xÞ þ xð@=@yÞ; Xx ¼ y; X2 x ¼ x; X3 x ¼ y; X4 x ¼ x; X5 x ¼ y; . . . ; Xy ¼ x; X2 y ¼ y; X3 y ¼ x; X4 y ¼ y; X5 y ¼ x; . . . ;          x ¼ x 1  a2 =2! þ . . .  y ða=1!Þ  a3 =3! þ . . . ¼ x cosa  y sina;   2     3    y ¼ y 1  a =2! þ . . . þ x ða=1!Þ  a =3! þ . . . ¼ y cosa þ x sina: (2.17) The same result follows from Eqs. (2.12):  1=2  ¼ R2  x2 d x=ð yÞ ¼ d  y=x ¼ da;  x2 þ  y2 ¼ R 2 ; y ;  2 1=2 2 R  x d x ¼ da; arcsinð x=RÞ ¼ arcsinðx=RÞ  a;  x=R ¼ sin½ arcsinðx=RÞ  a;  1=2  x ¼ x cosa  y sina;  y ¼ R2   x2 ¼ x sina þ y cosa:

(2.18)

2.2.3. The Functions Invariant with Respect to a Given Group and Invariant Manifolds We call a function f(x, y) invariant with respect to the group (2.1) if f ð x;  yÞ ¼ f ðx; yÞ:

(2.19)

By using the expansion (2.9), we can see that the necessary and sufficient condition of the invariance takes the form of the identity Xf ¼ xf;x þ Z f;y  0:

(2.20)

Thus, the invariant with respect to the group (2.1) is determined by the solution of the first-order partial differential equation (2.20), which is equivalent to the ordinary differential equation x1 dx ¼ Z1 dy:

(2.21)

Exact Solutions to the Beam Equations

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As soon as Eq. (2.21) has a unique solution that depends on one arbitrary constant, each group with two independent variables possesses a unique invariant I(x, y). Any arbitrary function f(I) also represents an invariant for (2.1). The invariant manifolds for the group of rotations are circles. In 3D space, Eq. (2.21) should be replaced in this case by the differential system dx=ðyÞ ¼ dy=x ¼ dz=0;

(2.22)

the integrals of which are I1 ¼ R ¼ const, I2 ¼ z ¼ const, whereas the general solution to Eq. (2.20) is an arbitrary surface of revolution F1(R, z) ¼ 0, z ¼ z(R). In addition to the surfaces, the invariant manifolds of lesser dimensionality may exist in 3D space; those are the lines described by the equations F1(I1, I2) ¼ 0, F2(I1, I2) ¼ 0. Obviously, for the group of rotations those lines are the circles that result from the intersection of two surfaces of revolution.

2.2.4. Extended Group Let us consider a curve y ¼ y(x) that is transformed to the curve y ¼ yðxÞ using the transformations in (2.1). We consider the value p ¼ dy/dx, which is transformed into the value  p ¼ d y=dx, as the third variable      (2.23) p ¼ dc=dj ¼ c;x þ c;y p = j;x þ j;y p ¼ wðx; y; p; aÞ: The transformations acting on the element (x, y, p) as  x ¼ jðx; y; aÞ;

 y ¼ cðx; y; aÞ;

 p ¼ wðx; y; p; aÞ

(2.24)

determine a group called the extended group. From (2.1) we have the following for the extended group:  x ¼ x þ ða=1!Þx þ . . . ;  y ¼ y þ ða=1!ÞZ þ . . . ; h   ih   i1  p ¼ d y=d  x ¼ dy þ a Z;x dx þ Z;y dy þ . . . dx þ a x;x dx þ x;y dy þ . . . h   ih   i1 ¼ p þ a Z;x þ Z;y p þ . . . 1 þ a x;x þ x;y p þ . . . h   i ¼ p þ a Z;x þ Z;y  x;x p  x;y p2 þ . . . ¼ p þ ða=1!Þzðx; y; pÞ þ . . . (2.25) We now can see that the infinitesimal operator of the extended group is e ¼ xð@=@xÞ þ Zð@=@yÞ þ zð@=@pÞ; X

(2.26)

where z can be expressed through x and Z and quadratically depends on p.

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Exact Solutions to the Beam Equations

2.2.5. Main Group and Defining Equations Let us consider a system of equations (S) with respect to the sought functions uk(k ¼ 1, . . ., m) of the independent variables xi(i ¼ 1, . . ., n  m). In this case, the pair (x, u) determines a point in the n-dimensional Euclidean space ℰn. Consider in ℰn a manifold F determined by the equations uk ¼ Fk ðxÞ:

(2.27)

Let the continuous r-parametric group aj(j ¼ 1, . . ., r) be described as  xi ¼ fðixÞ ðx; u; aÞ;

k ¼ fðkuÞ ðx; u; aÞ: u

(2.28)

Let us introduce into consideration the derivatives on the manifold F, which can be obtained by differentiation of the functions (2.27): pik ¼ duk/ dxi. If (x, u) varies according to (2.28), the derivatives pik are transformed according to the law  pki ¼ fðkpÞi ðx; u; p; aÞ:

(2.29)

e , N ¼ n þ m(n  m) deterEquations (2.28) and (2.29) in the space ℰ N mine an extended group of transformations. These relations represent an obvious generalization of the relations (2.24) to case a multidimensional space. Using the rules of tensor summation, we can represent the operator of the extended group as       e ¼ xi @=@xi þ Zk @=@uk þ zk @=@pk : (2.30) X i i The expression for zik in the general case can be conveniently introduced using the auxiliary operators Di:   (2.31) zki ¼ Di Zk  pkj Di xj ; Di ¼ @=@xi þ pki @=@uk : Similar to z in Eq. (2.26), the functions zik depend on the coordinates of the operator X and quadratically depend on p. With no loss of generality, we may consider the system (S) as consisting of the first-order quasilinear equations. Any system of practical interest can be transformed to the similar form by introducing some supplementary sought functions: Cl ðx; u; pÞ ¼ clik ðx; uÞpki þ cl ðx; uÞ ¼ 0;

l ¼ 1; . . . ; M:

(2.32)

If, by analogy with (2.20), e l ¼ 0; XC

l ¼ 1; . . . ; M:;

(2.33)

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Exact Solutions to the Beam Equations

e , which then Eqs. (2.32) determine a hypersurface in the extended space ℰ N represents an invariant manifold for the group with the operator (2.30). The group G, which contains all its one-parametric subgroups preserving the system (S), is called the main group. Obviously, the main group can be constructed if we can determine the most general form of the e for which the requirements (2.33) are satisfied [we consider operator X Eqs. (2.33) as the equations for the coordinates of the operator X]. To do so, we need only to use Eq. (2.31) and the fact that M variables of the variables pik can be expressed by means of Eqs. (2.32) through the rest of the m(n  m)  M variables that are independent and take arbitrary values. After substituting Eqs. (2.31) into Eqs. (2.33) with regard to Eqs. (2.32), the LHSs of Eqs. (2.33) take the form of the nonuniform quadratic forms with respect to the independent pik. All coefficients in those quadratic forms should be identically equal to zero with respect to x, u. The equations for the coordinates x,Z derived in this manner are called the defining equations of the group G for the system (S).

2.3. CALCULATION OF THE MAIN GROUP FOR THE EQUATIONS DESCRIBING A ONE-DIMENSIONAL NONSTATIONARY BEAM 2.3.1. Beam Equations and Coordinates of the Operater of the Extended Group The particle motion between the parallel electrodes in the nonstationary case can be described by a system of the first-order quasilinear partial differential equations as u;t þ uu;x ¼ j;x ;

r;t þ ðruÞ;x ¼ 0;

j;x ¼ E;

E;x ¼ r;

u  vx : (2.34)

Let us introduce new notations for the independent and dependent variables x1 ¼ t;

x2 ¼ x;

u1 ¼ u;

u2 ¼ j;

u3 ¼ r;

u4 ¼ E;

pki ¼ @uk =@xi : (2.35)

The system (2.34) now takes the form p11 þ u1 p12 ¼ u4 ;

p31 þ u3 p12 þ u1 p32 ¼ 0;

p22 ¼ u4 ;

p42 ¼ u3 :

(2.36)

Equations (2.36) can be resolved with respect to the pik as follows: p11 ¼ u4  u1 p12 ;

p31 ¼ u3 p12  u1 p32 ;

p22 ¼ u4 ;

p42 ¼ u3 :

(2.37)

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Exact Solutions to the Beam Equations

Thus, the derivatives p21, p12, p23, p14 are independent. By applying e to Eqs. (2.37), we come to the equations similar to the operator X Eq. (2.33): z11 þ u1 z12 þ Z1 p12 ¼ Z4 ; z22 ¼ Z4 ; z42 ¼ Z3 :

z31 þ u3 z12 þ u1 z32 þ Z3 p12 þ Z1 p32 ¼ 0;

(2.38)

With regard to Eqs. (2.37), the operators Di take the form D1 ¼

  @   @ @ @ @ þ u4  u1 p12 þ p 21 2  u3 p12 þ u1 p32 þ p 41 4 ; @x1 @u1 @u @u3 @u

D2 ¼

@ @ @ @ @ þ p 12 1 þ u4 2 þ p 32 3 þ u3 4 : 2 @x @u @u @u @u

(2.39)

The coordinates zik of the extended operator, which are included in Eq. (2.38), can be expressed through xi,Zk by means of the following formulas:   z11 ¼ D 1 Z1  u4  u1 p12 D 1 x1  p12 D 1 x2 ;   z12 ¼ D 2 Z1  u4  u1 p12 D 2 x1  p12 D 2 x2 ;   z31 ¼ D 1 Z3 þ u3 p12 þ u1 p32 D 1 x1  p32 D 1 x2 ; (2.40)   z32 ¼ D 2 Z3 þ u3 p12 þ u1 p32 D 2 x1  p32 D 2 x2 ; z22 ¼ D 2 Z2  p21 D 2 x1  u4 D 2 x2 ; z42 ¼ D 2 Z4  p41 D 2 x1  u4 D 2 x2 :

2.3.2. Splitting the Equations Calculation of the main group should be started from the simplest equations of system (2.40); the information thus obtained will facilitate studying the more complicated equations. The last equation of Eqs. (2.38), with z24 substituted from the last equation of Eqs. (2.40), takes the form Z4;x2 þ p12 Z4;u1 þ u4 Z4;u2 þ p32 Z4;u3 þ u3 Z4;u4   p41 x1;x2 þ p12 x1;u1 þ u4 x1;u2 þ p32 x1;u3 þ u3 x1;u4   u3 x2;x2 þ p12 x2;u1 þ u4 x2;u2 þ p32 x2;u3 þ u3 x2;u4 ¼ Z3 :

(2.41)

Here and henceforward, we use the following notations for the partial derivatives: @x/@xi  x,xi; @x/@uk  x,uk; and @ 2x/@xi @uk  x,xiuk. Equation (2.41) represents a quadratic form with respect to the independent variables pik, the coefficients of which should identically vanish

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Exact Solutions to the Beam Equations

with respectto xi, uk (the combination of pik is positioned from the left of the symbol , the corresponding coefficient – from the right)   (2.41a,b,c) p12 p41  x1;u1 ¼ 0; p32 p41 k x1;u3 ¼ 0; p12  Z4;u1  u3 x2;u1 ¼ 0;   (2.41d,e) p32  Z4;u3  u3 x2;u3 ¼ 0; p41  x1;x2 þ u4 x1;u2 þ u3 x1;u4 ¼ 0;    (2.41f) 1 Z4;x2 þ u4 Z4;u2 þ u3 Z4;u4  x2;x2  u4 x2;u2  u3 x2;u4 ¼ Z3 : The effect of Eq. (2.41) generating a substantially larger number of relations is called splitting. It should be emphasized that, owing to the condition (2.41b), Eq. (2.41e) splits with respect to u3: x1;u4 ¼ 0; this leads to splitting of Eq. (2.41e) with respect to u4: x1;u2 ¼ 0, x1;x2 ¼ 0, which, in combination with (2.41a,b), gives   (2.42) x1 ¼ x1 x1 ; D 2 x1 ¼ 0; D 1 x1 ¼ x1;x1 : With regard to Eqs. (2.42), the third equation in Eqs. (2.38) appears as Z2;x2 þ p12 Z2;u1 þ u4 Z2;u2 þ p32 Z2;u3 þ u3 Z2;u4    u4 x2;x2 þ p12 x2;u1 þ u4 x2;u2 þ p32 x2;u3 þ u3 x2;u4 ¼ Z4 : Splitting with respect to pik results in three equations:   p12  Z2;u1  u4 x2;u1 ¼ 0; p32 Z2;u3  u4 x2;u3 ¼ 0;    1 Z2;x2 þ u4 Z2;u2 þ u3 Z2;u4  u4 x2;x2 þ u4 x2;u2 þ u3 x2;u4 ¼ Z4 :

(2.43)

(2.43a,b) (2.43c)

Now consider the first equation of Eqs. (2.38):     Z1;x1 þ u4  u1 p12 Z1;u1 þ p21 Z1;u2  u3 p12 þ u1 p32 Z1;u3 þ p 41 Z1;u4 h      u4  u1 p12 x1;x1  p12 x2;x1 þ u4  u1 p12 x2;u1 þ p 21 x2;u2 i h    u3 p12 þ u1 p32 x2;u3 þ p41 x2;u4 þ u1 Z1;x2 þ p12 Z1;u1 þ u4 Z1;u2 þ p32 Z1;u3  i þu3 Z1;u4  p12 x2;x2 þ p12 x2;u1 þ u4 x2;u2 þ p32 x2;u3 þ u3 x2;u4 þ Z1 p12 ¼ Z4 : (2.44) We start the splitting with regard to pik:  1 2  3 2 p2  u x;u3 ¼ 0:

(2.44a)

It follows from (2.41d) that Z4;u3 ¼ 0. This fact implies splitting with respect to u3 in Eq. (2.41c): x2;u1 ¼ 0, Z4;u1 ¼ 0. By taking into consideration (2.43a), the first of these zeros gives the equality Z2;u1 ¼ 0, while

56

Exact Solutions to the Beam Equations

Eqs. (2.44a) and (2.43b) give Z2;u3 ¼ 0. For further splitting of Eq. (2.44), we take into account that x2;u1 ¼ x2;u3 ¼ 0 :   p12 p 21   x2;u2 ¼ 0 ; p12 p41   x2;u4 ¼ 0:

(2.44b,c)

Thus, we have   x 2 ¼ x 2 x1 ; x2 ;

D 1 x2 ¼ x2;x1 ;

D 2 x2 ¼ x2;x2 :

We continue to split (2.44):  p12   u3 Z1;u3 þ u1 x1;x1  x2;x1  u1 x2;x2 þ Z1 ¼ 0;   p21 Z1;u2 ¼ 0; p41 Z1;u4 ¼ 0;    1Z1;x1 þ u4 Z1;u1  x1;x1 þ u1 Z1;x2 ¼ Z4 :

(2.45)

(2.44d) (2.44e,f) (2.44g)

According to the equalities Z2;u3 ¼ Z4;u3 ¼ 0 and Eq. (2.45), Eq. (2.43c) can be split with respect to u3: Z2;u4 ¼ 0.

2.3.3. Intermediate Conclusion Thus, we now know that x1 ¼ x1(x1), x2 ¼ x2(x1, x2), while Zk do not depend on some of the variables, because Z1;u2 ¼ Z1;u4 ¼ 0, Z2;u1 ¼ Z2;u3 ¼ Z2;u4 ¼ 0, Z4;u1 ¼ Z4;u3 ¼ 0. Besides,   Z1 ¼ x2;x1 þ u1 x2;x2  x1;x1 þ u3 Z1;u3 ; (2.44d)   Z3 ¼ Z4;x2 þ u4 Z4;u2 þ u3 Z4;u4  x2;x2 ;

(2.41f)

  Z4 ¼ Z2;x2 þ u4 Z2;u2  x2;x2 ;

(2.43c)

  Z4 ¼ Z1;x1 þ u1 Z1;x2 þ u4 Z1;u1  x1;x1 :

(2.44g)

We only need to consider the second equation of Eqs. (2.38) as follows:     Z3;x1 þ u4  u1 p12 Z3;u1 þ p 21 Z3;u2  u3 p12 þ u1 p32 Z3;u3   þ p 41 Z3;u4 þ u3 p12 þ u1 p32 x1;x1  p32 x2;x1   þ u3 Z1;x2 þ p12 Z1; u1 þ p32 Z1; u3  p12 x2;x2 (2.46)   1 3 1 3 4 3 3 3 3 3 3 2 þ u Z;x2 þ p2 Z;u1 þ u Z;u2 þ p2 Z;u3 þ u Z;u4  p2 x;x2 þ Z3 p12 þ Z1 p32 ¼ 0:

Exact Solutions to the Beam Equations

Splitting with respect to pik gives    p12  u3 x1;x1  Z3;u3 þ Z1;u1  x2;x2 þ Z3 ¼ 0;  p21  Z3;u2 ¼ 0;

 p32  u1 x1;x1  x2;x1 þ u3 Z1;u3  u1 x2;x2 þ Z1 ¼ 0;  p41  Z3;u4 ¼ 0;  1 Z3;x1 þ u4 Z3;u1 þ u3 Z1;x2 þ u1 Z3;x2 ¼ 0:

57

(2.46a) (2.46b) (2.46c) (2.46d) (2.46e)

If we compare Eqs. (2.46c) and (2.44d), we see that they differ only in the sign of the term containing u3; therefore Z1;u3 ¼ 0. Inasmuch as, according to Eq. (2.46d), the function Z3 does not depend on u4, Eq. (2.46e) can be split with respect to u4: Z3;u1 ¼ 0. Equating the expressions for Z4 from Eqs. (2.43c) and (2.44g) and taking into consideration that Z1;u4 ¼ Z2;u4 ¼ 0, we obtain Z1;x1 þ u1 Z1;x2 ¼ Z2;x2 ;

Z1;u1  x1;x1 ¼ Z2;u2  x2;x2 :

(2.47)

2.3.4. Solving the Defining Equations Let us combine and renumber the equations we have yet to solve, bearing in mind that x1 ¼ x1(x1), x2 ¼ x2(x1, x2), Z1;u2 ¼ Z1;u3 ¼ Z1;u4 ¼ 0, Z2;u1 ¼ Z2;u3 ¼ Z2;u4 ¼ 0, Z3;u1 ¼ Z3;u2 ¼ Z3;u4 ¼ 0, and Z4;u1 ¼ Z4;u3 ¼ 0:   Z1 ¼ x2;x1 þ u1 x2;x2  x1;x1 ; (2.48)   Z3 ¼ Z4;x2 þ u3 Z4;u4  x2;x2 þ u4 Z4;u2 ;

(2.49)

  Z3 ¼ u3 Z3;u3  Z1;u1 þ x2;x2  x1;x1 ;

(2.50)

  Z4 ¼ Z2;x2 þ u4 Z2;u2  x2;x2 ;

(2.51)

Z1;x1 þ u1 Z1;x2 ¼ Z2;x2 ;

(2.52)

Z1;u1  x1;x1 ¼ Z2;u2  x2;x2 ;

(2.53)

Z3;x1 þ u1 Z3;x2 þ u3 Z1;x2 ¼ 0:

(2.54)

Equations (2.48)–(2.51) and (2.54) are, in fact, nothing more than Eqs. (2.44d), (2.41f), (2.46a), (2.43c), and (2.46e) renumbered and

58

Exact Solutions to the Beam Equations

simplified by using the results obtained. Equations (2.52) and (2.53) are just the relations (2.47) with the new numbers. Using Eq. (2.51), let us exclude Z4 from Eq. (2.49) as follows:     Z3 ¼ Z2;x2x2 þ u4 2Z2;x2u2  x2;x2x2 þ u4 Z2;u2u2 þ u3 Z2;u2  2x2;x2 :

(2.55)

Since all functions entering into this equation do not depend on u4, it can be split with respect to this variable:  4 2  2 u  Z;u2u2 ¼ 0;

   1 Z3 ¼ Z2;x2x2 þ u3 Z2;u2  2x2;x2 :

 u4  2Z2;x2u2 ¼ x2;x2x2 ;

(2.56) 2

The first two equations determine Z : Z2 ¼

 i   1h 2 x;x2 þ H2 x1 u2 þ H20 x1 ; x2 ; 2

(2.57)

where H2, H20 are arbitrary functions of their arguments. Let us now substitute Z2 into the equation for Z3, which we obtained by splitting with respect to u4: i 1 1h   Z3 ¼ x2;x2x2x2 u2 þ H20;x2x2 þ H2 x1  3x2;x2 u3 : 2 2

(2.58)

Since Z2;u2 ¼ 0, we have x2;x2x2x2 ¼ 0 and x2,Z2,Z3 take the form     1   x2 ¼ f2 x1 x2ð2Þ þ f1 x1 x2 þ f x1 ; 2       1   Z2 ¼ f2 x1 x2 þ f1 x1 þ H2 x1 u2 þ H20 x1 ; x2 ; 2     

1  1 H2 x  3 f2 x1 x2 þ f1 x1 u3 þ H20;x2x2 ; Z3 ¼ 2

(2.59)

with f1, f2, f3 being arbitrary functions of x1. Now we need to substitute Z3 into Eq. (2.50) and perform splitting with respect to u3: Z1;u1 ¼ f2 x 1 þ f1  x1;x1 ; H20;x2x2 ¼ 0;     Z1 ¼ f2 x2 þ f1  x1;x1 u1 þ H1 x1 ; x2 ;

(2.60)

Here H1 is an arbitrary function resulting from the integration on u1. Let us substitute the expressions for Z1,Z2 into Eq. (2.52) and perform splitting with respect to u1, u2:

59

Exact Solutions to the Beam Equations

h

i         f 0 2 x1 x2 þ f 0 1 x1  x1;x1x1 u1 þ H1;x1 þ f2 x1 u1 þ H1;x2 u1 1   ¼ f2 x1 u2 þ H20;x2 ; 2      2 2   1  u  f2 x ¼ 0; u1  f 0 1 x1  x1;x1x1 þ H1;x2 ¼ 0; 1 H1;x1 ¼ H20;x2 :

(2.61) Therefore, with regard to H20, x2x2 ¼ 0, we have for H1, H20 h  i    00  H1 ¼ x1;x1x1  f 0 1 x1 x2 þ H11 x1 ; x1;x1x1x1  f 1 x1 ¼ 0;   H20 ¼ H11;x1 x2 þ S x1 :

(2.62)

Thus, for x2, Z1, Z2, Z3, we obtain     x2 ¼ f 1 x 1 x 2 þ f x1 ; h h   i  i   Z1 ¼ f1 x1  x1;x1 u1 þ x1;x1x1  f 0 1 x1 x2 þ H11 x1 ; Z2 ¼

    1   1 f1 x þ H2 x1 u2 þ H11;x1 x2 þ S x1 ; 2

Z3 ¼

  1   1 H2 x  3f1 x1 u3 : 2

(2.63)

Let us tie together H2 and f1, x1 by substituting Z1, Z2 into (2.53):     H2 x1 ¼ 3f1 x1  4x1;x1 : (2.64) The relation for Z3 in Eqs. (2.63) takes the form Z3 ¼ 2x1;x1 u3 :

(2.65)

1

Let us now substitute Z from Eqs. (2.63) into Eq. (2.48) and perform splitting with respect to x2 in the resulting relation:   1 f 0 1 x1 ¼ x1;x1x1 ; 2

    H11 x1 ¼ f 0 x1 :

(2.66)

Substituting Z3 from Eq. (2.65) into Eq. (2.54) with regard to Eqs. (2.66), we have   x1;x1x1 ¼ 0; x1 ¼ a1 x1 þ a0 ; f1 x1 ¼ const; a1 ; a0 ¼ const: (2.67) Combining all these results, we come to the following solution of the system of defining equations:

60

Exact Solutions to the Beam Equations

    x1 ¼ a1 x1 þ a0 ; x2 ¼ f1 x2 þ f x1 ; Z1 ¼ ðf1  a1 Þu1 þ f 0 x1 ;     00 Z2 ¼ 2ðf1  a1 Þu2 þ f x1 x2 þ S x1 ; Z3 ¼ 2a1 u3 ;  00  Z4 ¼ ðf1  2a1 Þu4 þ f x1 :

(2.68)

2.3.5. Main Group Equations (2.68) contain three arbitrary constants a0, a1, f1, which determine a three-parametric group with the operators X1 ¼

@ ; @t

X3 ¼ x

X2 ¼ t

@ @ @ @ @ u  2j  2r  2E ; @t @u @j @r @E

@ @ @ @ þu þ 2j þE @x @u @j @E

(2.69)

and two arbitrary time-dependent functions f, S, which specify an infinite subgroup of the main group X4 ¼ f ðtÞ

@ @ @ @ 00 00 þ f 0 ðtÞ þ f ðtÞx þ f ðt Þ ; @x @u @j @E

X5 ¼ SðtÞ

@ : @j

(2.70)

To obtain Eqs. (2.69) and (2.70), it suffices to put to zero all the arbitrary elements except one and to write the resulting operator, the general view of which is X ¼ xi(@/@ xi) þ Zi(@/@ uk). Simultaneously, we have returned to the original notations of system (2.34). Looking at Eqs. (2.14) and (2.16), it is easy to see that the operators (2.69) correspond to the translations in time and scaling of the dependent and independent variables (a0 ¼ b1, a1 ¼ b2  1, f1 ¼ b3  1):  ¼ E; t ¼ t þ b 1 ;   ¼ u; j  ¼ j; r  ¼ r; E x ¼ x; u 2 2  ¼ b2 E; (2.71) t ¼ b 2 t;   ¼ b1   x ¼ x; u u; j ¼ b j; r ¼ b E 2 2 2 r; 2 2  ¼ b 3 E: t ¼ t;   ¼ b 3 u; j  ¼ b 3 j; r  ¼ r; E x ¼ b 3 x; u Using the concepts in Section 2.2, it is possible to ensure that the operators (2.70) define a group of transformations for any particular f(t), S(t), with the finite equations as follows: t ¼ t;   ¼ u þ b4 f 0 ðtÞ; x ¼ x þ b4 f ðtÞ; u 00  ¼ E þ b4 f 00 ðtÞ;  ¼ j þ b4 f ðtÞx; r  ¼ r; E j t ¼ t;   ¼ u; j  ¼ r;  ¼ j þ b5 SðtÞ; r x ¼ x; u

 ¼ E: E

(2.72)

The second transformation in Eqs. (2.72) represents the well-known feature of gradient invariance for the scalar potential: The field E ¼ rj does not change if an arbitrary function of time is added to j. The physical meaning of the first of the transformations (2.72) is not as obvious.

Exact Solutions to the Beam Equations

61

It signifies the invariance of the equations with respect to the transition to a non-inertial coordinate system that arbitrarily moves along the x-axis. The particular cases of this transformation are (1) the translations along the  ¼ u) and (2) the Galileo transforx-axis with f(t) ¼ a ¼ const ( x ¼ x þ a; u  ¼ j, E¯ ¼ E. mation with f(t) ¼ at,  x ¼ x þ at, u¯ ¼ u þ a for which t ¼ t, j

2.4. GROUP PROPERTIES OF THE BEAM EQUATIONS 2.4.1. Coordinates of Infinitesimal Operators in Different Systems It can be shown that the coordinates xi of the operators X represent a contravariant vector that is transformed from one system to another according to Eq. (1.2). For this reason, calculation of the main group that the system of equations (S) admits is commonly performed in the Cartesian coordinates to ensure the simplest form of the relevant equations. Concerning the case of the beam, those are Eqs. (1.101). At the same time, constructing the partial solutions can be substantially facilitated if we know the appearance of the elementary transformation operators in some curvilinear coordinate systems. Consider, for example, the operators (2.15) and (2.16) in polar coordinates R, c:  x ¼ xR;x þ ZR;y ¼ R1 ðxx þ yZÞ;  ¼ xc;x þ Zc;y ¼ R2 ðyx þ xZÞ; Z X ¼ yð@=@xÞ þ xð@=@yÞ ¼ @=@c; X ¼ xð@=@xÞ þ yð@=@yÞ ¼ Rð@=@RÞ:

(2.73)

From (2.73) it can be seen that the transformation (2.15) is, in fact, a translation with respect to the angle, while the homogeneous scaling (2.16) is reduced to a scaling in R.

2.4.2. Preliminary Considerations Before using the general method for main group calculation (represented in Section 2.3), it is helpful to determine whether the system of equations under investigation is invariant with respect to certain elementary transformations, including translations, scalings, and rotations. The information thus gained may be of profound interest by itself; needless to say, the already known transformations represent excellent tests to verify the correctness of the calculations needed to derive the defining equations. We say that the system (S) admits translations along a certain coordinate if this coordinate does not explicitly enter into (S). It can be seen that the beam equations (1.101) are invariant with respect to the translations in

62

Exact Solutions to the Beam Equations

time and the translations along each of the Cartesian coordinates yk with the operators X1 ¼ @=@t;

X2þk ¼ @=@yk ;

k ¼ 1; 2; 3:

(2.74)

Let us now obtain all groups of scaling that preserve Eq. (1.101). The finite equations for such transformations appear as t ¼ at;

 y h ¼ b h yh ;

h ¼ ch uh ; u

 ¼ kj; j

 h ¼ fh Hh ; (2.75) H

 ¼ er; r

Here uh and Hh are, respectively, velocity components and magnetic field components in the Cartesian coordinate system yk in which the covariant, contravariant, and physical components coincide; h is a fixing index, uh ¼ fu; v; wg. By transforming Eqs. (1.101) to the new variables and requiring the preservation of these equations, which implies the equality of the corresponding coefficients at each of the terms in these equations, we derive from the first of the motion equations a=c 1 ¼ b 1 =c21 ¼ b 2 =ðc 1 c2 Þ ¼ b 3 =ðc 1 c 3 Þ ¼ b 1 =k ¼ 1=ðc 2 f 3 Þ ¼ 1=ðc 3 f 2 Þ: (2.76)

These equations may be represented in the form k ¼ c21 ;

a ¼ b 1 =c 1 ¼ b 2 =c2 ¼ b 3 =c3 ;

b 2 =c 1 ¼ 1=f3 ;

b 3 =c 1 ¼ 1=f2 : (2.77)

Similar considerations, as applied to the second and third of the motion equations, result in the new relations k ¼ c22 ;

b 3 =c2 ¼ 1=f1 ;

b 1 =c3 ¼ 1=f2 ;

b 1 =c2 ¼ 1=f3 ;

k ¼ c23 ;

b 2 =c3 ¼ 1=f1 :

(2.78)

Thence, it may be concluded that k ¼ c21 ¼ c22 ¼ c23 ¼ c2 ;

b 1 ¼ b 2 ¼ b 3 ¼ b ¼ ac;

f 1 ¼ f 2 ¼ f3 ¼ f ¼ c=b ¼ 1=a:

(2.79) The current conservation equation gives the already obtained expression for a, while the Poisson equation allows combining e and a: b2 =k ¼ 1=e;

e ¼ k=b2 ¼ 1=a2 :

(2.80)

As a result, Eqs. (1.101) allow a two-parametric group of scaling transformations with the finite equations t ¼ at;

 yh ¼ acyh ;

h ¼ cuh ; u

 ¼ c2 j; j

 ¼ a2 r; r

 h ¼ a1 Hh : H (2.81)

Exact Solutions to the Beam Equations

63

Two infinitesimal operators X5 ¼ t

@ @ @ @ þ yk k  2r  Hk ; @t @r @y @Hk

X6 ¼ yk

@ @ @ þ uk k þ 2j @yk @u @j

(2.82)

correspond to this group. It is easy to see that X2 in Eqs. (2.69) is equal to the difference X5  X6. Note that Eqs. (1.104) and (1.108) do not explicitly depend on the azimuth angle c. Given Eqs. (2.73), this means that the beam equations are invariant with respect to the rotations around the z-axis in the (x, y)-plane. However, the Cartesian coordinates do not differ from each other and Eqs. (1.101) are quite symmetrical. As an example, the substitution   ¼ v; v ¼ u; x ¼ y;  y ¼ x; z ¼ z; u  y ¼ Hx ; H  z ¼ Hz  x ¼ Hy ; H H

 ¼ w; w

(2.83)

transforms the first motion equation into the second one (and conversely) with no changes in the current conservation equation, Poisson equation, and external magnetic field equations. Similar statements are valid for the transformations of the pairs (x, z) and (y, z). This implies that the 3D beam equations should allow not only the rotations in the (x, y), but also in the (x, z) and (y, z)-planes. It can be expected that, similar to (2.72), the transformations with arbitrary functions of time are valid in three dimensions with respect to the y- and z- axes. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi The appearance of the relativistic radicals 1  V 2 in the motion equations makes velocity scaling impossible: Only one of the two operators in Eqs. (2.82) is meaningful. Besides, it is known that the relativistic motion equations for particles and the Maxwell equations in the nonstationary case are invariant with respect to the Lorenz transformations representing the rotations in the (it, x),(it, y), and (it, z) planes. The first of them is associated with the operator X ¼ tð@=@xÞ þ xð@=@tÞ:

(2.84)

It should be noted that the Lorenz transformation involves not only time and coordinates but also velocity, electromagnetic field, and density. This is why the operator (2.84) should contain some additional terms. These considerations allow extrapolation of the results that have been obtained for simpler systems (S) (1D nonstationary flows, planar stationary flows) to the general case. The proof of the fact that the group thereby constructed is the main group (or, in other words, includes all subgroups of the transformations preserving (S)), can be obtained only by means of rigorous solution of the defining equations.

64

Exact Solutions to the Beam Equations

2.4.3. Non-Relativistic Beam Studying the group properties of the system (1.101) and (1.102) for a nonuniform external magnetic field shows that the main group includes a finite nine-parametric group of translations, rotations, and scalings, as well as an infinite group, which is induced by the gradient invariance of the scalar potential j: X1 ¼ t

@ @ @ @ ; þ yk k  2r  Hk @t @r @y @Hk

@ @ @ @ @ þ uk k þ 2j þ 2r þ Hk ; @t @u @j @r @Hk 0 1 @ @ @ A; ¼ eikl @yk i þ uk i þ Hk @y @u @Hi

X2 ¼ t X2þl

X5þl ¼

@ ; @yl

X9 ¼

@ ; @t

X10 ¼ SðtÞ

@ ; @j

(2.85)

l ¼ 1; 2; 3:

It should be remembered that the elements of the tensor eikl in the Cartesian coordinates yk are equal either 0 or  1. The operators X2 þ l describe a rotation by the same angle in the relevant planes belonging to the spaces of coordinates yk, velocities uk, and magnetic field components Hk. For a uniform magnetic field, only the sum X1 þ X2 ¼ yk

@ @ @ ; þ uk k þ 2j @yk @u @j

(2.86)

but none of these operators taken separately is meaningful. The translations X5 þ l are replaced by more complicated transformations containing the arbitrary functions of time f k(t) which, however, should not be changing too fast to ensure the opportunity to consider the electric field as a potential field: X10þh ¼ f h

h 00 i  h 0 @  h 0 @ h h i k @ þ f þ f y þ f e y H ; ikh @j @yh @uh

(2.87)

h ¼ 1; 2; 3: For the chosen set of f(k), the finite equations of the group appear as t ¼ t;  x ¼ x þ a11 f 1 ;  y ¼ y þ a12 f 2 ; z ¼ z þ a13 f 3 ;  1 0  2 0  0  ¼ u þ a11 f ;   ¼ w þ a13 f 3 ; u v ¼ v þ a f ; w 12 h 00  0  i  ¼ j þ a11 f 1 x þ f 1 yHz  zHy j h 00 i  0 þ a12 f 2 y þ f 2 ðzHx  xHz Þ h 00  0  i  ¼ r: þ a13 f 3 z þ f 3 xHy  yHx ; r

(2.88)

Exact Solutions to the Beam Equations

65

Thus, the transition to a non-inertial frame of reference, the origin of which is arbitrarily moving in space, is possible not only for electrostatic flows, but also in the presence of a uniform magnetic field. If Hk ¼ 0, the main group includes the transformations (2.85) and (2.87). Certain interest represents the propagation of a beam against the motionless background with a constant density r0 of definite sign. This case can be accounted for by the corresponding term in the RHS of the Poisson equation. It is clear that the scaling operators in Eqs. (2.85) should be replaced in this case by Eq. (2.86), although the magnetic field can be nonuniform.

2.4.4. The Relativistic Beam For the relativistic beam, the system (S) is invariant with respect to the transformations with the operators @ @ @ @ @  2r ; þ yk k  E k k  H k @t @r @y @E @Hk 0 1 @ @ @ @ A; ¼ eikl @yk i þ uk i þ Ek i þ Hk @y @u @E @Hi 0 1 @ @ @ ¼ yl þ t l þ @dkl  ul uk k A @t @y @u 0 1 @ @ @ þ H k i A þ ul r ; þeikl @Ei k @E @r @H

X1 ¼ t X1þl

X4þl

X7þl ¼

@ ; @yl

X11 ¼

(2.89)

@ : @t

Here dik is the Kronecker symbol; the operators X1 þ l correspond to the rotations about the same angle in the spaces yk, uk, Ek, and Hk; and the operators X4 þ l determine the Lorenz transformation. For example, the finite equations of a group with the operator X5 are described as t ¼ tch a þ xsh a;  x ¼ t sh a þ x ch a; y ¼ y; z ¼ z;  ¼ ðu ch a þ xsh a;Þ=D;   ¼ w=D; u v ¼ v=D; w  x ¼ Ex ; E  y ¼ Ey ch a þ Hz sh a; D ¼ ush a þ ch a; E  x ¼ Hx ; H  y ¼ Hy ch a  Ez sh a;  z ¼ Ez ch a  Hy sh a; H E   ¼ rD: Hz ¼ Hz ch a þ Hy sh a; r

(2.90)

66

Exact Solutions to the Beam Equations

2.5. INVARIANT SOLUTIONS 2.5.1. The Concept of Invariant Solutions The main property of the solutions of the system (S) admitting the group G is that any transformation from G carries any solution of the system (S) again into a solution of the system (S). Indeed, let u ¼ u(x) be some solution of (S). With the variables changed  x ¼ xðx; uÞ, u¯ ¼ u¯(x, u), this     solution turns into the solution u ¼ u1 ðxÞ of the transformed system  S .  are Since (S) is invariant with respect to group G, the systems (S) and S identical and u ¼ u1(x) is a solution of (S). Let H be a subgroup of the main group. The solution u ¼ u(x) of the system (S) is invariant with respect to H (H-solution) if the equation u ¼ u(x) determines an invariant manifold in the space ℰn of the independent variables and sought functions—in other words, a manifold—the points of which are not removed from the manifold under any transformation from H. The function I(x) is an invariant for the p-parametric subgroup determined by the set Xl, l ¼ 1, . . ., p of p operators if Xl I ðxÞ ¼ 0; l ¼ 1; . . . ; p:

(2.91)

This requirement represents the obvious generalization of Eq. (2.20). The task of constructing all invariants of the group H can be reduced to constructing a full set of the functionally independent invariants of this group—namely, to constructing a set of the invariants Jk(x), k ¼ 1, . . ., t— so that, for any given invariant I(x), a function f(y(k)), k ¼ 1, . . ., t can be found to satisfy the equality I(x) ¼ f(J(k)(x)), k ¼ 1, . . ., t identically with respect to x. Calculating the invariant J(k) is equivalent to jointly solving the system of p linear equations (2.91), which, in turn, is reducible to integrating a system of ordinary differential equations. Let R be the rank of the matrix kxlik(l ¼ 1, . . ., p; i ¼ 1, . . ., n) of the system (2.91), which consists of the coordinates of the operators Xl. The number of the functionally independent invariants is t ¼ n  R. We have denoted by m the number of the unknown functions uk in the system (S). Therefore, the invariant manifold u ¼ u(x) is described by m equations Fk ðJ a Þ ¼ 0; k ¼ 1; . . . ; m; a ¼ 1; . . . ; t;

(2.92)

provided that the rank of the matrix k @ Fk/@ Jak is equal to m. Subject to this condition, the Eqs. (2.92) can be resolved with respect to m invariants   (2.93) J k ¼ Fk J mþ1 ; . . . ; J t ; k ¼ 1; . . . ; m:

Exact Solutions to the Beam Equations

67

Thus, as many as t  m invariants are independent on the manifold u ¼ u(x). The number s ¼ t  m ¼ n  R  m < n  m;

(2.94)

which is called the invariant solution rank, is less than the number n  m of the independent variables xi(i ¼ m þ 1, . . ., n) of the system (S). Therefore, after substituting Eqs.(2.93) into (S), we obtain a system denoted by the symbol (S/H), with fewer independent variables. In particular, at s ¼ 1 we come to the ordinary differential equations.

2.5.2. An Example: The H-Solution Let us construct a solution that is invariant with respect to the subgroup with operator X2 from (2.69). In this case p ¼ R ¼ 1, and the invariants are constructed by means of integrating a system of ordinary equations, which leads to following result: dt=t ¼ dx=0 ¼ du=ðuÞ ¼ dj=ð2jÞ ¼ dr=ð2rÞ ¼ dE=ð2EÞ; J 2 ¼ ut; J 3 ¼ jt2 ; J 4 ¼ rt2 ; J 5 ¼ Et2 : J1 ¼ x;

(2.95)

For the solution under consideration, m ¼ 4, n ¼ 6, s ¼ 1. Equations (2.93) take the form Jk ¼ Fk(J1), k ¼ 1, . . ., 4: u ¼ t1 F1 ðxÞ; j ¼ t2 F2 ðxÞ; r ¼ t2 F3 ðxÞ; E ¼ t2 F4 ðxÞ:

(2.96)

On substituting these expressions into (2.34), we reach a system of ordinary differential equations with respect to the functions Fk:  0  0  0  0  0 F1 þ F1 F1 ¼ F2 ; 2F3 þ F1 F3 ¼ 0; F2 ¼ F4 ; F4 ¼ F3 : (2.97)

2.5.3. Substantially Different Invariant Solutions The expression      Xa ; Xb ¼ Xa Xb  Xb Xa ¼ xi Zk;i  Zi xk;i @=@xk

(2.98)

is called the commutator of two operators Xa ¼ xi(@/@ xi), Xb ¼ Zi(@/@ xi). The operators Xa(a ¼ 1, . . ., p) determine a p-parametric group Gpn of the transformations in ℰn, with a commutator of any two operators being a linear combination of the operators belonging to the same set:   (2.99) Xa ; Xb ¼ cgab Xg : The constants cgab are called the structural constants.

68

Exact Solutions to the Beam Equations

We will verify this property by some particular examples. The operators Xa represent the elements of a linear space ℒp called the Lie algebra, in which the group Gpn corresponds to the adjoint group with the operators   Eb ¼ cabg Xa @=@Xg : (2.100) The finite equations related to the operators Eb show how Xa varies in ℒp under the transformations of the group Gpn in the Euclidean space ℰn. As already noted, the transformations from the group G turn the solution u ¼ u(x) into the new solution u ¼ u1(x) of the system (S). Two invariant solutions are called essentially different if one of them cannot be transformed into another by means of any transformation from G; otherwise we will discuss the inessentially different solutions. Obviously, it suffices to indicate all essentially different solutions. For Gpn, this problem is reduced to constructing the optimal systems of 1, . . ., (p  1)-parametric dissimilar subgroups of the group Gpn. If the operators Xa of the group Gpn are known, the formulas (2.99) allow calculation of the structural constants; then by using the constants found, the most general transformation of the adjoint group can be calculated. By exposing the operators Xb of the subgroup H to such general transformation, we obtain the most general view of the operators Xb corresponding to all subgroups similar to H. By specifying the transformation of the adjoint group, it is possible to select the simplest general of the subgroup H in the optimal system.

2.6. OPTIMAL SYSTEMS OF THE SUBGROUPS FOR A THREE-DIMENSIONAL STATIONARY BEAM 2.6.1. The Adjoint Group According to Eqs. (2.85), the main group G911 is determined by the operators X1 ¼ x

@ @ @ @ @ @ @  Hy  Hz ; þ y þ z  2r  Hx @x @y @z @r @Hx @Hy @Hz

X2 ¼ u

@ @ @ @ @ @ @ @ þ Hy þ Hz ; þv þw þ 2j þ 2r þ Hx @u @v @w @j @r @Hx @Hy @Hz

X3 ¼ y

@ @ @ @ @ @ þ Hx ; þx v þ u  Hy @x @y @u @v @Hx @Hy

69

Exact Solutions to the Beam Equations

X4 ¼ z

@ @ @ @ @ @ þx w þu  Hz þ Hx ; @x @z @u @w @Hx @Hz

X5 ¼ z

@ @ @ @ @ @ þy w þv  Hz þ Hy ; @y @z @v @w @Hy @Hz

X6 ¼

@ ; @x

X7 ¼

@ ; @y

X8 ¼

@ ; @z

X9 ¼

(2.101)

@ : @j

The calculations according to (2.98) are summarized in Table 1. The first line contains the commutators (X1, Xb). Since the commutators possess the property of antisymmetry (Xa, Xb) ¼  (Xb, Xa), it is sufficient to find the values (Xa, Xa þ k) positioned above the main diagonal. The diagonal contains the commutators (Xa, Xa) ¼ 0. The lines in Table 1 allow us to write the adjoint group operators according to (2.100): E1 ¼ X6

@ @ @  X7  X8 ; @X6 @X7 @X8

E3 ¼ X5

@ @ @ @ þ X4  X7 þ X6 ; @X4 @X5 @X6 @X7

E4 ¼ X 5

@ ; @X9

@ @ @ @  X3  X8 þ X6 ; @X3 @X5 @X6 @X8

E5 ¼ X4

TABLE 1

E2 ¼ 2X9

@ @ @ @ þ X3  X8 þ X7 ; @X3 @X4 @X7 @X8

Commutators Xb

Xa

X1

X2

X3

X4

X5

X6

X7

X8

X9

X1 X2 X3 X4 X5 X6 X7 X8 X9

0 0 0 0 0 X6 X7 X8 0

0 0 0 0 0 0 0 0 2 X9

0 0 0 X5 –X4 X7 –X6 0 0

0 0 –X5 0 X3 X8 0 –X6 0

0 0 X4 –X3 0 0 X8 –X7 0

–X6 0 –X7 –X8 0 0 0 0 0

–X7 0 X6 0 –X8 0 0 0 0

–X8 0 0 X6 X7 0 0 0 0

0 –2X9 0 0 0 0 0 0 0

70

Exact Solutions to the Beam Equations

E 6 ¼ X6

@ @ @ þ X7 þ X8 ; @X1 @X3 @X4

E 7 ¼ X7

@ @ @  X6 þ X8 ; @X1 @X3 @X5

E 8 ¼ X8

@ @ @  X6  X7 ; @X1 @X4 @X5

(2.102) E9 ¼ 2X9

@ : @X2

The finite equations of the adjoint group, which can be derived from the equations similar to Eqs. (2.13), are given below (the identical part  l ¼ Xl of the transformations Ak is omitted). X A1 A2 A3 A4 A5 A6 A7 A8 A9

 6 ¼ a1 X6 ; X  7 ¼ a1 X7 ; X  8 ¼ a1 X8 ; :X 1 1 1 2  : X 9 ¼ a2 X 9 ;  4 ¼ X4 cos a3  X5 sin a3 ; X  5 ¼ X4 sin a3 þ X5 cos a3 ; :X   X6 ¼ X6 cos a3  X7 sin a3 ; X7 ¼ X6 sin a3 þ X7 cos a3 ;  3 ¼ X3 cos a4  X5 sin a4 ; X  5 ¼ X3 sin a4 þ X5 cos a4 ; :X   X6 ¼ X6 cos a4  X8 sin a4 ; X8 ¼ X6 sin a4 þ X8 cos a4 ;  3 ¼ X3 cos a5  X4 sin a5 ; X  4 ¼ X3 sin a5 þ X4 cos a5 ; :X   X7 ¼ X7 cos a5  X8 sin a5 ; X8 ¼ X7 sin a5 þ X8 cos a5 ;  1 ¼ X1 þ a6 X6 ; X  3 ¼ X 3 þ a6 X 7 ; X  4 ¼ X4 þ a6 X8 ; :X    5 ¼ X5 þ a7 X8 ; : X1 ¼ X1 þ a7 X7 ; X3 ¼ X3  a7 X6 ; X    5 ¼ X5  a8 X7 ; : X1 ¼ X1 þ a8 X8 ; X4 ¼ X4  a8 X6 ; X  : X2 ¼ X2 þ 2a9 X9 :

(2.103)

2.6.2. Optimal System of One-Parametric Subgroups The operators Xk, owing to their linear independence in the space ℒ9, play the role of the basis operators; therefore, any one-parametric subgroup corresponds to a linear combination of the operators Xk with constant coefficients X ¼ ek Xk :

(2.104)

Let us subject the operator (2.104) to transformation A1, the result of this transformation—to transformation A2—and so forth up to transformation A9. For example, the result of applying A3 to the operator X from (2.104) reads as   A3 X ¼ e1 X1 þ e2 X2 þ e3 X3 þ e4 cos a3 þ e5 sin a3 X4     þ e4 sin a3 þ e5 cos a3 X5 þ e6 cos a3 þ e7 sin a3 X6 (2.105)  6  7 8 9 þ e sin a3 þ e cos a3 X7 þ e X8 þ e X9 :

Exact Solutions to the Beam Equations

71

Thus, we have A 9 . . . A 1 X ¼ ek Xk ; e1 ¼ e1 ; e2 ¼ e2 ; e3 ¼ A cos a5 þ B sin a5 ; e4 ¼ A sin  a5 þ B cos a5 ; 5 4 e5 ¼ e3sin a þ e cos a  e sin a cos a4 ;  4 3 3 6  7 e6 ¼ a1 e cos a þ e sin a þ e8 sin a4 þ a6 e1  a7e3  a8e4 ; cos a 3 3 4 1 7 5 1 1 e ¼ a8e þ a7 e þ a1 ðC cos a5 þ D sin a5 Þ þ a6e3 ; 9 2 e8 ¼ a8 e1 þ a7e5 þ a1 sin a5 Þ þ a6e4 ; e9 ¼ a2 1 ðD cos a5  C  2 e þ 2a9 e ; 3 5 4 4 5 A ¼ e cos a4 þ e cos a3  e sin a3  sin a4 ; B ¼ e cos a3 þ e sin a3 ; D ¼ e8 cos a4  e6 cos a3 þ e7 sin a3 sin a4 ; C ¼ e7 cos a3  e6 sin a3 :

(2.106) Let e 6¼ 0. Then, by varying a6, it is always possible to make the coefficient e¯6 equal zero; the same may be done with e¯7, e¯8 by varying a7, a8. Varying a5 allows equating e¯4 to zero. The coefficient e¯5 turns to zero by varying a4. Thus, having used the arbitrary parameters a4, . . ., a8, we have transformed the operator from Eq. (2.106) to the form 1

e1 X1 þ e2 X2 þ e3 X3 þ e9 X9 :

(2.107)

Varying a9 at e¯2 6¼ 0 gives e¯9 ¼ 0; if e2 ¼ 0 and e9 6¼ 0, by way of varying a2, it is possible to make e¯9 ¼ 1. These two cases correspond to the following subgroups: X1 þ aX2 þ bX3 ; X1 þ aX3 þ X9 :

(2.108)

These subgroups are obtained by dividing the expression (2.107) by e¯1 ¼ 6 0 and subsequent re-denotation of the constants. Since the invariant manifolds are constructed by using Eqs. (2.91), the constant multiplier at the operator X is unessential. It makes no sense to introduce, by way of a multiplier at the operator X9, a constant that may vanish because such case can be obtained from the first subgroup at a ¼ 0. It should be noted that the operator X3 can be replaced by X4 or X5; this is another consequence of the above-mentioned equality among the Cartesian axes x, y, and z and the rotations around them. Now let e1 ¼ 0, e2 6¼ 0. As before, operating with the values a4, a5, let us make e¯4, e¯5 equal to zero. It follows from e¯4 ¼ 0 that e¯3 6¼ 0; therefore, it is possible to make e¯6 ¼ 0 by varying a7 and e¯7 ¼ 0 by varying a6. Varying a9 gives e¯9 ¼ 0, while varying a1 leads to e¯8 ¼ 1 if e6, e7, and e8 are not equal to zero simultaneously. Both of those cases are taken into account in the subgroup X2 þ aX3 þ bX8 ; b ¼ 0; 1 with the possibility of replacing X3, X8 by X4, X7; X5, X6.

(2.109)

72

Exact Solutions to the Beam Equations

At e1 ¼ e2 ¼ 0, e3 ¼ 6 0, it is possible, as above, to make e¯4, e¯5, e¯6, e¯7 equal zero. Varying a1, a2 gives e¯8 ¼ e¯9 ¼ 1 if these coefficients differ from zero, and leads to the subgroup X3 þ aX8 þ bX9 ; a; b ¼ 0; 1:

(2.110)

As before, the pair X3, X8 allows introducing symmetrical combinations. We still must consider the case e1 ¼ . . . ¼ e5 ¼ 0, e6 6¼ 0. Varying a3, a4 ensures the equalities C ¼ D ¼ 0 and, as a sequence, the equalities e¯7 ¼ e¯8 ¼ 0, while varying a2 gives e¯9 ¼ 0. As a result, we have X6 þ aX9 ; a ¼ 0; 1:

(2.111)

Similar subgroups exist with the operators X7, X8 replaced for X6. The combinations of the constructed operators must be supplemented by the basis operators (2.101) if those operators cannot be obtained from their combinations by specifying the arbitrary constants. This is not necessary in our case. Thus, the optimal system of one-parametric subgroups is as follows (symmetrical combinations of 6–15 are omitted): 1: X1 þ aX2 þ bX3 ; 2: X1 þ bX3 þ X9 ;

3: X2 þ bX3 þ bX8 ; 5: X6 þ aX9 ; 4: X3 þ aX8 þ bX9 ða; b ¼ const; a; b ¼ 0; 1Þ: (2.112)

2.6.3. Some Comments As mentioned in Section 2.4, the information concering the transformations preserving the system (S), which can be derived at the stage of preliminary investigation without using the general approach, may be very helpful in solving the defining equations. The same can be said about constructing the optimal systems of groups. The approach outlined in the previous section certainly yields the result, if we can avoid the mistakes or omissions possible in the first stage. This is why any considerations that may verify the correctness of intermediate results, supplemented by the perception of the true physical sense of the operations made, are of exceeding importance when applying the general approach. In the case of elementary transformations (2.101), such considerations are not difficult to establish. Indeed, it is clear that the result of two rotations is equivalent to some third rotation, so that the combination X3 þ aX4 hardly represents an invariant solution with any new properties. The superposition of translations X6 þ aX7 results in a linear combination of the Cartesian coordinates x and y, which can be reduced to x by means of the rotation around the z-axis and scaling. Adding X9 to X2 is meaningless because the beam equations admit a shift with respect to j, whereas adding X6 to X3 is compensated by the translation X7. All these properties

Exact Solutions to the Beam Equations

73

follow from Eq. (2.106) if we make the corresponding ek equal to zero and introduce possible simplifications by varying the parameters ap of the group. Therefore, before constructing the optimal system, it is necessary to clearly realize the geometric meaning of the transformations (2.101), and it is quite useful to ‘‘process’’ Eqs. (2.106) as mentioned above.

2.6.4. Invariant Solutions of the Rank 2 The invariant solutions of the rank 2 can be constructed on the subgroups (2.112) satisfying the system (S/H) with two independent variables. It suffices to consider only the subgroups 1-5, because the subgroups 6-15 do not provide any new flow configurations from the geometric standpoint. When using (2.73) for the subgroup 1, it is convenient to pass over to the polar coordinate system R, c, and introduce similar coordinates in the planes ðu; vÞ and (Hx, Hy), denoting them V, cV and H, cH, correspondingly. The operator 1 appears as1 X ¼ X1 þ aX2 þ bX3 ¼ R þ aw

@ @ @ @ @ þb þ z þ aV þb @R @c @z @V @cV

@ @ @ @ @ @ þ ða  1ÞHz ; þ 2aj þ 2ða  1Þr þ ða  1ÞH þb @w @j @r @H @cH @Hz (2.113)

and the system (2.91) can be reduced to a single equation XJ ¼ 0. The equivalent system of ordinary differential equation appears as dR dc dz dV dcV dw dj ¼ ¼ ¼ ¼ ¼ ¼ ¼ b R b z aV aw 2aj ¼

dr dH dc dHz : ¼ ¼ H¼ b ða  1ÞHz 2ða  1Þr ða  1ÞH

(2.114)

In this case n ¼ 11, m ¼ 8, and t ¼ n  1 ¼ 10. The 10 independent invariants can be chosen as follows: 1 3 4 b lnR  c ¼ J ; R=z ¼ J 2 ; Ra V ¼ J ; Ra w ¼ J ; 5 6 7 R2a j ¼ J ; R2ða1Þ r ¼ J ; Rða1Þ H ¼ J ; 8 9 10 ða1Þ R Hz ¼ J ; cV  c ¼ J ; cH  c ¼ J :

(2.115)

  We emphasize that uð@=@vÞ  vð@=@uÞ is not equal to vR @=@vc  vc ð@=@vR Þ, as it may seem at first. It is important to remember that X corresponds to the rotation by the same angle in the planes ðx; yÞ; ðu; vÞ, and (Hx, Hy), which leaves the angle between the vectors r and v, as well as the angle between the vectors r and H, everywhere unchanged. This implies that vR , vc , HR, and Hc preserve their values, whereas the projections onto the x, y axis do change. 1

74

Exact Solutions to the Beam Equations

Eight equations (2.92) transformed to the form (2.93) can be written as 3 4 5 6 V ¼ RaJ ; w ¼ RaJ ; j ¼ R2aJ ; r ¼ R2ða1ÞJ ; 7 8 9 10 H ¼ Ra1J ; Hz ¼ Ra1J ; cV  c ¼ J ; cH  c ¼ J ;  1 2 k k 1 J ¼ J J ; J ; J 1 ¼ b 1J ¼ q; b ¼ b 2 =b1 :

(2.116)

The velocity components in the systems x, y; R, c; and p, q are interconnected by the relations that can be obtained either from general formulas or by differentiation with respect to the time of the equations that interconnect the coordinates in those systems. Now we must change from the contravariant velocity components to the physical ones as below: vR ¼ u cosc þ v sinc ¼ V cosðcV  cÞ; vc ¼ u sinc þ v cosc ¼ V sinðcV  cÞ;  1=2 vp ¼ b 1 vR  b 2 vc ; vq ¼ b 2 vR þ b 1 vc ; bk ¼ b21 þ b22 bk :

(2.117)

With regard to these relations, let us introduce the spiral coordinates in Eq. (2.116) a J 4 ; w ¼  pa J 3 ; vq ¼ p pa J 5 ; j ¼  p2a J 6 ; r ¼ p2a2 J 7 ; vp ¼  a1 8 a1 9 a1 10 p J ; Hq ¼  p J ; Hz ¼  p J ; p ¼ expðb 1 pÞ: Hp ¼ 

(2.118)

Here and below Jk ¼ Jk(J1, J2); for example, for J3 in (2.118) we have   3 9 9 J 3 ¼ eab 2 qJ b 1 cosJ  b 2 sinJ : (2.119) The system of ordinary differential equations for the subgroup 2 can be obtained from Eqs. (2.114) at a ¼ 0, and dj/(2aj) replaced by dj. All argumentation can be repeated; thus, the H-solution follows from (2.118) if we put a ¼ 0 and represent j in the form   (2.120) j ¼ b 1 p þ J5 J1 ; J2 : For Eqs. (2.118), (2.120) at b2 ! 0 we have J1 ! c, p ! ln R. In case 3, we obtain dR dc dz dV dcV dw dj dr dH dcH dHz ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ : (2.121) b b Hz 0 b b V w 2j 2r H The functionally independent invariants are  ¼ J 3 ; cw  ¼ J 4 ; c  2 j ¼ J 5 ; c  2 r ¼ J 6 ; R ¼ J 1 ; bc  bz ¼ J 2 ; cV 7 8 9 10  ¼ J ; cH  z ¼ J ; c  c ¼ J ; c  c ¼ J ; c  ¼ ec=b : cH V H (2.122)

Exact Solutions to the Beam Equations

75

The corresponding invariant solution appears as vR ¼ eac J 3 ; vc ¼ eac J 4 ; vz ¼ eac J 5 ; j ¼ e2ac J 6 ; HR ¼ eac J 8 ; Hc ¼ eac J 9 ; Hz ¼ eac J 10 ; a ¼ b1 :

r ¼ e2ac J 7 ;

(2.123)

The system of ordinary differential equations and the functionally independent invariants for subgroup 4 are given by the expressions dR dz dV dw dj dr dH dHz ; ¼ dc ¼ ¼ ¼ dcV ¼ ¼ ¼ ¼ ¼ dcH ¼ 0 0 a 0 0 b 0 0 3 4 5 R ¼ J 1 ; ac  z ¼ J 2 ; V ¼ J ; w ¼ J ; j  bc ¼ J ; 8 9 10 Hz ¼ J ; cV  c ¼ J ; cH  c ¼ J :

6 r ¼ J ;

H ¼ J ; 7

(2.124) The H-solution appears as vR ¼ J 3 ; vc ¼ J 4 ; w ¼ J 5 ; j ¼ bc þ J 6 ; HR ¼ J 8 ; Hc ¼ J 9 ; Hz ¼ J 10 :

r ¼ J7 ;

(2.125)

The system of ordinary differential equations, the independent invariants, and the corresponding H-solution for subgroup 5 are presented below: dx ¼

dy dz du dv dw dj dr dHx dHy dHz ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ; 0 0 0 0 0 a 0 0 0 0

y ¼ J 1 ; z ¼ J 2 ; u ¼ J 3 ; v ¼ J 4 ; w ¼ J 5 ; j  ax ¼ J6 ; r ¼ J 7 ; (2.126) Hx ¼ J 8 ; Hy ¼ J 9 ; Hz ¼ J 10 : u ¼ J 3 ; v ¼ J 4 ; w ¼ J 5 ; j ¼ ax þ J 6 ; r ¼ J 7 ; Hx ¼ J 8 ; Hy ¼ J 9 ; Hz ¼ J 10 :

2.6.5. Two-Parametric Subgroups and Invariant Solutions of the Rank 1 Any two-parametric subgroup p ¼ 2 is determined by two operators of the form given by Eq. (2.104). Overall, the procedure of constructing the optimal system of subgroups is similar to the case p ¼ 1. Obviously, any operator from the system (2.112) can be taken as one of the operators required. Examination of the general representation for the adjoint group (2.106) will show which of the values ak should be fixed so as not to ‘‘spoil’’ such a representation. The remaining arbitrary parameters are to be used to simplify the second operator. The problem of constructing the solution of the rank 1 on the two-parametric subgroup thus constructed is reduced to finding a simultaneous solution to the equations similar to Eqs. (2.91), with one unknown function I(x). Such a solution may not exist; therefore, we must deal with the arbitrary constants of the

76

Exact Solutions to the Beam Equations

first and second operators to determine when the simultaneous solution is possible. (Note here that the paired combinations of the optimal system (2.112) give essentially different solutions of the rank 1, although generally speaking, those combinations do not completely exhaust the optimal system of the two-parametric subgroups.) We now consider some examples. Let X8 be taken by way of one of the operators. Then all ek except e8 are equal to zero, and from Eq. (2.106) we obtain the coefficients that e8 generates by means of the transformation (2.106): 8 e6 ¼ a1 1 e sin a 4 ;

8 e7 ¼ a1 1 e cos a 4 sin a 5 ;

8 e8 ¼ a1 1 e cos a 4 cos a 5 :

(2.127) This implies that the transformations with the parameters a4, a5 should be fixed (a4 ¼ a5 ¼ 0), while the remaining parameters should be chosen to simplify the second operator. Notice that the equation X8I ¼ I, z ¼ 0 assumes no dependence on z; therefore, to ensure the existence of a simultaneous solution, we must exclude X4, X5, which explicitly contain z, as well as X8, from the second operator: e4 ¼ e5 ¼ e8 ¼ 0. At e1 6¼ 0, varying a6, a7, a8 yields the equalities e¯6 ¼ e¯7 ¼ e¯8 ¼ 0. Depending on whether e2 is zero or not, we derive the combinations X1 þ aX2 þ bX3 and X1 þ aX3 þ bX9. Thus, we have constructed a two-parametric subgroup as follows: X1 þ aX2 þ bX3 ; X8 ; X1 þ aX3 þ X9 ; X8 :

(2.128)

The solution being invariant with respect to X1 þ aX2 þ bX3 is determined by Eqs. (2.118), whereas the invariance with respect to X8 indicates2 the absence of the dependence on z. Obviously, Eqs. (2.118), with the dependence on J2 omitted, satisfy both requirements. The solution invariant with respect to the subgroup HhX1 þ aX3 þ X9, X8i can be derived in the same manner. Let us now show how the H-solution can be constructed on the subgroup HhX1 þ aX2, X2 þ bX3i, the operators of which correspond to 1,3 in Eqs. (2.112). Transition to the spherical coordinates r, y, c; V, yV, cV; and H, yH, cH in the spaces yk ; vk ; Hk allows us to represent these operators in the form @ @ @ @ @ þ aV þ 2aj þ 2ða  1Þr þ ða  1ÞH ; @r @V @j @r @H 0 1 (2.129) @ @ @ @ @ @ @ A : þ 2j þ 2r þ H þ b@ þ þ X2 þ bX3 ¼ V @V @j @r @H @c @cV @cH

X1 þ aX2 ¼ r

2 The system of ordinary differential equations on X8 is dx/0 ¼ dy/0 ¼ dz ¼ du1/0 ¼ du2/0 ¼ . . ., where uk are dependent variables, so that the functionally independent invariants represent x, y and all uk. The corresponding H-solution is given by the relations uk ¼ uk(x, y).

77

Exact Solutions to the Beam Equations

The systems of ordinary differential equations corresponding to (2.91) and the functionally independent invariants for those equations take the form dr dy dc dV dyV dcV dj dr dH dyH dcH ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ; r 0 0 aV 2aj ð2a  2Þr ða  1ÞH 0 0 0 0 I 1 ¼ y; I 2 ¼ c; I 3 ¼ ra V; I4 ¼ yV ; I5 ¼ cV ; I 6 ¼ r2a j; I7 ¼ r2ða1Þ r; I8 ¼ rða1Þ H; I 9 ¼ yH ; I 10 ¼ cH ; dr dy dc dV dyV dcV dj dr dH dyH dcH ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ; 0 0 b V 2j 2r H 0 b 0 b  J 4 ¼ yV ; J 5 ¼ c  cV ; J 1 ¼ r; J 2 ¼ y; J 3 ¼ cV;   ¼ ec=b : J 9 ¼ yH ; J10 ¼ c  cH ; c J 8 ¼ cH;

 j; J6 ¼ c 2

 r; J7 ¼ c 2

(2.130) As an example, we consider the expression for the velocity module, which follows from these two systems of invariants: V ¼ raI3(y, c) and V ¼ ec/bJ3(r, y). The identity of the LHSs is ensured owing to the specification I3 ¼ ec/bJ(y), J3 ¼ raJ(y), which gives V ¼ ra ec=b J ðyÞ:

(2.131)

The remaining flow parameters can be represented in a similar way. It is now convenient to proceed from the spherical coordinates in the spaces vk ; Hk to the projections on the axes r, y, c. For vr ; vy ; vc , we have u ¼ V sin yV cos cV ;

v ¼ V sin yV sin cV ;

w ¼ V cos yV ;

vr ¼ ðu cos c þ v sin cÞ sin y þ w cos y ¼ ¼ V ½ sin y sin yV cosðcV  cÞ þ cos y cos yV ; vy ¼ ðu cos c þ v sin cÞ cos y  w sin y ¼ ¼ V ½ cos y sin yV cosðcV  cÞ  sin y cos yV ; vc ¼ ðu sin c þ v cos cÞ sin y ¼ V sin y sin yV sinðcV  cÞ:

(2.132)

The relationship between the velocity components in the coordinate systems r, y, c and x, y, z can be obtained in a manner similar to that used previously for the polar coordinates, as applied to the H-solution for subgroup 1. The simultaneous analysis of the systems of invariants gives yV ¼ J4(y), cV  c ¼ J5(y); therefore, the velocity module in the expressions for vr ; vy ; vc should be multiplied by some functions of y, which results in the velocity components having the same functional form as its module.

78

Exact Solutions to the Beam Equations

2.7. RESULTS OF CONSTRUCTING THE INVARIANT SOLUTIONS 2.7.1. System of Three-Parametric Subgroups For 3D nonstationary flow, the invariant solutions of the rank 1 can be constructed on three-parametric subgroups. Below we consider threeparametric subgroups of the group (2.85), which correspond to 3D stationary flows 1-11 [@/@t ¼ 0; the numeration of the rotation operators in Eqs. (2.85), (2.101) is different] and to the nonstationary flows 12-26 with translational (@/@z ¼ 0), axial (@/@c ¼ 0), and spherical symmetry: 1: aX2 þ X7 ; X8 ; X9 ; 2: aX2 þ X1 ; X8 ; X9 ; 3: bX2 þ X5 ; aX2 þ X8 ; X9 ; 4: aX2 þ X1 þ aX5 ; X8 ; X9 ; 5: aX2 þ X1 ; bX2 þ X5 ; X9 ; 6: X4 ; X5 ; X9 ; 7: X7 þ aX10 ; X8 ; X9 ; 8: X1 þ aX10 ; X8 ; X9 ; 9: X5 þ bX10 ; X8 þ aX10 ; X9 ; 10: X1 þ aX5 þ aX10 ; X8 ; X9 ; 11: X1 þ aX10 ; X5 þ bX10 ; X9 ; 12: X1 þ aX2 ; X7 ; X8 ; 13: X1 þ X2 þ aX9 ; X7 ; X8 ;

14: X2 ; X7 ; X8 ; 15: X5 þ aX1 ; X2 ; X8 ; 16: X5 þ aX2 ; X1 þ bX2 ; X8 ; 17: X5 þ aX9 ; X1 þ X2 þ bX9 ; X8 ; 18: X1 ; X2 ; X8 ; 19: X1 þ aX10 ; X5 þ bX10 ; X8 þ cX10 ; 20: X7 þ aX10 ; X2 ; X8 ; 21: X7 þ aX10 ; X1 þ X2 þ aX9 ; X8 ; 22: X7 þ aX10 ; X1 þ aX2 ; X8 ; 23: X1 ; X8 þ aX9 ; X5 ; 24: X1 ; X2 ; X5 ; 25: X2 ; X4 ; X5 ; 26: X1 þ aX2 ; X4 ; X5 : (2.133)

For the subgroups 19–22, the function S in X10 is defined, accordingly, as S ¼ 1; t 2; et/a; tg with g ¼ (2a  1)/(1  a); a, b, a, b, c ¼ const. At Hk ¼ const, the extension of the main group at the expense of the operators (2.87) results in the H-solution on the subgroup 23 no longer being essentially different with respect to the solution 5 at a ¼ b ¼ 0.

2.7.2. Stationary Flows Table 2 shows a functional view of the invariant solutions constructed on the subgroups 1–11 of the system (2.133). The v and H columns contain the expressions for physical components of the vectors in one of four orthogonal coordinate systems given in Section 1.5: Jv ¼ fJxi g, JH ¼ {Kxi}, where Jv ; J4 ; J5 ; JH are functions of x, a, b are arbitrary constants, and a ¼ b1/b2 for solution 4. We have seen earlier that the transition to a curvilinear coordinate system may be useful in constructing the functionally independent invariants of the subgroup under investigation. Table 2 also implies that the argument of the function Jxi determines a surface on which the thermoemission condition can be satisfied. Hence, the four

Exact Solutions to the Beam Equations

TABLE 2

79

H-Solutions of the Rank 1: Stationary Flows

No.

x

v

j

r

H

1 2 3 4 5 6 7 8 9 10 11

x c R p y r x c R p y

eay Jv Ra Jv eazþbc Jv eab2 q Jv ra ebc Jv vr ; 0; 0 Jv Jv Jv Jv Jv

e2ayJ4 R2aJ4 e2(az þ bc)J4 e2ab2qJ4 r2ae2bcJ4 J4 ay þ J4 a ln R þ J4 az þ bc þ J4 ab2q þ J4 a ln r þ bc þ J4

e2ayJ5 R2(a  1)J5 e2(az þ bc)J5 e2(a  1)b2qJ5 r2(a  1)e2bcJ5 J5 J5 R 2J5 J5 e 2b2qJ5 r 2J5

eayJH Ra  1JH eaz þ bcJH e(a  1)b2qJH ra  1ebcJH Hr, 0, 0 JH R 1JH JH e b2qJH r 1JH

systems introduced in Section 1.5 prove to be singled out from the infinite set of coordinate systems. Solutions 1–5 can describe both the vortex and potential flows. A special group consists of solutions 7–11 that determine the vortex beams: The scalar potential and velocity representations for those beams make impossible the existence of energy integral for the entire flow. The external magnetic field satisfies the homogeneous Maxwell equations, a solution of which, for H taken from the last column of Table 2, is given in Table 3, where Kxi are functions of x and d, H0, H01, H02, H03 are constants. In Table 3, Zn ¼ c1Jn þ c2Yn, Pa ¼ c1Pa þ c2Qa, and Pam ¼ c1Pam þ c2Qam are the Bessel, Legendre, and adjoint Legendre functions, respectively; Pam, Qam in 5.5 are the complex values with pure imaginary m; therefore, the linear combination Pam should be constructed using both real and imaginary parts of Pam, Qam. Case 2.1 corresponds to a homogeneous magnetic field, the projection of which onto the plane z ¼ const makes the angle p/2  d with the emitter c ¼ 0. Case 5.1, at a ¼ 1 and Kr ¼ H0P1(cos y), determines a homogeneous magnetic field directed along the axis of the conical emitter y ¼ y0 ¼ const. The magnetic fields for solutions 7–11 easily follow from the specification of the constants in the expressions 1–5 for Н. Now let us proceed to a brief review of exact solutions of the stationary beam equations.

2.7.3. Electrostatic Beams Solutions 1-5 with magnetic field equal to zero are given in Kirstein and Kino (1958) based on the separation of variables in the beam equations. The ordinary differential equations that determine the flows 3 (b ¼ 0), 4, 5

80

Exact Solutions to the Beam Equations

TABLE 3 The Components Kxi Determining the External Magnetic Field Requirements No. for constants

Kx2

Kx3



Kx1

1.1 a ¼ 0

–

H01

H02

H03

1.2 a 6¼ 0

ax þ d

 H0 sin 

H0 cos 

0

2.1 a ¼ 1

cþd

H0 sin 

H0 cos 

H03

2.2 a 6¼ 1

ac þ d

H0 sin 

H0 cos 

0

3.1 a ¼ b ¼ 0

–

3.2 a ¼ 0, b 6¼ 0 b ln R þ d 3.3 a 6¼ 0, b ¼ 0 aR 3.4 a, b 6¼ 0, n ¼  ib 4.1 a ¼ 1

5.2 5.3 5.4 5.5

5.6

1

H01R

H02R

H03

 H0R 1  sin  Z1()

H0R 1 cos 

0

0

Z0() Zn()

0

aR

Z n()

b  Zn ðÞ

b2p þ d

 H0e b1p  sin   H0e b1p  sin  H01

H0e b1p cos  H03

H0e b1p cos  pffiffiffiffiffiffiffiffiffiffiffiffiffi H01 2  1 a¼b¼0 csc y þH02  y H0 sin  a ¼ 0, b 6¼ 0 b ln tan þ d 0 b siny 2 y a ¼  1, – H0 ln tan H0 csc y 2 b¼0 1 d Pa ðÞ a 6¼ 0, 1; cos y Pa() a dy b¼0 d  Pm0 ðÞ a ¼ 1; cos y P0m() dy b 6¼ 0, m ¼  ib 1 d m P ð Þ a, b 6¼ 0; cos y Pam() a dy a m ¼  ib

4.2 a 6¼ 1 5.1

1

ab2p þ d

0 H03 H0 0

cos siny

0 

b m P ðÞ siny 0

b 1 Pm ðÞ a siny a

have been numerically integrated for some values of a, b1/b2 and y. Flows 1-5 were studied in Vashkovski and Syrovoy (1983), including 3D flows at b 6¼ 0. The existence of exact solutions describing the flows between the tilted planes c ¼ const (2, a ¼ 0) and coaxial cones y ¼ const with a common apex (5, a ¼ b ¼ 0) was discussed by Walker (1950); the first case was investigated in detail in Ivey (1952, 1953). A numerical solution determining a flow from the cone y ¼ const in y-direction along the circles was obtained by Waters (1959). The only currently known analytical solution (Meltzer, 1956) for the flow from the equipotential emitter c ¼ 0 with the curvilinear trajectories R ¼ const (2, a ¼  1) is described

Exact Solutions to the Beam Equations

81

by Eq. (1.158). This solution can be expressed in terms of elliptic integrals with regard to arbitrary emission conditions (Ogorodnikov, 1973c). The devices with planar or spherical geometry and rectilinear trajectories (1, a ¼ 0; 3, a ¼ b ¼ 0; 6) are investigated to the fullest extent. The emission in r-mode is described by the classic Child–Langmuir solution (Child, 1911; Langmuir, 1913). Different modes, which can be realized under arbitrary emission condition, are investigated in Fay et al. (1938), Page and Adams (1949), Ivey (1949), and Copeland et al. (1956). The first of these works is distinguished by its level of detail and abundance of illustrations. Page and Adams (1949) show that the entire variety of the planar diode modes can be reduced to two certain universal curves. Polynomial approximations for arbitrary emission mode are presented in Copeland et al. (1956). A solution in the form of a series for a cylindrical diode in r-mode is given in Langmuir and Blodgett (1923) and Bottenberg and Zinke (1964). Page and Adams (1945) obtained an expansion that converges much faster compared with that in Langmuir and Blodgett (1923) when the collector radius is substantially larger than the emitter radius. The general case was considered by Crank et al. (1939), Gold (1957b), Page and Adams (1949), Van der Ziel (1948), and Von Tschopp (1961); the solution can be represented either in the form of a series or found by numerical integration of the beam equations. The expansion parameter in Bottenberg and Zinke (1964) is the expression s ¼ 1  R 1 (R ¼ 1 is the emitter equation). The computation of 200 coefficients of the series and investigation of their behavior allows definite conclusions regarding the convergence properties of that expansion at s < 1. It is shown that the oscillating behavior of the Langmuir–Blodgett b-function, never physically explained before, in fact does not take place: The function b monotonically approaches unity after reaching its first maximum. Crank et al. (1939) create a pleasant impression in their paper, which, to our knowledge, is the only work that uses the qualitative theory of differential equations to study the dense beam properties. Some solutions in the form of series are known for the flows between the concentric spheres (Langmuir and Blodgett, 1924; Gold, 1958). Nevertheless, as shown in Kan (1948), Poplavsky (1950), and Itzkan (1960), the spherical diode equations allow a closed solution in terms of the Bessel functions of real and pure imaginary argument under arbitrary conditions on the emitter. The results from Kan’s work (1948) were partly repeated in Abdelkader (1963): The solution was expressed in quadratures. Simple approximate solutions for the potential in cylindrical and spherical diodes are given in Soudack and Zien (1967). In Syrovoy (1967a), the recurrent relations for the coefficients of a series are obtained to describe the transit time in cylindrical and spherical diodes with arbitrary emission conditions and arbitrary function of radius as the spatial coordinate.

82

Exact Solutions to the Beam Equations

2.7.4. Beams in a Magnetic Field The solutions (3, b ¼ 0) and (5, b ¼ 0), with a magnetic field different from zero, were considered in Kirstein (1959a). A system of ordinary differential equations determining the second of those solutions for the homogeneous magnetic field and emission in r-mode was numerically integrated in Dryden (1962) and Waters (1963); the flows in T-mode were studied in Manuilov and Tsimring (1978) and Manuilov (1981). Some flows from a conical emitter in nonhomogeneous magnetic fields (5, b ¼ 0, a 6¼ 1) were investigated in Dryden (1963). Solution 4 in magnetic field (4.2, a ¼  1) was discussed in Danilov (1968a). Different modes in planar and cylindrical magnetrons were comprehensively studied in Braude (1935, 1940, 1945) and Grinberg and Volkenstein (1938); the planar diode solution was given in elementary functions. A planar magnetron in T-mode with magnetic field tangent to emitter was considered in Syrovoy (1965d); the case of nonzero normal component of magnetic field was presented in Tsimring (1977). Analytical expressions for the supercritical-mode parameters in the case of planar and cylindrical magnetrons under the assumption of unambiguity of the velocity vector were obtained in Hull (1924) and Brillouin (1945). An analytical solution with a discrete spectrum of the multiple-loop trajectories in the near-critical mode was constructed in Danilov (1964). The collector current in the vicinity of the ‘‘cutoff’’ point represents an eigenvalue of the problem in question and takes discrete values located in the range between the collector current immediately before the ‘‘cutoff’’ point and the collector current equal to zero after the complete lockdown. The one-loop solution (Braude, 1935) corresponds to the ‘‘cutoff’’ current density; the solution in Brillouin (1945) corresponds to zero current. Thus, the results in Danilov (1964) explain the transition from the first mode to the second. The Brillouin-type solutions, when the moving particles do not intersect the equipotential surfaces, as applied to the case of axial symmetry (3, a ¼ b ¼ 0), have been addressed by many authors. A solution for the magnetic field 3.1 at H01 ¼ 0 was given in Cook (1959). The Brillouin-type flow (5, a ¼ b ¼ 0) in the magnetic field 5.2 was studied in Syrovoy (1965d). Solutions 7–11 describe the direct-heat diodes with linear or logarithmic potential distribution on the emitter. It can be shown (Syrovoy, 1965d) that the r-mode conditions result in an infinite value of tangential current density. Physical interpretation of this fact seems difficult; nevertheless, solution 7 for r- and U-modes was comprehensively studied by Von Hintringer et al. (1966). Solutions 7–11 for temperature-restricted emission, with the current density everywhere finite, were considered in Syrovoy (1965d).

Exact Solutions to the Beam Equations

83

2.7.5. Relativistic Flows The early works on relativistic beams (a comprehensive review is given in Belov, 1978), address the planar or spherically symmetrical electrode geometries with the self-magnetic field not taken into account, and the corresponding solutions having, as a rule, a form of the slowly converging series. In Gold (1957a), a solution for the planar relativistic diode in r-mode was expressed in parametric form through the Jacobi integral. In the work by Ignatenko (1962), which is a generalization of the results from Fay et al. (1938) to the relativistic case, the same problem was solved in terms of elliptic integrals. In Bradshow (1958), the self-magnetic field in the planar magnetron problem was not completely taken into account; in Danilov (1966a,b) this problem was reduced to elliptic integrals. As shown in Lucas (1958), the beams with rectilinear trajectories cannot start from the equipotential cathode when the external magnetic field is absent and the self-magnetic field is regarded accurately. According to Buneman (1958), the trajectory rectilinearity requirement implies space-charge homogeneity in the x-dependent solution. Some analytical solutions for the simplest Brillouin-type flows with the circular (Buneman, 1954; De Packh and Ulrich, 1961) and straight-line trajectories (Buneman, 1954) are also known. All of these 1D flows are described by the invariant solutions that follow from Table 2 (1, a ¼ 0; 3, a ¼ b ¼ 0; 6). More interesting are the 2D flows (Syrovoy, 1963; Vashkovskii and Syrovoy, 1991), which can be obtained from solutions 2 and 4 at a ¼ 0 and solution 5 at a ¼ b ¼ 0. The flows between the tilted planes (c ¼ const) or spiral cylinders (p ¼ const) are described by the solutions that allow the introduction of the emitter surface on condition that the self-magnetic field is accurately regarded. Solution 5 at a ¼ b ¼ 0 determines a Brillouin-type flow with vy  0. With the additional constraint vc ¼ 0, the corresponding system of ordinary differential equations proves to be underdetermined, so that the flow parameters can be expressed through the radial velocity given by any function y; jvr ðyÞj < 1:  0 sin yKc ¼ vr J5 sin y; j0 ¼ vr Kc ; ð sin yj0 Þ0 ¼ J5 sin y; qffiffiffiffiffiffiffiffiffiffiffiffiffi (2.134)   j0 ¼ aur = sin y; r ¼ au0 r = r2 sin y ; Hc ¼ a 1 þ u2r =ðr sin yÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffi where ur ¼ vr = 1  v2r ; a ¼ const: Energy homogeneity of the flow under consideration is identical to its potentiality; Eqs. (2.134) describe a vortex flow for any function vr . Requiring the energy integral to exist for the entire flow, we obtain   1 vr ¼ T 2  1 T 2 þ 1 ; r ¼ a2 ðj þ 1Þðr sinyÞ2 ; Hc ¼ aðj þ 1Þðr sinyÞ1 ;

j¼

2 1 1  T 2 T 1 ; 2 (2.135)

y T ¼ b tana ; 2

a; b ¼ const > 0:

84

Exact Solutions to the Beam Equations

For any pair of the values a, b there exists a cone y ¼ y0, on which the conditions tan(y0/2) ¼ b 1/a for the total space charge are satisfied. At y < y0, the flow is directed toward the coordinate origin; at y > y0, the flow is directed toward the increasing r.

2.7.6. Nonstationary Flows The invariant solutions that correspond to the subgroups 12-26 from Eq. (2.133) are listed in Table 4, where Jv , J4, J5, and JH depend on x, and b0 ¼ b00 þ 1 ¼ b1(b12 þ b22) 1 þ 1, and a0 ¼ a00 ¼  b2(b12 þ b22) 1, 1 b ¼ (1  a) . The external magnetic field in the last column of Table 4 can be easily obtained from Table 3. As an example, solution 15.1 corresponds to solution 4 at a ¼ 1. In Table 3, this case corresponds to the expression 4.1, in which H0t 1, H03t 1 should be taken instead of H0, H03. The exception is solution 23, which has no analog in Table 3. For this solution, we have  1=2  1=2 ; Kc ¼ H02 ; Kz ¼ H0 a2 þ x2 : Kr ¼ ðH0 =aÞx a2 þ x2

(2.136)

Some of the invariant solutions in Table 4 can be expressed in terms of elementary functions (Syrovoy, 1964a). Most of the solutions describe the planar, cylindrical, rotating, or spiral space-charge waves; the rest describe the nonstationary modes in the diode-type constructions with the emitting surfaces similar to those in stationary flows. In these constructions, the nonstationarity is caused by the collector potential variation described by a decaying power-like function. The oscillating mode has not been discovered in this case. Solutions 12.2 and 16.5 in Table 4 can describe the relativistic flows if the self-magnetic field is regarded strictly. The functional representations of the electric field E and magnetic field H for such flows are similar.

2.8. SOLUTIONS INVARIANT WITH RESPECT TO THE TRANSFORMATIONS WITH ARBITRARY FUNCTIONS OF TIME 2.8.1. H-Solutions of the Rank 1 00

The solutions in Table 5, with f k ¼ {f, g, h} and g ¼ a(a  1)/2, f ¼ f =f , f 0 ¼ f 0 =f , are constructed on the two-parametric subgroups using the transformations (2.87). Solutions 1–4 possess the translational symmetry @/@ z ¼ 0 in the magnetic field directed along the z-axis, whereas in case 5 only the component Hy is zero. 00

TABLE 4 H-Solutions of the Rank 1: Nonstationary Solutions No.

12.1 12.2 12.3 13 14 15.1 15.2 16.1 16.2 16.3 16.4 16.5 17.1 17.2 17.3 18 19 20 21 22.1 22.2 23 24 25 26.1 26.2 26.3

Requirements for constants

a 6¼ 0, a 6¼ 1 a¼0 a¼1 a 6¼ 0 – a ¼ b2/b1 a¼0 a ¼ a0 , b ¼ b0 a 6¼ 0, b ¼ 1 a ¼ 0, b ¼ 1 a ¼ 0, b 6¼ 1 a ¼ 0, b ¼ 0 a ¼ a00 , b ¼ b00 a ¼ 0, b 6¼ 0 a 6¼ 0, b ¼ 0 – – a 6¼ 0 a 6¼ 0, a 6¼ 0 a 6¼ 0, 1, a 6¼ 0 a ¼ 0, a 6¼ 0 a 6¼ 0 – – a 6¼ 0, a 6¼ 1 a¼0 a¼1

x

v a1

tx t 1x t xae t x p R ln t þ p teac t tRb  1 t 1R pt b ln R  t t  ac c t 1R x xae t txa  1 t 1x R 1(az  t) y r tra  1 t 1r t

t1 xJv Jv xJv xJv t1 Jv t1 eb2 q Jv t1 Jv t1 RJv t1 RJv RJv t1 RJv Jv RJv RJv RJv t1 RJv Jv t1 Jv xJv tab Jv Jv Jv t1 rJv t 1Jr, 0, 0 t 1rJr, 0, 0 Jr, 0, 0 rJr, 0, 0

j 2 2

t x J4 J4 x2J4 x2J4 t 2J4 t 2e 2b2qJ4 t 2J4 t 2R2J4 t 2R2J4 R2J4 t 2R2J4 J4 R2J4 R2J4 R2J4 t 2R2J4 a ln R þ bc þ cz þ J4 t 2(ay þ J4) aet/ay þ x2J4 t2ab(at by þ J4) at 1y þ J4 J4 t 2r2J4 t 2J4 t 2r2J4 J4 r2J4

r 2

t J5 t 2J5 J5 J5 t 2J5 t 2J5 t 2J5 t 2J5 t 2J5 J5 t 2J5 t 2J5 J5 J5 J5 t 2J5 t 2J5 t 2J5 J5 t 2J5 t 2J5 R 2J5 t 2J5 t 2J5 t 2J5 t 2J5 J5

H

t 1JH t 1JH JH JH t 1JH t 1JH t 1JH t 1JH t 1JH JH t 1JH t 1JH JH JH JH t 1JH t 1JH t 1JH JH t 1JH t 1JH R 1JH t 1JH t 1Kr, 0, 0 t 1Kr, 0, 0 t 1Kr, 0, 0 Kr, 0, 0

TABLE 5

H-Solutions of the Rank 1

Subgroups

Conditions

1 x ¼ tya  1, f ¼ ta, X1 þ aX2, X11, X8 Hz ¼ H0t 1 2 x ¼ y, f ¼ ta, X2, X11, X8 Hz ¼ H0t 1 3 x ¼ yae t, f ¼ eat, X1 þ X2 þ aX9, X11, X8 Hz ¼ H0

4 X1 þ X2, X11 þ X12, X8

5 X11, X12, X13

x ¼ t;  ¼ gx  fy g ¼ 1; Hz 6¼ 0; Hz ¼ H0

u

w

j

r

t 1(ax þ yJ1) t 1yJ2

t 1yJ3

t 2(gx2 þ aH0xy þ y2J4)

t 2J5

t 1(ax þ J1)

t 1J2

t 1J3

t 2(gx2 þ aH0xy þ J4)

t 2J5

ax þ yJ1

yJ2

yJ3

1 2 2 2a x

f 0 x þ J1

x ¼ t; g ¼ f 1 ; Hz 6¼ 0; 0 f x þ J1 h ¼ f ; Hz 6¼ 0

v

 g0 y þ J2 ZJ3

 g0 y þ J2

0

h z þ J3

þ aH0 xy þ J4

J5

1 h00 00 g  Hz gf 1  f  g2 f 2  2  i  00 0 J   g0 þ f x2 þ gf 1  gþ 5  0 þ Hzf xy þ 2 J4  00 1 00 2 00 g y2 þ  h z2 þ f x þ 2 0

þHzf xy þ Hx  g0 yz

J5

87

Exact Solutions to the Beam Equations

Let us consider in detail solution 5, with the functions Jk satisfying the equations  0   fJ1 Þ ¼ Hz fJ2 ; gJ2 Þ0 ¼ gðHx J3  Hz J1 Þ; hJ3 Þ0 ¼ Hx hJ2 ; (2.137) 00 00 00 J5 ¼ f þ  g þ h ¼ r0 ðfghÞ1 ; r0 ¼ const: At Hx ¼ 0, we have 00

02 J5 ¼  h þ 2f ¼ r0 h1 ;

J3 ¼ w0 h1 ;

w0 ¼ const:

(2.138)

A mode of interest, periodically varying h and r, arises at f ¼ eat: u ¼ ax þ A cosðot þ d1 Þ; w ¼ h1 ðw0 þ h0 zÞ;

v ¼ ay þ A cosðot þ d2 Þ;

00     2j ¼ a2 x2 þ 2Hz a1 xy þ y2 þ h z2 ; h ¼ r0 2a2 1 r ¼ r0 h1 ;   pffiffiffi  þ B cos 2at þ d ; d2  d1 ¼ arccos aHz1 ; o2 ¼ Hz2  a2 > 0;

(2.139) where d, A, B ¼ const. Equations (2.139) are somewhat peculiar. First, the functions h, r, w are everywhere finite on the condition that the corresponding constants are properly chosen. In this case, Eqs. (2.139) describe some oscillating state of the beam in the homogeneous external field Hz, with a homogeneous beam density changing in time. Second, at B ¼ 0, h ¼ const Eqs. (2.139) imply that a nonstationary flow, with harmonically oscillating velocity components u; v and homogeneous space-charge distribution r ¼ 2a2, can be realized in a stationary electromagnetic field. Owing to the condition o2 > 0, the equipotential surfaces are hyperbolas. Finally, at A ¼ B ¼ 0 the solution determines a stationary vortex nonmonoenergetic flow (Syrovoy, 1984a), thereby revealing the existence of the stationary solutions being invariant with respect to transformations (2.87), which preserve the nonstationary beam equations. In this case, the arbitrary functions in Eqs. (2.87) are specified as exponents. Solutions 4 and 5 also represent certain interest in that they can describe the evolution of an ellipsoidal bunch (Syrovoy, 2004b) that is considered as a flow fragment restricted in all three coordinates.

2.8.2. H-Solutions of the Rank 3 The importance of studying the H-solutions of the rank s > 1 arises from the fact that the system (S/H) of partial differential equations for such solutions may admit a nontrivial group that does not represent an obvious truncation of the main group constructed for the original system.

88

Exact Solutions to the Beam Equations

The example of such truncation is the elimination of z-scaling from the operator X1 in Eqs. (2.101) for solution in Eq. (2.126) at a ¼ 0. The existence of nontrivial transformations allows us to reduce the dimensionality consequentially by unity by means of consequent analysis of the oneparametric subgroups, so we eventually derive the solutions satisfying the ordinary differential equations, notwithstanding that those solutions are not invariant with respect to any (n  m  1)-parametric subgroup of the system (S) with n  m independent variables. Let us consider, for example, a solution of the rank 3 that is invariant with respect to the one-parametric subgroup HhX11 þ X12i and satisfies the system of equations with three independent variables t, x, z  0 y þ J1 ; u¼F

 0 x þ J2 ; v¼G

w ¼ J3 ;

j¼

 00 i 1 h00 0   0  G g þ Hzf x2 f  Hz G 2

 00     0  Hyf 0 xz þ J4 ; r ¼ J5 ; ℒJ  J;t þ ðgJ1  fJ2 ÞJ;x þ J3 J;z ;  þ Hzf 0 xy þ Hx G þ G    00  00 0  F  0 J2  Hy J3 þ f 1 F 0   G ℒJ1 ¼ gJ4;x þ Hz  F g0  Hzf x    0  Hyf 0 z; þ Hx G      0 þ Hz x;  0 J1 þ Hx J3 þ f 1 F 0 G ℒJ2 ¼ fJ4;x  Hz þ G 1  0 ℒJ3 ¼ J4;z þ Hy J1  Hx J2   Hy f F x; ℒJ5 þ ðgJ1  fJ2 Þ;x þ J3;z J5 h¼ 0;  00 i 00  2  0  G  G  þ Hzf 0 ; f þ g2 J4;xx þ J4;zz ¼ J5  f  Hz G x ¼ gx  fy;

(2.140) ð kÞ

ð kÞ

 ; k ¼ 0; 1; 2:  ; gðkÞ =f  G where f ðkÞ =g  F The system (2.140) has a solution invariant with respect to a subgroup with the operator  1 ð@=@J2 Þ þ ðF2 x þ F3 zÞð@=@J4 Þ; X ¼ @=@x þ F1 ð@=@J1 Þ þ GF

(2.141)

where the function of time Fk is expressed through f, g, with J1, J2 being linear and J4 quadratic in x. This operator determines the following solution of the original system: u ¼ ax þ by þ U; v ¼ cx þ dy þ V; 1 2 1 2 j ¼ Ax þ Bxy þ Cy þ Dx þ Ey þ F: 2 2

(2.142)

Here a, b, c, d, A, B, C are functions of t; D, E are dependent on t and linear in z; and U, V, w, F, r are functions of t, z. When there is no dependence on z, this solution is the solution given in Pease (1960). Some specifications of this flow are investigated below. We do not study in detail the group properties of the system (2.140). Let us introduce a new sought function, which eliminates the dependence on x in the second and third motion equations

Exact Solutions to the Beam Equations

   0 þ Hz x2  Hy f 1 F J 4 ¼ J4  1 f 2 F 0 G 0 xz; 2

89

(2.143)

and consider the conditions to allow the x-translation for the system that follows from system (2.140). Obviously, we must make zero the coefficient before x in the first motion equation:   00 00 (2.144) ff  gg þ ff 0 gg0 f 2  g2 ¼ 0: In particular, this condition is satisfied at g1 f 0 ¼ a; f 1 g0 ¼ b; f 1 f ¼ g1 g ¼ a2 ; f ¼ eat ; g ¼ bf ; a; b ¼ const;     J4 ¼ J4  1 aðb þ Hz Þ bx  y 2  aHy bx  y z; a2 ¼ ab; b2 ¼ a1 b: 2 00

00

(2.145) Assuming the conditions (2.145) are satisfied, we study the H-solution of the rank 2 of the system (2.140), constructed on the subgroup of x-translations. Such a solution satisfies the system ℒJ1 ¼ ðHz  aÞJ2  Hy J3 þ bHx z; ℒJ2 ¼ ðHz þ bÞJ1 þ Hx J3  aHy z; ℒJ  J;t þ J3 J;z ; ℒJ3 ¼ J 4;z þ Hy J1  Hx J2 ; J5;t þ ðJ3 J5 Þ;z ¼ 0; J 4;zz ¼ J5  2ab  Hz ða  bÞ

(2.146)

with the flow hydrodynamic parameters expressed in terms of the function Jk(t, z): u ¼ ay þ J1 ; v ¼ bx þ J2 ; w ¼ J3 ; r ¼ J5 ; b a 2 2 j ¼  ðHz  aÞx þ ðHz þ bÞy þ bHx xz  aHy yz þ J 4 : 2 2

(2.147)

It is curious that at ab < 0 the functions f, g from (2.145) prove complex, so no physical sense whatsoever can be attached to the transformations (2.87), although the corresponding invariant solutions to Eqs. (2.146) are real. Below, we restrict ourselves to the case Hy ¼ 0. In this case, the system (2.146) contains four parameters a, b, Hx, Hz, which determine the group properties. The combinations of these parameters represent as many as 12 different cases (Syrovoy, 1984a). We illustrate the characteristic peculiarities of the corresponding solutions using only three of these cases [at a ¼ 0 we have f ¼ 1; g ¼ bt; J4 ¼ J4 in (2.145) instead of the exponents]: ð4Þ Hz 6¼ 0; ð5Þ Hz ¼ 6 0; ð6Þ Hz ¼ 6 a;

Hz 6¼ a; Hz 6¼ a  b; Hz 6¼ b; Hx 6¼ 0; Hz 6¼ b; Hx ¼ 0:

Hz 6¼ b; a ¼ 0;

Hx 6¼ 0;

V0 ¼ aHx =ða þ bÞ;

(2.148)

90

Exact Solutions to the Beam Equations

Further, we use the notations as follows: u1 ¼ J1 ; u3 ¼ J3 ; u 5 ¼ J5 ; u2 ¼ J2 þ cz; 1 2  u4 ¼ J 4 þ cHx z ; c ¼ bHx ðHz  aÞ1 : 2

(2.149)

Below we represent the operators comprising the main group of modifications (4)–(6) for system (2.146); the repeating operators are merely listed but not unfold. @ @ @ þ ua þ 2u4 ; @z @ua @u4 ð   @ @ @ @ @ 00 þ V0 A þ A0 þ V 0 Hx A þ A z ; X4 ¼ A  aHx Adt @z @u1 @u2 @u3 @u4

ð4 Þ X1 ¼

@ ; @t

X2 ¼ S

X5 ¼ ðHz  aÞt

@ ; @j

X3 ¼ z

@ @ @ þ þ Hx z ; @u1 @u2 @u4

X6 ¼ FðzÞ

@ ; @u1

z ¼ u2  V0 z;

ð5Þ X1 ;

X2 ; X3 ; X5 ; z ¼ ðHz  aÞtu2  u1 ; @ @ @ 00 0 @ þA z ; Y6 ¼ Fðz; u2 Þ ; Y4 ¼ A þ A @z @u3 @u4 @u1

ð6Þ X1 ;

X2 ;

Z3 ¼ z

Y4 ;

Y6 ;

z ¼ ðHz  aÞtu2  u1 ;

@ @ @ þ u3 þ 2u4 ; @z @u3 @u4

2

Z5 ¼ Gðz; u2 Þ4ðHz  aÞt

3 @ @ 5 þ ; @u1 @u2

(2.150) Here A(t), S(t) are arbitrary functions of time and a ¼ 1, 2, 3; F, G are arbitrary functions of their arguments. These formulas show that, for cases (4)–(6), the system (2.146) allows some elementary transformations, including the translation in time (X1), the gradient invariance transformation (X2), and the scaling (X3 ; Z3). The structure of the transformations with operators X4, Y4 is similar to (2.87) and contains the arbitrary function of time A(t). Very similar is the transformation X5, which determines a finite subgroup and consists of adding a term linear in t to u1, a constant to u2, and a term linear in z to u4. All these cases are characteristic of the existence of the infinite groups, the operators of which include the arbitrary functions F, G of one or two variables. Let us consider some invariant solutions of Eqs. (2.146) constructed on the one-parametric subgroups with the operators from Eqs. (2.150). In case (4), on the subgroup Hha 1X1 þ Hx 1X5i we have (Ik ¼ Ik(z)): 1 ðHz  aÞt2 þ I1 ; u1 ¼ a 2 u4 ¼ atz þ I4 ;

u5 ¼ I 5 ;

t þ I2 ; u2 ¼ a  ¼ aHx1 : a

u3 ¼ I3 ;

(2.151)

Exact Solutions to the Beam Equations

91

This solution describes a perturbation of the stationary flow caused by adding the power functions of time to u; v, and the homogeneous electric field E linear in time. In case (5), the subgroup Hha 1X1 þ bX3 þ Y6  X5i at Fðz; u2 Þ ¼ ðu2 Þ determines the solution (Ik ¼ Ik(x), x ¼ ze abt): z=u2 þ F u1 ¼ zI1 I2 þ b1 I1 ; u4 ¼ z2 I4 þ Hx b1 z;

u2 ¼ zI2 þ b1 ; u5 ¼ I5 :

u3 ¼ zI3 ;

(2.152)

In case (6), when there are two operators with arbitrary functions, let us consider the subgroup Hha 1X1 þ Z3 þ Z5 þ Y6i on condition that G(z, u2) ¼ G(u2), F(z, u2) ¼  zu2 1G(u2). As a result, we derive the solution (Ik ¼ Ik(x), x ¼ ze at, l ¼ ln z þ I2) u1 ¼ I1 LðlÞ;

u2 ¼ LðlÞ;

u3 ¼ zI3 ;

u4 ¼ z2 I4 ;

u5 ¼ I5 ;

(2.153)

with L being the inverse function to the function resulting from the integration of G 1(u2) over u2. The function L does not enter into the system of equations determining Ik(x); therefore, this function is not constrained by any relations and remains arbitrary. Of note, the spatiotemporal structure of solutions (2.151) through (2.153) is more complicated compared with the invariant solutions given in Table 5. The investigation of the group properties of the system (2.140) is incomplete; we have restricted our analysis to reducing the dimensionality by the translations in x-direction. We now perform one more particular consideration to derive a stationary solution with a non-solenoid velocity field rv ¼ 6 0. The system (2.140), with f, g taken from (2.145), allows the translation in t, with the corresponding scaling x determined by the operator pffiffiffiffiffi Y ¼ @=@t þ ab xð@=@xÞ: (2.154) In the solution invariant with respect to the subgroup HhYi, allpthe ffiffiffiffiffi sought functions are dependent on z, Z ¼ entx ¼ mx  y; n ¼  ab, pffiffiffiffiffiffiffi m ¼ b=a, so that such solution describes a stationary flow determined by the following equations: ℒJ1 ¼ mJ 4;Z þ ðHz  aÞJ2  Hy J3 þ bHx z; ℒJ2 ¼ J 4;Z  ðHz þ bÞJ1 þ Hx J3  aHy z; ℒJ3 ¼ J 4;z þ Hy J1  Hx J2; ℒJ  ðmJ1 J2 þ nZÞJ;Z þ J3 J;z ; ℒJ5 þ J3;z þ ðmJ1  J2 Þ;Z J5 ¼ 0; 1 þ m2 J 4;ZZ þ J 4;zz ¼ J5 þ 2n2 þ ðb  aÞHz :

(2.155) On the condition that the corresponding coefficients are properly chosen and a linear combination of Z, z is added to J 4 , the transformation representing a shift in Z, J1, J2 preserves the system (2.155):

92

Exact Solutions to the Beam Equations

Z¼

@ @ @ @ þ g1 þ g2 þ ð g3 Z þ g 4 z Þ  ; @Z @J1 @J2 @ J4

g1 ¼ ac1 ; g2 ¼ nc2 ; c1 ¼ ðHz  aÞ=ða þ bÞ;

  g3 ¼ g1 ðHz þ bÞ; g4 ¼ g1 mc2 Hx =c1  Hy ; c2 ¼ ðHz þ bÞ=ða þ bÞ: (2.156)

The solution invariant with respect to the subgroup HhZi appears as (Ik ¼ Ik(z)): J1 ¼ g1 Z þ I1 ; J2 ¼ g2 Z þ I2 ; J3 ¼ I3 ; J 4 ¼ 1 g3 Z2 þ g4 Zz þ I4 ; J5 ¼ I5 : 2

(2.157)

Let us specify the complete functional view of the velocity components u; v for (2.157): u ¼ nc1 x þ ac2 y þ I1 ðzÞ;

v ¼ bc1 x  nc2 y þ I2 ðzÞ:

(2.158)

A solution of this type is constructed in Danilov (1968a) at I1 ¼ I2 ¼ Hx ¼ Hy ¼ 0, although in this work the coefficients at x, y depend on four constants, whereas in Eqs. (2.158) those coefficients are ‘‘construed’’ from a, b, Hz. The fourth parameter can be introduced if the subgroup HhX11 þ aX12i is originally taken. In this case, the parameters a, b in Eqs. (2.158) should be replaced by a/a, ab. The system (2.146) allows the stationary solution v ¼ bx þ J2 ; w ¼ J3 ; r ¼ J5 ; Jk ¼ Jk ðzÞ; u ¼ ay þ J1 ; 1 1 j ¼  bðHz  aÞx2 þ aðHz þ bÞy2 þ bHx xz  aHy yz þ J 4 ; 2 2

(2.159)

It should be noted that solution (2.159) at J1 ¼ J2 ¼ J3 ¼ H ¼ 0 describes a vortex electrostatic flow with hyperbolic trajectories (Meltzer, 1949a), whereas solution (2.137) at f ¼ eat, g ¼ ebt, h ¼ e (a þ b)t describes a 3D potential flow (Meltzer, 1949b). At J1 ¼ J2 ¼ J3 ¼ Hx ¼ Hy ¼ 0, Hz ¼ a  b in solution (2.159), we come to a potential flow with the particle trajectories in the form of ellipses or hyperbolas (Kirstein, 1958a; Po¨schl 6 0 we obtain a vortex-free flow and Veith, 1962; Walker, 1955), while at J3 ¼ with periodically varying w as investigated in Kent (1960). Thus, all the stationary solutions that are known from the literature, with additive separation of variables in the velocity components, are invariant with respect to the transformations that contain the arbitrary functions of time, specified as the exponents. The condition (2.144) holds true not only for the exponential functions from Eqs. (2.145), but also at f(t) ¼ g(t). This case, along with the H-solution constructed on the subgroup HhX13i, is considered in

Exact Solutions to the Beam Equations

93

Syrovoy (2003a,b). In Syrovoy (2003b), the results are represented in the form of extensive tables that include as many as 41 solution types, with various time dependencies and homogeneous magnetic field components. The axisymmetric degenerations of those solutions are investigated in Syrovoy (2008b). The solutions of the rank 3, invariant with respect to the transformations with the operator X13 (Syrovoy, 2003a, 2008b), generate several 3D and axisymmetric stationary flows aside from a substantial number of the nonstationary oscillatory and aperiodical modes, some of which can be described in terms of elementary functions or Bessel functions. For the solutions such as u ¼ uðt; x; yÞ;

v ¼ vðt; x; yÞ;

w¼

h0 z þ W ðt; x; yÞ; h (2.160)

00

j¼

 1 h 2 h0  z þ Hy x  Hx y z þ Fðt; x; yÞ; h 2h

r ¼ rðt; x; yÞ

by means of consecutive use of the operators Y ¼ @/@ y þ nW @/@ W, n ¼ const, and Z ¼ @/@ t at Hx ¼ Hy ¼ 0, h ¼ exp(at) , we obtain u ¼ uðxÞ;

v ¼ vðxÞ;

w ¼ az þ eny W ðxÞ;

1 j ¼ a2 z2 þ FðxÞ; 2

r ¼ rðxÞ: (2.161)

If the conditions t ¼ 0;

x ¼ 0;

u ¼ dx=dt  x_ ¼ 0;

v ¼ 0;

and F ¼ F0 ¼ 0

(2.162)

are satisfied, solution (2.162) can be described by elementary functions x¼

J0 J0 J0 cosðot þ dÞ  2 eat þ 2 ; Do2 aD ao

v ¼ Hz x; 82F ¼ u2 þ v2 ;2 r u ¼ J0 eat ; 39 < = 1 1 1 W ¼ W0 exp at þ nHz J0 4 2 t  sinðot þ dÞ þ 2 2 eat 5 ; 3 : ; ao Do aD cosd ¼ a=D;

sind ¼ o=D;

D 2 ¼ a2 þ o 2 ;

o2 ¼ Hz2 þ a2 ;

J0 ; W0 ¼ const:

(2.163) The asymptotics x t , u t , F t , z  z0 t, and y t are valid at small t, and the r-mode conditions are satisfied in the plane x ¼ 0. This implies that this plane may be considered a virtual emitter with parabolic potential distribution with respect to z. In this plane, the particles are traveling along the straight lines y ¼ const, while in the planes x ¼ const at small x the particles have parabolic orbits y (z  z0)4. 3

2

4

4

94

Exact Solutions to the Beam Equations

By sequentially applying the operators Y ¼ x@=@x þ y@=@y þ u@=@u þ v@=@v þ 2F@=@F þ nW@=@W; Z ¼ @=@t

(2.164)

at Hx ¼ Hy ¼ 0 and n ¼ 2, solution (2.161) gives a new solution in elementary functions, which describes an axially symmetric nonmonoenergetic flow: 1 vc ¼  Hz R; vz ¼ 2az þ W0 R2 ; 2   1 1 j ¼ a2 2z2  R2 þ rR2 ; r ¼ 6a2 þ Hz2 : 4 2

vR ¼ aR;

(2.165)

At Hz ¼ 0, W0 ¼ 0 we obtain a known monoenergetic electrostatic flow (Meltzer, 1949b; Syrovoy, 2003b) with hyperbola-like trajectories. At Hz 6¼ 0, W0 ¼ 0 we come to a monoenergetic flow with the same geometry in the meridian plane, but it is twisted and the value of r is increased. This solution, similar to its electrostatic analog, simultaneously satisfies the exact beam equations and the paraxial equation. For the general case, we arrive at the flow with the trajectories x ¼ const: z ¼ a R2 þ x=R2 ;

1 a ¼ W0 =a: 4

(2.166)

This flow is displayed in Figure 1, with the parabolic separatrix z ¼ a R2. The elliptical equipotentials appear as

1 2 1 2 2 (2.167) a þ Hz R þ 2a2 z2 ¼ j: 2 4 R

x = -64

x = 64

-16 -8 -2 -10

-5

x=0 0

2

8

16

5

10

15

FIGURE 1 Trajectories of the axisymmetric non-monoenergetic flow with parabolic separatrix.

z

Exact Solutions to the Beam Equations

95

In the half-space x 0, z > 0, the turning point z* of the trajectory and the corresponding radius R* are determined as pffiffiffiffiffiffi R4∗ ¼ x=a; z∗ ¼ 2 ax: (2.168) In the half-space x 0, z < 0, the flex point is determined by the values pffiffiffiffiffiffiffiffiffiffiffiffiffi (2.169) R4∗ ¼ 3jxj=a; z∗ ¼ 2 ajxj=3: Stationary variants of solution (2.142) have been comprehensively investigated in Syrovoy (2009a), which resulted in many new solutions to elementary functions for 3D and axially symmetric flows. The properties of those solutions depend on the velocity vector behavior in the plane z ¼ const. The flows, for which u;x þ v;y ¼ 0;

(2.170)

v;x  u;y ¼ Hz ;

(2.171)

may be called z  solenoidal. If

we use the term z  potential flows.

2.8.3. z-Solenoidal Flows The solution appears as u ¼ ay þ J1 ðzÞ; v ¼ bx þ J2 ðzÞ; w ¼ w ðzÞ; r ¼ rðzÞ; 2j ¼ bðHz  aÞ x2 þ aðHz þ bÞ y2 þ 2bHx xz  2aHy yz þ FðzÞ; a; b ¼ const:

(2.172)

The z-dependent functions satisfy the following equations: :: 2 €z þ l 2€z  Lz ¼  o ðJ0 t þ E0 Þ;  2 € 1 þ o K1 ¼ B ðHz  aÞHy z þ Hx w ; rw ¼ J0 ; K  1 _ K1  BHx z ; F0 ¼ J0 t  O2 z þ E0 ; K2 ¼ Hz  a

K1 ¼ J1 þ Hy z; K2 ¼ J2  Hx z; B ¼ Hz  a þ b: l2 ¼ O2 þ o2 þ Hx2 þ Hy2 ; L ¼ bðHz þ bÞHx2  aðHz  aÞHy2  o2 O2 ; o2 ¼ ðHz  aÞ ðHz þ bÞ; O2 ¼ 2ab þ Hz ða  bÞ: (2.173) Here E0 ¼ (d’/dz)0 is the longitudinal electric field in the starting plane z ¼ t ¼ 0 at x ¼ y ¼ 0, and J0 is the z-component of the current density. At zero starting velocity (w0 ¼ 0), the tangential current density

96

Exact Solutions to the Beam Equations

at z ¼ 0 is infinite; nevertheless, the current takes finite values within any band adjoining the plane z ¼ 0 (the zero index corresponds to this plane). Syrovoy (1984a) shows five different cases of behavior of the longitudinal velocity w, depending on the signs of the determinant D of the secular equation and the parameters l2, L, and m2: m4 þ l2 m2  L ¼ 0;

pffiffiffiffi 1 m2 ¼  l2  D; 2

1 D ¼ l4 þ L: 4

(2.174)

In Syrovoy (2009a), the existence domains are constructed in the plane x ¼ Hx2 þ Hy2, Z ¼ Hx2  Hy2, where the set D ¼ 0 represents a quadratic parabola, the set L ¼ 0 is a straight line, and the set l2 ¼ 0 is a vertical line. Physical meaning has the domain restricted by the rays Z ¼  x; we have D < 0 inside the parabola and D > 0 outside it. Figure 2 shows an example of the existence domain for a solution of the type I D > 0;

L > 0;

m21 > 0;

m22 < 0;

m21 ¼ m2 ; m22 ¼ n2 :

(2.175)

Studying the non-monoenergetic flows reveals the existence of the singular surfaces with the infinite space-charge density and the emission conditions, which can be termed as r- and T- pseudo-modes (or Pr, PT–modes). L=0 L>0 L<0

H 2x - H 2y

D=0

L=0

0

D<0 D=0 45°

FIGURE 2

D > 0, L > 0,type I

D>0 H 2x + H 2y

Existence domain for the solutions of the type I.

Exact Solutions to the Beam Equations

97

These pseudo-modes are by the analytical dependence of pffifficharacterized pffiffiffi ffi the flow parameters on 3 z and z, similar to r- and T- modes, with the difference that E0 ¼ 6 0 for Pr-mode and E0 ¼ 0 for PT -mode. For the solution satisfying the conditions in (2.175), the dependencies z(t) and w(t) are described by the following expressions (U-mode):  0 ð1  costÞ  g sint þ t; z ¼ g1 ½expða tÞ  cost þ g2 ½expða tÞ  cost  E  0 sint  g cost þ 1 ;  ¼ g1 ½a expða tÞ þ sint þ g2 ½a expða tÞ þ sint þ E w 2 z w  0 ¼ nE0 ; w ¼  o J0 ;  ; w ; E t ¼ nt; a ¼ m=n; z  ðw =nÞ J0 L w (2.176) where g1, g2, and g are arbitrary constants. The T- and PT -modes are determined by the formulas g ¼ 1 þ aðg1  g2 Þ; z ¼ g1 ½expðatÞ  cost  a sint þ g2 ½expða tÞ  cost þ a sint  0 ð1  costÞ  sint þ t ; þE  ¼ g1 ½a expða tÞ  a cost þ sint þ g2 ½a expða tÞ þ a cost þ sint w  0 sint  cost þ 1: þE (2.177) For r- and Pr-modes, we have g2 ¼ g1 

1  E0 ; 1 þ a2

9 8 = <   1 2 0 1  z ¼ 2g1 ð shat  a sintÞ þ E  sint þ t; exp ð a t Þ þ a cost þ a sint ; : 1 þ a2  0 a ½ expða tÞ þ a sint  cost  cost þ 1:  ¼ 2ag1 ð chat  costÞ þ E w 1 þ a2

(2.178) The distinctive features of the solution of type I can be conveniently illustrated using its simplest representative described by Eqs. (2.176) at g1 ¼ g2 ¼ E¯0 ¼ 0: z ¼ t  g sint;

 ¼ 1  g cost: w

(2.179)

 ðzÞ represents a curtailed cycloid, with w  never At g < 1 the function w turning into zero, while at g ¼ 1 is a cycloid that forms a regular set of virtual emitters. The same z-dependencies are peculiar to the potential beam with a straight axis and elliptical cross-section in the homogeneous magnetic field Hz (Kent, 1960), although the relationships between the frequency n and magnetic field H for those two cases are different.

98

Exact Solutions to the Beam Equations

At o2 > 0, by solving the equation for K1 from Eqs. (2.173), we obtain J 1, J 2: J1 ¼ a1 cosot þ a2 sinot þ J11 ; o J2 ¼ ða1 sinot þ a2 cosotÞ þ J21 ; Hz  a J11 ¼

aðHz  aÞ  B a  xt þ B H  x;  x cost  g Hy sint  2 H gH 2 2 2 nðo  n Þ o n nðHz þ bÞ o2

bðHz þ bÞ  B b  xt þ B H  y;  y cost  g Hx sint  2 H gH n ð o 2  n2 Þ o  n2 nðHz  aÞ o2  x ¼ w Hx ; H  y ¼ w Hy ; B ¼ Hz  a þ b: H J21 ¼

(2.180) The flow tube configuration results from integration of the equations u ¼ x_ ¼ ay þ J1 ;

v ¼ y_ ¼ bx þ J2 ; 1 € x  abx ¼ aJ2 þ J_ 1 ; y ¼ ðx_  J1 Þ: a

(2.181)

At ( ab) ¼ w2 > 0 we have  x ¼ x  X ¼ a1 sinwt þ a2 coswt; w  y ¼ y  Y ¼ ða1 coswt  a2 sinwtÞ; a

(2.182)

where X(t) and Y(t) correspond to a partial solution of the nonhomogeneous equation for x from Eqs. (2.181). The flow tube geometry is determined by Eqs. (2.181) written at z ¼ 0: € x0  abx0 ¼ aJ20 ;

1 y0 ¼ ðx_ 0  J10 Þ; a

1  x0 ¼ x0 þ J20 ¼ a1 sinwt þ a2 coswt; b 1 w  y0 ¼ y0 þ J10 ¼ ða1 coswt  a2 sinwtÞ: a a

(2.183)

Equations (2.183) show that the trajectories at z ¼ 0 in the system  0 are ellipses (at b ¼  a, the ellipses become circles): x0 ; y  a   2  y ¼ a21 þ a22 ; x20 þ   (2.184) b 0 which retain their parameters in any plane z > 0 in the system x; y determined by Eqs. (2.182). The trajectory shape depends on the sign of the product ab (at ab > 0, ellipses are replaced by hyperbolas) but not on the

99

Exact Solutions to the Beam Equations

signs of the multipliers. At a > 0, b < 0, the motion occurs along the contour in a negative direction, whereas at a < 0, b > 0, motion is in a positive direction. This also requires that the origins of the coordinate systems x; y y0 be superposed at t ¼ 0, which is provided by the equalities and  x0 ;  1 1  J20 ¼ X0 ;  J10 ¼ Y0 : b a

(2.185)

The functions X, Y can be obtained from Eqs. (2.181) as follows: o ða2 cosot  a1 sinotÞ; ðHz  aÞB  ¼ Y  Y1 ¼  1 ða1 cosot þ a2 sinotÞ; Y B ! (2.186)  g H þ b 1 z  y cos t   x t  Hy ;  x sin t þ X1 ¼ 2 H H H o2 o  n2 n nðHz  aÞ ! g 1  y sin t þ  y t þ Hx :  x cos t þ Hz  a H H H Y1 ¼  2 o  n2 n nðHz þ bÞ o2  ¼ X  X1 ¼  X

The constants a1, a2 follow from Eqs. (2.185): o  y; a2 þ  g BH Hz  a o  y ; Y0 ¼ 1 a1 þ gH  x; a2   gH X0 ¼  ðHz  aÞB B g  x ; oa2 ¼   y ; g ¼ 1  gBH gðHz  aÞBH : a1 ¼  2 2 o o  n2  x; gBH J10 ¼ a1 þ 

J20 ¼

(2.187)

These formulas allow us to provide a complete geometric description of the flow (2.172), (2.179). With this in mind, let us introduce two nonorthogonal coordinate systems  x ¼ x  X, y ¼ y  Y, z and x1 ¼ x  X1 ,  y1 ¼ y  Y1 , z. In the system  x;  y; z, the beam represents a cylinder in the same sense that a torus represents a cylinder in the orthogonal system associated with the central circle. The difference is that in the orthogonal case the cross section (circle) remains unchanged in the plane normal to the axis of the torus, while in our case the corresponding cross section (ellipse) does not change in the plane z ¼ const. This cross section is translated along the axis X(t), Y(t), z(t); herewith a regular sequence of virtual emitters with r-mode singularity exists at g ¼ 1 in the planes z ¼ nt ¼ 2pk. As follows from Eqs. (2.186), the spatial axis of the beam in the y1 ; z is located on the elliptic cylinder surface, coordinate system  x1 ; 

Hz  a 2 Hz  a  2 2 2 2  (2.188) x þ H ; y1 ¼  g Hx þ Hz þ b 1 Hz þ b y

100

Exact Solutions to the Beam Equations

constructed by translation of the contour (2.188) along the spatial axis X1(t), Y1(t), z(t). Figure 3 shows this construction. The dependencies of the longitudinal velocity on z, more complicated than those in Eqs. (2.179), are shown in Figure 4 (PT is the mode without formation of virtual emitters but with strong deceleration) and Figure 5 (infinitely increasing velocity, T-mode). The solution of type II is determined by the conditions 1 D > 0;  l4 < L < 0; 4

l2 > 0;

m21 ¼ m2 < 0;

m22 ¼ n2 < 0: (2.189)

 on t for Figure 6 shows the dependence of the longitudinal velocity w an elliptical beam in r-mode very close to formation of a virtual emitter which, at the same time, is never realized. A regular sequence of virtual emitters may exist at the multiple roots: pffiffiffiffi 1 n2 ¼  l2 þ D; 2

pffiffiffiffi 1 m2 ¼  l2  D; 2

n ¼ km;

k ¼ 2; 3; . . .

(2.190)

 ðtÞ are presented in Figure 7. The corresponding dependencies w y y1 x

x1 y

x

FIGURE 3 Displacement of the elliptical beam’s axis across the elliptical cylinder’s surface the ( x1 , y1 ) plane of the coordinate system x1 , y1 , z. w 2

1

0

2

4

6

8

10

12 t

FIGURE 4 The change of longitudinal velocity in PT - mode without formation of virtual emitters, but with deep deceleration.

Exact Solutions to the Beam Equations

101

w 5.0 2.5

1

0

FIGURE 5

2

3

4

t

Unlimited increase of the longitudinal velocity in T-mode. w 2 1

0

2

4

6

8

10

12

t

FIGURE 6 Longitudinal velocity in r-mode with approaching to the virtual emitter that is never realized. w 2 k=2

3

1

0

p /2

p

3p / 2

2p t

FIGURE 7 Different regularities of the longitudinal velocity change when forming the regular sequence of virtual emitters (multiple roots of characteristic equation).

2.8.4. z-Potential Flows Consider a solution of the type (Syrovoy, 1985a): u ¼ by þ ax þ J1 ðzÞ; v ¼ cx þ ay þ J2 ðzÞ; w ¼ wðzÞ;       2j ¼ a2 þ c2 x2 þ  a2 þ b2 y2 þ 2aO xy þ cHx  aHy xz þ aHx  bHy yz þ FðzÞ; ð   1   1 0 0 r ¼ r0 þ ðww Þ ; r0 ¼ 2 a2 þ O2 þ Hz2 ; F ¼ w2  Hy J1  Hx J2 dz; 2 2 1 1 b ¼ O þ Hz ; c ¼ O  Hz ; a; O ¼ const: 2 2

(2.191)

102

Exact Solutions to the Beam Equations

Equations (2.191) describe a vortex, non-monoenergetic, non-solenoidal flow. The longitudinal field E0 and the current density J0 at the coordinate origin take the form   dF dz € 0 ; w ¼  z_ : (2.192) ¼ w_  Hy J1 þ Hx J2 0 ; J0 ¼ r0 w0 þ w E0 ¼ dz 0 dt It follows from the formula for E0 in Eqs. (2.192) that, in contrast to r-mode, the conditions for T-, PT -, and Pr-modes can be satisfied at the nonzero tangent magnetic field. The functions w, J1, J2 satisfy the equations ½r0 w þ wðww0 Þ0 þ 2a½r0 þ ðww0 Þ ¼ 0; wK0 2 þ bK1 þ aK2 ¼ 0; wK0 1 þ aK1 þ cK2 ¼ 0; K1 ¼ J1 þ Hy z; K2 ¼ J2  Hx z; o2 ¼ bc:

(2.193)

After transforming to the variable t(z), the solution for system (2.193) in parametric form appears as z ¼ Zs sinnt þ Zc cosnt þ Ze expð2atÞ þ Z0 ; w ¼ n Zs cosnt  nZc sinnt  2aZe expð2atÞ;    1  ; Zc ¼  1 w_ 0 þ 2aC  ; Ze ¼  1 C;  Zs ¼ w0  C 2 n n 2a    ¼ J0 = 4a2 þ n2 ; Z0 ¼ ðZc þ Ze Þ;n2 ¼ r þ H2 þ H2 ; C 0

x

(2.194)

y

J1 ¼ Hy z þ expðatÞ ða1 cosot þ a2 sinotÞ; o J2 ¼ Hx z þ expðatÞ ða2 cosot þ a1 sinotÞ: c In contrast to z-solenoidal flows, the RHS of Eq. (2.170) differs from zero in the case under consideration and equals 2a, which, at o2 > 0, makes impossible the existence of the closed streamlines in the plane where the longitudinal velocity vanishes. The flow picture at z ¼ 0 is similar to that of the vortex source or the vortex sink (a<>0), whereas the particle trajectory in the coordinates  x0 ;  y0 represents a spiral-like curve that turns into a logarithmic spiral in axisymmetric case (Figure 8):  x0 ¼ x0 

bJ20  aJ10 cJ10  aJ20 2 0 þ y20  R2 ; y0 =x0  tanc; ;  y 0 ¼ y0  2 ; x 2 2 a þo a þ o2

 x0 ¼ expðatÞða1 cosot þ a2 sinotÞ;  y ¼ expðatÞða2 cosot  a1 sinotÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 01 2 2 a1 þ a2 expðab=oÞ a a2 R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp@ cA ; tanb ¼ : a1 o cos2 c  ðb=cÞ sin2 c (2.195)

Exact Solutions to the Beam Equations

103

y0 0.5

-0.25

0

0.25

0.50

0.75

1.00 x0

FIGURE 8 Projection of the spatial trajectory onto the plane z ¼ 0.

At o2 > 0 we have ( b/c) > 0, which implies that the expression under the radical sign in the denominator is always positive. The parametric equations of 3D trajectories are determined by the relations below and the formula for z(t) from Eqs. (2.194):  x ¼ x  X ¼ expðatÞ ða1 cosot þ a2 sinotÞ; o  y ¼ y  Y ¼ expðatÞ ða2 cosot  a1 sinotÞ: b

(2.196)

The stream tubes in the system  x;  y represent the surfaces with an elliptical cross section, the semi-axes of which are changing in accordance with the law exp( at): 2   b 2   y ¼ a21 þ a22 expð2atÞ: (2.197) x þ o The projection of the 3D trajectory, which is located on the surface (2.197), onto the plane z ¼ const by means of the curves congruent to the beam axis is described by Eqs. (2.195). At o2 < 0, there exists the constraint (b/c) tan2c < 1 on the angle c in Eqs. (2.195), which means that the flow fills only a restricted part of the space and hence does not represent any interest. Thus, due to the relation o2 ¼  O2 < 0, the electrostatic flows of the type represented by Eqs. (2.191) fall outside our consideration. The functions X(t), Y(t) describing the beam axis appear as  ¼ X  X1 ¼ expðatÞ ðXce cosot þ Xse sinotÞ; X  ¼ Y  Y1 ¼ o expðatÞ ðYce cosot þ Yse sinotÞ ; Y b X1 ¼ Xc cosnt þ Xs sinnt þ Xe expð2atÞ þ X0 ; Y1 ¼ Yc cosnt þ Ys sinnt þ Ye expð2atÞ þ Y0 :

(2.198)

104

Exact Solutions to the Beam Equations

The coefficients in Eqs. (2.198) take the form 

   a2 þ o2  n2 Xc  2anXs ¼  aHy þ bHx Zc  nHy Zs ;  2    2anXc þ a þ o2  n2 Xs ¼ nHy Zc þ aHy þ bHx Zc ;   1 Xe ¼  2 3aHy þ bHx Ze ; 9a þ o2

X0 ¼ Yc ¼

a2

 1  aHy þ bHx ðZc þ Ze þ Zce Þ ; 2 þo

 1 Hy Zc  aXc þ nXs ; b

 1 Hy Zs  nXc  aXs ; b

 1 Hy ðZc þ Ze þ Zce Þ  aX0 ; b 3  2  1 2aoO 4ðaF1 þ oF2 Þ  2a þ bHz a1  a2 5; Xce ¼ 4aða2 þ o2 Þ c Ye ¼ 

 1 Hy Ze þ 3aXe ; b 2

Ys ¼

Y0 ¼

2 3  2  1 2aoO 4ðoF1 þ aF2 Þ þ a1  2a þ bHz a2 5 ; Xse ¼ 4aða2 þ o2 Þ c 8 9 <1     = 1 1 o 2 2 2 2a þ o F1  aoF2 þ 2aO a1 þ 2a  cHz a2 ; Yce ¼  Hy Zce þ ; b 4aða2 þ o2 Þ :b c 8 9 <1  =    o 2  1 1 aoF1 þ 2a2 þ o2 F2 þ 2a  cHz a1  2aOa2 ; Yse ¼ Hy Zse  2 2 : ; b 4aða þ o Þ b c     F1 ¼ 2aHy þ bHx Zce þ oHy Zse ; F2 ¼ 2aHy þ bHx Zse  oHy Zce :

(2.199) The constants a1, a2 in the formulas for J1, J2 from Eqs. (2.194) are determined in a manner similar to Eqs. (2.185) and (2.187). In the coordinate y1 ¼ y  Y1 , the beam axis is positioned on the system  x1 ¼ x  X1 ;  ellipse, the semi-axis of which is changing as exp( at) and which, in contrast to Figure 2, is turned by the angle W with respect to the axis x1 : tan2W ¼

2ðXce Yce þ Xse Yse Þ  : 2 þ X2 Y2ce þ Y2se  Xce se

(2.200)

Solution (2.191) becomes quasi-symmetrical at O ¼ 0, b ¼  c ¼  Hz/2. In this case, in the cylindrical coordinate system R, c, z, we have

Exact Solutions to the Beam Equations

105

vR ¼ aR þ J1 ðzÞcosc þ J2 ðzÞsinc; 1 vc ¼  Hz R  J1 ðzÞ sinc þ J2 ðzÞcosc; w ¼ wðzÞ; 2 0 1 0 1 1 1 2j ¼ @a2 þ Hz2 AR2  2@aHc þ HR Hz ARz þ F; 4 2 1 r ¼ r0 þ ðww Þ ; r0 ¼ n ¼ 2a þ Hz2 ; 2  1 2 v þ v2c þ w2  j; ℋ¼ 2 R 0 0

2

HR ¼ Hx cosc þ Hy sinc;

2

ð   1 2 F ¼ w  Hy J1  Hx J2 dz; 2

Hc ¼ Hx sinc þ Hy cosc; (2.201)

where ℋ is full energy. The curvilinear axis of the beam is determined by the formulas given above with the constants specified in accordance for the case of quasisymmetry. The beam radius in the plane x; y is changing in time in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi accordance with the law R ¼ a21 þ a22 expðatÞ. At Hx ¼ Hy ¼ 0, we obtain an axisymmetric monoenergetic flow, which according to Eqs. (2.192), allows the possibility of r-mode: 0 1 1 1 vR ¼ aR; vc ¼  Hz R; w ¼ wðzÞ; 2j ¼ @a2 þ Hz2 A R2 þ w2 ; 2 4 r ¼ r0 þ ðww0 Þ0 ;

1 r0 ¼ n2 ¼ 2a2 þ Hz2 : 2 (2.202)

The stream tubes that correspond to solution (2.202) may be represented in the explicit form z ¼ z(R). Let us investigate the case when the r-mode conditions are satisfied for the z-dependent functions: 2

3 2a  ¼ R ¼ expðatÞ; w ¼ C4 sinnt  cosnt þ expð2atÞ5 ; R n R0 8 0 2 0 13 0 19 1 <1 = nz 1 2 2  A5  sin@ lnR  A ; a ¼ 2a=n: @1  A þ a41  cos@ lnR z ¼ ¼ :a ; 2 C a a R

(2.203)

106

Exact Solutions to the Beam Equations

Turning w into zero signifies the formation of a virtual emitter and, as a consequence, possible distortion of the beam laminarity. The expression for w in Eqs. (2.203) can be represented in the form  w¼C

pffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ 1½cosb expðatÞ  cosðb  tÞ ;

t ¼ nt:

1 cosb ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 a þ1 (2.204)

At t ¼ 0 both functions in the square bracket have equal values and derivatives; nevertheless, they have no points of intersection at a > 0, whereas at a < 0, due to the presence of the rapidly decreasing exponent, the locations of virtual emitters practically coincide with the zeros of the function cos(b  t). Figure 9 shows the flow with no limitations along the longitudinal coordinate at a ¼  1, a ¼ 1, n ¼ 2, Hz ¼ 2. The particles leave the starting surface in the tangential direction. At w0 6¼ 0, the beam slope W with respect to the z-axis is determined by the condition tan W ¼ a/w0. Figure 10 illustrates the flow geometry when the virtual emitter is formed at a ¼ 1, a ¼  1, n ¼ 2, Hz ¼ 2. The closer the values R0 ¼ R1 and R0 ¼ R2 for any two trajectories, the longer is the laminar fragment between them because the fragment length along the z-axis is determined by the point of intersection of these two trajectories. The fact that the injection velocity w0 is nonzero does not eliminate the reflection surface but moves it away from the emitter. The trajectories for the solution (2.202) are dissimilar. It is noteworthy that the asymptotics of these curves is still all different from the asymptotics of the trajectories in the thermocathode vicinity.3   1 t z1=3 Within the starting surface vicinity in r-mode, we have R on the axis. For the thermocathode at H ¼ 0, the stream tube generatrix   1 t6 z2 , represents an analytic curve with the asymptotics R R 2

1

0

R0 = 2 1.5 1

1

0.5 0.3

2

3

4

z

FIGURE 9 Trajectories of the axisymmetric laminar monoenergetic flow that starts from the singular plane.

3

See Chapter 3.

Exact Solutions to the Beam Equations

z

107

R0 = 0.2 0.3

3

0.5 2 1 1

0

2 1

2

3

4

5 R

FIGURE 10 Trajectories of the axisymmetric monoenergetic flow when forming the virtual emitter.

whereas at H 6¼ 0 the stream tube generatrix is nonanalytic, with   1 t4 z4=3 . R

2.8.5. Laminated Flows The laminated flows are characterized by the absence of flowing between the planes x ¼ const, while the flow parameters depend on the x-coordinate and other spatial coordinates: u ¼ 0; v ¼ ax þ by þ J2 ðzÞ; w ¼ wðzÞ; 2j ¼ Hz2 x2 þ b2 y2 þ Hx2 z2 þ w2  bHz xy  2Hx ðHz x  byÞz þ 2V0 ðHz x þ by þ Hx zÞ; 1 r ¼ r0 þ ðww0 Þ0 ; r0 ¼ n2 ¼ b2 þ Hx2 þ Hz2 ; ℋ ¼ V02 ; 2 curl P ¼ 0; div v ¼ b þ w0 ; a ¼ Hz ; H ¼ fHx ; 0; Hz g ; € þ r0 w ¼ CexpðbtÞ; J2 ¼ Hx z þ V0 ; V0 ¼ const: w

(2.205) Solution (2.205) describes a potential monoenergetic flow, which is z-solenoidal at b ¼ 0. The longitudinal velocity is determined by Eqs. (2.194), with the substitution 2a ! b and the new n2 value taken from (2.205). The dependence y(t) is given by the expression  y ¼ y  Y ¼ y0 expðbtÞ; Y1 ¼

Y ¼ Y1 þ Yc cosnt 

  1 Hz x  V0  Yc b2  n2 ; b

Yc ¼

n2 Yc expðbtÞ; 2b2

 C Hx : n2

(2.206)

At Hx ¼ 0, we have Y ¼ (Hzx  V0)/b, which means that the coordinate y in the plane x ¼ const gains a linear displacement in x. The solution

108

Exact Solutions to the Beam Equations

; zÞ has trajectories of the same type as the axially (2.205) in the plane ðy symmetric beam (2.202) in the meridian plane (R, z):

 



  y y 1 y0 1 1 z ¼  þ a 1  cos  sin ; a ¼ b=n: 1 ln ln  y a a y0 a y0 (2.207) A monoenergetic electrostatic flow with similar geometry is possible at Hx ¼ Hz ¼ 0, but with no displacement in y: u ¼ 0; v ¼ by; r ¼ r0 þ ðww0 Þ0 ;

w ¼ wðzÞ; 2j ¼ b2 y2 þ w2 ; r0 ¼ b2 ¼ n2 ; a ¼ 1:

(2.208)

At b ¼ 0, solution (2.205) determines the z-solenoidal flow with a different law of longitudinal velocity variation, which, up to normalization, coincides with the dependence (2.179): z ¼

z ¼ nt  g sinnt; ðJ0 =n3 Þ

¼ w

w ¼ 1  g cos nt: ðJ0 =n2 Þ

(2.209)

In the normalization (2.209), the coordinate  y on the trajectory appears as

1 g  y ¼ ða x þ V0 Þt þ Hx nt2 þ cosnt þ y0 : (2.210) 2 n At g ¼ 1, the formation of virtual emitters does not result in nonlaminarity because of w 0. The trajectories in the planes x ¼ const are dissimilar and may be obtained, one from another, by means of the translation along the y-axis. In Figure 11, the flow path is displayed in x ¼ 0, and ax=Hx ¼ 1: the coordinates ðn y=Hx ; zÞ at V0 ¼ 0,  n y a x 1 z ¼ t  sint; t þ t2 þ cost: ¼ (2.211) Hx Hx 2 On the condition that Hx ¼ 0, V0 ¼ 0, the particles at x ¼ 0 are traveling along the straight lines y ¼ y0, whereas at x 6¼ 0 the trajectory has the form of curve 3 in Figure 11, with the explicit equation z ¼  y  sin y;

a x ¼ 1: n

(2.212)

This trajectory is qualitatively similar to the curves for Eqs. (2.211); however, it has more inflexion points at  y ¼ kp and more sharply passes through the virtual emitter plane (shown in Figure 11 by the dashed line). In contrast to solution (2.205), its z-solenoidal analog has no electrostatic variant.

Exact Solutions to the Beam Equations

109

vy/Hx,y 50

40

30

20 2 1

10

0

FIGURE 11

5

10

3

15

z

Trajectories of the x-laminated flow.

In the solutions considered above, the non-interacting layers coincide with the planes x ¼ const. The solution with the velocity components vR ¼ RJ1 ðcÞ;

vc ¼ RJ2 ðcÞ;

w ¼ az þ RJ3 ðcÞ;

a ¼ const;

(2.213)

for an arbitrarily oriented homogeneous magnetic field are described by a system of equations with a partial solution that determines a laminated monoenergetic solenoidal flow with no flow between the half-planes c ¼ const: a vR ¼  R; vc ¼ 0; w ¼ az þ RHc ; 2 20 1 3 1 4@1 2 3 j¼ a þ Hc2 AR2 þ a2 z2 5 þ aHc Rz; r ¼ a2 ; 2 4 2

ℋ ¼ 0; curl P ¼ 0; div v ¼ 0; H ¼ Hx ; Hy ; 0 :

(2.214)

The equipotential surfaces of solution (2.214) are ellipsoids. In the plane c ¼ const, the particle trajectories are described by the equation z¼

C 2Hc R:  3a R2

(2.215)

110

Exact Solutions to the Beam Equations

R

y = const C>0

C<0 R* C=0 z* 0

FIGURE 12

z

Trajectories of the c-laminated flow.

The trajectory separatrix C ¼ 0 represents a straight line, the slope of which is varied with c varying (Figure 12). In the absence of a magnetic field, Eqs. (2.214) determine a known axisymmetric solution with hyperbola-like trajectories (Meltzer, 1949b).

2.8.6. H-Solutions Inessentially Different with Respect to Infinite Subgroups Recalling Section 2.5, we are reminded that any transformation of the main group of the system (S) transforms any (not necessarily invariant) solution of the system (S) again into a solution of this same system. From the standpoint of group classification, it suffices to indicate all the essentially different H-solutions, although some inessentially different H-solutions may also be of interest if the relevant transformation is not elementary. For example, if we apply the transformations of translation, rotation, and scaling, we thereby represent the same solution in another coordinate system. Owing to the infinite subgroup of transformations defined by the operators (2.87), a curious situation arises for a non-relativistic beam in a homogeneous magnetic field. By subjecting any stationary solution of the beam equations to such transformations, we arrive at a certain nonstationary ‘‘image solution’’ that cannot be obtained by means of any manipulations with the nonstationary solutions given in Table 4. The simplest interpretation allows constructing the ‘‘images’’ of 1D flows, particularly of the r-mode solution for the planar diode, under appropriate normalization, appears as u ¼ x2=3 ;

r ¼ x2=3 :

2j ¼ x4=3 ;

(2.216)

By subjecting (2.216) to the transformations (2.87) at g ¼ h ¼ 0, we have u ¼ ðx þ f Þ2=3  f 0 ;

00

2j ¼ ðx þ f Þ4=3  f x  f 4=3 ;

r ¼ ðx þ f Þ2=3 : (2.217)

Exact Solutions to the Beam Equations

111

Depending on the choice of f(t), the solution (2.217) can describe highfrequency oscillations in the planar diode (Syrovoy, 1965a) (the r-mode conditions are apparently violated). By considering solution (2.151), we can see (Syrovoy, 1984a) that z ¼ u2  V0 z ¼ aHx1 ½t  tðzÞ þ V0 ; V0 ¼ const

I3 ðzÞ ¼ dz=dt;

(2.218)

represents a nontrivial combination that cannot be reduced to a constant. Using the infinite group with the operator X6 from (2.150), we obtain from (2.151) a solution with the new value u¯1 ¼ u1 þ G(z). Thus, the same potential and density distribution in time and space correspond to a continuum (G is an arbitrary function) of dynamical states that differ from each other in the form of the x-velocity component. This does not signify the non-uniqueness of the solution because the combination (2.218) at z ¼ t ¼ 0 is reduced to a linear function of time, but not to a constant. For solution (2.152) expressed in terms of the functions Ik(x), the combination z ¼ ðHz  aÞtu2  u1 ¼ u2 ½ðHz  aÞt  I1 ðxÞ

(2.219)

is nontrivial. By subjecting this solution to a transformation with the operator Y6, we obtain the new value u¯1 ¼ u1 þ G(z, u2), where G is an arbitrary function of the arguments indicated.

2.9. INVARIANT SOLUTIONS OF THE GEOMETRIZED BEAM EQUATIONS 2.9.1. On the Exact Solutions of the Geometrized Beam Equations As mentioned in Section 2.4, the coordinates of the infinitesimal operators form a contravariant vector. This implies that the main group of the original system (S) does not depend on the reference system if the relationship between the old and new coordinates is expressed in terms of finite expressions. This statement becomes less obvious within the framework of the geometrized approach when the Cartesian and curvilinear coordinates are interconnected by a partial differential equation. It can be assumed that the geometrized equations determine in the space (x, u, p) a surface with some new group properties that allow constructing the solutions that have been previously unknown. As to the planar flows, this possibility was studied in Syrovoy (1982, 1985b); nevertheless, no new solutions of the beam equations have been revealed among the examples given in these works [the example given in Syrovoy (1985b) is

112

Exact Solutions to the Beam Equations

a mistake: Solution (13) coincides with (14) after transforming to local coordinates]. For axisymmetric flows, Eqs. (1.141) and (1.149) cannot be separated from Eqs. (1.140), which interrelate R, z and x1, x2, because h3 ¼ R in Eq. (1.149). The admissible group is thus narrowed. As a result, it is yet unclear whether solutions 3, 5, 9, and 11 at b ¼ 0 can be obtained. The same solutions at b 6¼ 0 describe 3D flows, but the prospect for realizing those solutions within the framework of the geometrized approach is greatly reduced. The reason for this is that constructing exact solutions requires verification of not only three Euclidean conditions (1.111), but also the condition (1.112), since the initial surface configuration cannot be set beforehand but follows (similar to the solutions given in Table 2) from the functional form of the solution found. This means that we are speaking about a system with an excessive number of equations. In summary, the number of presently known exact solutions for the ‘‘common’’ equations exceeds that for the geometrized equations.

2.9.2. Group Properties and Invariant Solutions Our aim here is to show how all planar electrostatic potential flows determined by Eqs. (1.141) and (1.149) at h3 ¼ 1 can be obtained with the geometrized approach. The problem of constructing the coordinate system to describe the above-mentioned flows in terms of ordinary differential equations was beyond the scope of the single-component flow theory. The description of the flows provided here is formulated somewhat differently: The sought functions are determined by the ordinary differential equations, whereas the relationship between the Cartesian and curvilinear coordinates is expressed by Eqs. (1.142), which represent a solution of the system of partial differential equations. The example below illustrates the methods used to interpret exact solutions of the geometrized equations. Equations (1.141) and (1.149) are invariant with respect to the transformations with operators     X1 ¼ x1 @=@x1  h 1 ð@=@h 1 Þ þ 2rð@=@rÞ; X2 ¼ f @=@x2  f 0 h 2 ð@=@h 2 Þ; X3 ¼ h 1 ð@=@h 1 Þ þ h 2 ð@=@h 2 Þ  4rð@=@rÞ; X4 ¼ @=@x1 : (2.220) Aside from the translation and scaling transformations, which we already know, Eqs. (2.220) also contain the transformation of ‘‘remarking’’   x2 ¼ F x2 . Let f(x2) ¼ b(x2)/b0 (x2). the coordinate surfaces x2 ¼ const as  Consider the solution that is invariant with respect to the subgroup HhX1 þ X2 þ bX3i, b ¼ const

Exact Solutions to the Beam Equations

  b1 h 1 ¼ x1 H1 ðxÞ; h 2 ¼ bb1 b0 H2 ðxÞ;  1 24b   r¼ x RðxÞ; x ¼ x1 b x 2 :

113

(2.221)

The functions H1, H2, R in Eqs. (2.221) satisfy the equations H2 R ¼ J0 x5b3 H1 ; xbþ1 H11 H0 2 þ xbþ1 H21 H0 1 ¼ c; J ; c ¼ const; h   0 i 0 0 4bþ4 4 b1 2bþ2 bþ1 2 1 0 0 H1 H2 x H1 þx x H1 H2 H 1 ¼ 2x3bþ1 H1 H2 R: x (2.222) The first of these equations follows from the current conservation equation; the second follows from the Poisson equation. To clarify (2.221), let us consider a vicinity of the emitter x1 ¼ 0, x ¼ 0 and construct a solution with power-like asymptotics to satisfy the total space-charge conditions (m, n, e, t ¼ const): H1 ¼ A 0 xn ð1 þ A 1 xe þ . . .Þ;

H2 ¼ B 0 xm ð1 þ B 1 xe þ . . .Þ;

R ¼ R 0 xt ð1 þ R 1 xe þ . . .Þ:

(2.223)

Let us calculate the arc length of the coordinate axes near the surface x1 ¼ 0, restricting ourselves to the main term, which coincides with the distance counted along the axes of the local coordinate system X, Y ð  bþn ; X ¼ h1 dx1 ¼ A 0 bn ðb þ nÞ1 x1 ð (2.224)  m Y ¼ h 2 dx2 ¼ B 0 bmb ðm  bÞ1 x1 : As follows from Eqs. (2.224), we must put m ¼ 0 to exclude the degeneration of the arc length along the x2-axis on the emitter into zero or infinity. Since X should tend to zero at x1 ! 0, we have b þ n > 0. Let us substitute the expansions (2.223) into the first of Eqs. (2.222). The balance condition between the terms in the left-hand and right-hand sides of this equation appears as t ¼ n þ 5b  3, B0R0 ¼ J0A0. Consider the Poisson equation from Eqs. (2.222). In the vicinity of the emitter, the terms in its left-hand and right-hand sides have the order xak, a1 ¼  3(b þ n) þ 1, a2 ¼  (b þ n) þ 1, a3 ¼ 2(b þ n)  2, with a2  a1 ¼ 2(b þ n) > 0. The first item in the LHS, characterizing the second potential derivative along  the emitter normal, counterbalances the RHS 2 B0 J01 . The terms in the LHS of the second of term: b þ n ¼ 35, A50 ¼ 25 Eqs. (2.222) are proportional to x (b þ n) þ e and xb þ n. Let us calculate the emitter curvature by applying Eq. (1.57):   1 nþe 1 ðbþnÞþe x : k1 ¼ h1 1 ð lnh 2 Þ;1 ¼ eB 1 A 0 b

(2.225)

114

Exact Solutions to the Beam Equations

To guarantee that the curvature is finite and does not vanish, the equality  n  b þ e ¼ 0 must necessarily hold true. Thus, the first term in the second of Eqs. (2.222) counterbalances the constant in its RHS: 3 e¼bþn¼ ; 5

eB 0 B 1 =A 0 ¼ c;

c 6¼ 0:

(2.226)

At c ¼ 0, the balance of the terms gives 6 e ¼ 2 ð b þ nÞ ¼ ; 5

eB 0 B 1 =A 0 þ nA 0 =B 0 ¼ 0;

As a result, for h 1, h 2, r we have   h 1 ¼ A 0 x1 2=5 bn ð1 þ A 1 xe þ . . .Þ;   r ¼ R 0 x1 2=5 b4n ð1 þ R 1 xe þ . . .Þ;

c ¼ 0:

(2.227)

h 2 ¼ B 0 bb1 b0 ð1 þ B 1 xe þ . . .Þ; 3 n ¼  b: 5 (2.228)

Now, we need to use Eqs. (1.142) to describe the transformation to Cartesian coordinates (x01 ¼ 0):   1 1 2=5 3=5 1 0 b ; h1 h 1 2 h 1;2 ¼ nA 0 B0 x 1 h 2;1 ¼ cb b ;   5 3=5 y ¼ nA 0 B1 þ c lnb; eiy ;1 ¼ ieiy y;1 ¼ ieiy h1 0 x 2 h 1;2 ; 3   1 b iy e ;1 ; h 1eiy ¼ iB 0n b  h 2 eiy x1 ¼0 ¼ hB 0 bb1þic b0 ¼ B 0 ðb þ icÞ1iðbbþic Þ0 ;   x þ iy ¼ iB 0 bb n1 eiy  bic  ðb  icÞ1 bic ; b  ic 6¼ 0;     b  ic ¼ 0: (2.229) x þ iy ¼ iB 0 n1 eiy  1 þ lnb ; The expression for x þ iy at x1 ! 0 in (2.229) represents the parametric equations of emitting surface in Cartesian coordinates. At b  ic 6¼ 0, we have   C0 ¼ B 0 b2 þ c2 1=2 ; tand ¼ b=c; x þ iy ¼ Reic ¼ C0 bb eiðc ln bþdÞ ;    ¼ c  d: c ¼ c lnb þ d; b ¼ ec=c ; R eðb=cÞc ; c R ¼ C0 bb ; (2.230) As can be seen from (2.230), the emitter represents a logarithmic spiral  with the emission current density J ¼ ru ¼ r=h1 b5n eð5n=cÞc . The spiral coordinate along the emitter is a liner combination of ln R and c; therefore, at x1 ¼ 0 we have c ¼ gq, g ¼ const. To compare this solution with solution 4 (see Table 2), it is necessary to switch in both solutions to the local Cartesian coordinate X, because the curvilinear coordinates x1 and p may have different dimensions. Equations (2.224) and a similar formula

Exact Solutions to the Beam Equations

115

for the spiral coordinates p, q imply that x1 b 5n/3X5/3, p e b2qX. Using these relations, let us transform the asymptotic expressions (2.228) and the corresponding expansions of the solution in spiral coordinates. Instead of (2.228), we derive 5n=3 2=3 X ; u ¼ h1 1 b

r b10n=3 X2=3 :

(2.231)

u eab 2 q p2=3 eða2=3Þb2 q X2=3 ; r e2ða1Þb2 q p2=3 e2ða2=3Þb2 q X2=3 :

(2.232)

For solution 4 (see Table 2), we have

The comparison of expressions (2.231) and (2.232) shows that the flow parameters along the emitter’s normal vary in accordance with the r-mode, while the dependences on the tangential coordinates differ only in the notations of the constants. Thus, solution (2.221) at c 6¼ 0 describes the same flow as solution 4. The temperature-limited emission is determined by the asymptotics j X, r X 1/2, which results in the following values of the parameters in Eqs. (2.221)–(2.223): b ¼ 0, n ¼ e ¼ 2/3. It follows from Eqs. (2.230) at b ¼ 0, c 6¼ 0 that solution (2.221) corresponds to the flow from the cylinder R ¼ const with the exponential current density dependent on the angle, which means that this solution is similar to solution 3 (see Table 2.1) at a ¼ 0. At c ¼ 0, b 6¼ 0, Equations (2.229) imply that the emitter is a plane, with the flow parameters being a power-like function of the tangent coordinate (Syrovoy, 1979). This is solution 2 (see Table 2). Finally, at c ¼ b ¼ 0, we find from (2.229) that b exp(y/B0), which leads to the flow described by the expressions in solution 1 (see Table 2). By considering different combinations of the parameters b, c, we have shown that solution (2.221) contains all possible H-solutions for planar flows. A similar approach can be used to study the solutions of a relevant structure (Syrovoy, 1978) in the vicinity of the trajectory x2 ¼ const.

2.10. THE EXACT SOLUTIONS, WHOSE RELATION TO THE GROUP PROPERTIES IS YET UNKNOWN 2.10.1. Terminology The solution of this type can be divided into two groups. The first group refers to the planar (@/@ z  0) potential solenoidal flows, which can be constructed using the complex formalism (Kirstein, 1958a; Ogorodnikov, 1969, 1972, 1973a; Syrovoy, 1984). The second group is associated with the concepts of degenerated flows (Danilov, 1968a) and generalized Brillouin flows (Danilov, 1963a,b, 1966a).

116

Exact Solutions to the Beam Equations

A flow is called solenoidal if rv ¼ 0. On the commonly accepted condition rA ¼ 0, this relation implies the solenoidity of the generalized momentum rP ¼ 0. Generally speaking, the solenoidity requirement makes the system of beam equations (1.70), (1.71), and (1.80) overdetermined. The potential solenoidal flow is described by the harmonic action function DW ¼ 0. The peculiarities of the degenerated flows are as follows: First, the existence of a cyclic coordinate on which the solution does not depend (z for the planar case, c for the axisymmetric case), and second, vanishing of one of the velocity components along the noncyclic direction. The generalized Brillouin flows represent a subclass of the degenerated flows for which the generalized momentum Pi is preserved along the trajectories. In particular, this condition is satisfied by the momentum Pi ¼ ui þ Ai  0, with the spatial components ui of the 4-velocity vector expressed through the vector potential  1=2 ua ¼ v a 1  V 2 ¼ Aa ;

a ¼ 1; 2; 3:

(2.233)

2.10.2. Planar Solenoidal Flows A complex representation of the equations for a planar (x, y) stationary solenoidal flow in the homogeneous magnetic field Hz is proposed in Kirstein (1958a) and generalized to the nonhomogeneous magnetic field Hx, Hy in Ogorodnikov (1969). Let us switch from the variables x, y to the independent complex variables t ¼ x þ iy, t* ¼ x  iy. Using the relationship between the derivatives with respect to x, y, and t, t*, we have W;x ¼ W;t þ W;t ;

  W;y ¼ i W;t  W;t ;

1 W ¼ ðf þ f Þ; 2

(2.234)

rW ¼ W;x þ iW;y ¼ f;t  Qðt Þ: The magnetic field Hz can be expressed in terms of the vector potential components Ax, Ay: 1 1 Ax þ iAy ¼ Hz ðy þ ixÞ ¼ iHz t: 2 2

(2.235)

The complex velocity can be found from the expression for the generalized momentum components Px þ iPy ¼ rW;

u þ iv ¼ Q þ iot;

Hz ¼ 2o:

(2.236)

Exact Solutions to the Beam Equations

117

As soon as P3 ¼ w þ Az ¼ W, z ¼ 0, and Az is a harmonic function, we have w ¼ Az ¼ w þ w :

(2.237)

Equations (2.236) and (2.237) give u2 þ v2 ¼ ðu þ ivÞðu  ivÞ ¼ QQ  io ðt Q  tQÞ þ o2 tt ; 2j ¼ QQ  io ðt Q  tQÞ þ o2 tt þ ðw þ w Þ2 :

(2.238)

The Poisson equation gives the following expression for the spacecharge density:   0 0 r ¼ Dj ¼ 4j;tt ¼ 2 Q0 Q þ 2w0 w þ o2 : (2.239) The current conservation equation with regard to the condition rv ¼ 0, h  i 1 ur;x þ vr;y ¼ Re ðu þ ivÞ r;x  ir;y ¼ ½ðu þ ivÞðrrÞ þ ðu  ivÞrr ¼ 0 2 (2.240) represents a basic equation in the solenoid flow theory (Ogorodnikov, 1969):  0 00 0 00 00 00  Q Q Q þ QQ0 Q þ io tQ Q  t Q0 Q þ  0 00   (2.241) 00 0 00 00  þ2 Q w w þ Qw0 w þ 2io t w w  t w0 w ¼ 0: Equation (2.241) allows as many as six solutions (Ogorodnikov, 1969, 1972, 1973a; Syrovoy, 1984b) describing the potential solenoidal flows (a, b, c, a, b ¼ const):     u ¼  o  aR2 y; v ¼ o  aR2 x; ð1Þ Q ¼ ia=t; w ¼ tbþ1 ; b 6¼ 1;  w ¼ 2 lnR;  b ¼ 1; w ¼ 2R bþ1 cosðb þ 1Þc; o ¼ 0; ð2Þ Q ¼ ia½t þ ðb þ icÞ=ð2aÞ; w0 ¼ exp ðb þ icÞt þ at2 ; u ¼ a½y þ c=ð2aÞ; v ¼ a½x þ b=ð2aÞ; w ¼ 2Rew; ð3Þ Q ¼ i tant; o ¼ 0; w ¼ 2Rew; w0 ¼ cosb t; u ¼ sh2y=ð cos2x þ ch2yÞ; v ¼ sin2x=ð cos2x þ ch2yÞ; ð4Þ Q ¼ iat þ b þ ic; w ¼ t; u ¼ ða þ oÞy þ b; v ¼ ða þ oÞx  c; w ¼ 2x; ð5Þ Q ¼ iot  ia; w ¼ ebt ; u ¼ 0; v ¼ 2ox þ a; w ¼ 2ebx cosby; ð6Þ Q ¼ cnðt; kÞ; w0 ¼ ia snðt; kÞdnðt; kÞ þ dn2 ðt; kÞ o ¼ 0; u þ iv ¼ Qðt Þ; w ¼ 2Rew: k2 sn2 ðt; kÞ; (2.242)

118

Exact Solutions to the Beam Equations

The most interesting of the solutions (2.242) is the last one; this solution can be expressed in terms of the elliptical Jacobi functions. The potential can be easily found from the energy integral (2.238), the spacecharge density from Eq. (2.239), and the magnetic field components in the x, y plane by differentiating Az from Eq. (2.237). Formulas (3) and (4) from Eqs. (2.242) at w ¼ 0 give the solutions found in Kirstein (1958a), the first of which [see Eq. (1.31)] was already considered in Section 1. Below we analyze those solutions proposed by Danilov (1966a, 1968a) that do not require use of the double-stream concept and discuss some examples of reducing the original problem to a linear partial differential equation, as given in these works. Strictly speaking, the last case cannot be referred to exact solutions.

2.10.3. Degenerate Flows Let us write the beam equations (1.77), (1.80), and (1.81) at @/@ t ¼ 0, assuming the existence of the cyclic coordinate (@/@ x3 ¼ 0), and the coordinate system orthogonality: ℋ;1 ¼ v2 R 3  v3 R 2 ;

v 1 R 2 ¼ v2 R 1 ; 1 R1 ¼ P 3;2 ; R2 ¼ P 3;1 ; R3 ¼ P 2;1  P 1;2 ; ℋ ¼ gik vi vk  j; 2  1  2 pffiffiffi 11  pffiffiffi 22  pffiffiffi sv ;1 þ sv ;2 ¼ 0; gg j;1 ;1 þ gg j;2 ;2 ¼ s; s ¼ g r; pffiffiffi1 pffiffiffi1 1 2 H3 ¼ const; H h ¼ g A 3;2 i ; hH ¼  g iA 3;1 ; pffiffiffi 1 pffiffiffi 1 g g22 A 3;1 þ g g11 A 3;2 ¼ 0: ;1

ℋ;2 ¼ v3 R 1  v1 R 3 ;

;2

(2.243) We are seeking the solution to Eqs. (2.243) in the form     v1 ¼ U x1 ; v3 ¼ V x2 ; A2 ¼ 0; v2 ¼ 0 ; A 1 ¼ 0;

  A 3 ¼ A x2 :

(2.244) From Eqs. (2.244) we obtain for Pi, Ri, Hi       P 1 ¼ U x1 ; P 3 ¼ P x2 ; R1 ¼ P0 x2 ; P 2 ¼ R2 ¼ R3 ¼ H2 ¼ H3 ¼ 0:

H1 ¼

pffiffiffi1 0  2  g A x ; (2.245)

For the coordinate systems (1.17) with the conformal metrics h1 ¼ h2 ¼ h [which include, in particular, the cylindrical (x1 ¼ z, x2 ¼ ln R, x3 ¼ c; h2 ¼ h3 ¼ R) and spiral cylindrical coordinates), Eqs. (2.243) take the form

Exact Solutions to the Beam Equations

ℋ;1 ¼ 0; ℋ;2 ¼ V ðV 0 þ A0 Þh 2 ðh 3 rUÞ;1 ¼ 0; 3 ;  1  2 2 2 ℋ ¼ h U þ h 2  j; 3 V 2  1 0      h 3 A ;2 ¼ 0; h 3 j;1 ;1 þ h 3 j;2 ;2 ¼ h2 h 3 r:

119

(2.246)

on the condition that (2.244) and (2.245) are satisfied. In cylindrical and spiral coordinates, the solution of the system (2.246) is given in Danilov (1968a) (a, b, c, J0 ¼ const). For the axisymmetric case we have vz ¼ UðzÞ;

vR ¼ 0;

1 R2 V 2 þ H0 V ¼ aR2 þ b; 2 1 P 3 ðRÞ ¼ V ðRÞ þ H0 R2 ; 2

vc ¼ R1 V ðRÞ;

U0 2 ¼ a þ 2J0 U1 þ cU2 ; 1 1 ℋðRÞ ¼ R2 V 2 þ aR2 þ b lnR; 2 4 1 1 j ¼  aR2  b lnR þ U2 ðzÞ; 4 2

(2.247)

00

1 rðzÞ ¼ ðU2 Þ  a; 2 H z ¼ H0 ;

rU ¼ J0 ;

HR ¼ Hc ¼ 0:

Accordingly, in the spiral coordinates the solution reads vp ¼ e1 e2 UðpÞ; vq ¼ 0; vz ¼ V ðqÞ; j ¼ e22 FðpÞ  Q 0 q; rU ¼ J0 e42 ; 1 2 2 1 2 1 00 1 1 2 F ¼ e1 U  b 2 r0 ; F þ 4b 22 F ¼ J0 e2 V ¼  r0 b 1 1 U ; 2 H 0 e2 þ Q 0 ; 2 4 2 1 1 ℋðqÞ ¼ V 2 þ b 2 r e2 þ Q0 q; H p ¼ H 0 e1 e2 ; P 3 ðqÞ ¼ V þ H0 q; 2 4 2 0 2 Hq ¼ Hz ¼ 0;

e1 ¼ eb2 p ;

e2 ¼ eb2 q ;

r0 ; J0 ; H0 ; Q0 ; b 1 ; b 2 ¼ const:

(2.248) The momentum P1(x1) in solutions (2.247) and (2.248) varies in the course of motion. In contrast to the invariant solution 4 at a ¼  1 (see Table 2), Eqs. (2.248) induce a different structure of the z-component of velocity and provide an item in the potential distribution, being linear in q.

2.10.4. Generalized Brillouin Flows Consider the relativistic flows (Danilov, 1966a) with the generalized momentum identically equal to zero and with the cyclic coordinate x3. The requirement (2.233) implies the flow potentiality. It can be easily seen that   ð2Þ ð2Þ ua ¼ gbg ub ug ¼ gbg A b A g ¼ V 2 1  V 2 1 ; uð2Þ ¼ ua þ 1     (2.249) u ¼ 1  V 2 1=2 ¼ 1 þ j: ¼ gbg A b A g þ 1 ¼ 1  V 2 1 ;

120

Exact Solutions to the Beam Equations

On introducing the vector potential with regard to the relation between u and j, Eqs. (1.90) can be reduced to three equations for curl H, the Poisson equation, and the expression for u(2) in terms of Aa. In the orthogonal coordinate system, those equations appear as follows:  1    1   L 3 A 2;1  A 1;2 ;2 ¼ L 1 sA 1 ; L 3 A 2;1  A 1;2 ;1 ¼ L 2 sA 2 ;  1        pffiffiffi L 2 A 3;1 ;1 þ L1 L 1 u;1 ;1 þ L 2 u;2 ;2 ¼ gsu; 1 A 3;2 ;2 ¼ L 3 sA 3 ;  2  1 2  1 2 þ h 2 A 2 þ h 3 A 3 þ 1; s ¼ ru1 ; uð2Þ ¼ h1 1 A1 L i ¼ hk h l =h i ; i 6¼ k 6¼ l: (2.250) Let us assume that in the coordinate systems x1 ¼ x, x2 ¼ y, x ¼ z (h1 ¼ h2 ¼ h3 ¼ 1) and x1 ¼ R, x2 ¼ c, x3 ¼ z (h1 ¼ h3 ¼ 1, h2 ¼ R) the following relations take place:   v1 ¼ 0; A 1 ¼ 0; A 2 ¼ A 2 x1 ; (2.251)      A 3 ¼ a x1 sh kx2 þ U x1 ; k ¼ const: 3

Let us consider the first case in detail, putting A2 ¼ A(x), a2 ¼ 1 þ A2. The first of Eqs. (2.250) is satisfied identically; the second equation and the expression for u give 00

s ¼ sðxÞ ¼ A =A;

u ¼ aðxÞch½ky þ UðxÞ:

(2.252)

Similar relations follow from the third and fourth of Eqs. (2.250) after putting zero coefficients at the hyperbolic sinuses and cosines:   00 00 2 a þ k2 þ U 0  s a ¼ 0 ; aU þ 2a0 U0 ¼ 0: (2.253) By integrating the second of these equations, we obtain   1  1  0 2   00 U0 ¼ U0 A2 þ 1 ; A1 A ¼ A2 þ 1 A þ U02 þ k2 A2 þ 1 ; U0 ¼ const: (2.254) Thus, we arrive at the exact solution A 2 ¼ AðxÞ; A 1 ¼ 0; u ¼ aðxÞch½ky þ UðxÞ;

A 3 ¼ aðxÞsh½ky þ UðxÞ; 00 s ¼ sðxÞ ¼ A =A:

a2 ¼ A2 þ 1; (2.255)

In the cylindrical coordinates, putting A2 ¼ RA(R) and introducing the differentiation with respect to Z ¼ ln R, we arrive at Eqs. (2.254) with k2 replaced by k2 þ 1. An additional difference is the new expression for

Exact Solutions to the Beam Equations

121

¨  1), A_ ¼ dA=d. It is interesting to note that the ultras ¼ R 2(A 1A relativistic solutions (2.251) do not have a nontrivial non-relativistic limit because their structure is adjusted to the relativistic energy integral. At low velocities, s does not depend on x1, while the current conservation equation, being automatically satisfied in relativistic case, induces some additional relations which, in particular, give a ¼ eikx for Eqs. (2.255).

2.10.5. Reducing the Problem to a Linear Partial Differential Equation Let us consider the generalized Brillouin flows with the single nonzero spatial component u3 of the 4-velocity vector. The self-magnetic field can be expressed through A3:       u3 ¼ U x1 ; x2 ; A 3 ¼ A x1 ; x2 ; P 3 ¼ u3 þ A 3 ¼ P x1 ; x2 ; ua ¼ A a ¼ Pa ¼ 0; a ¼ 1; 2: (2.256) These equations describe a vortex flow that is assumed to be nonmonoenergetic. Equations (1.90) for (2.256) take the form  2 3 3 ð2Þ ¼ h1 j; s ¼ ru1 ; ;1 ¼ v P;1 ;  ℋ;2 ¼v P;2 ; u 3 U  þ 1; ℋ ¼ ℋ1  u p ffiffiffi 1 L 2 A;1 ;1 þ L 1 A;2 ;2 ¼ L 3 sU; L 1 j;1 ;1 þ L 2 j;2 ;2 ¼ gsu: (2.257) In the non-relativistic limit (Danilov, 1963b, 1966a) at P3 ¼ v3 þ A3 , the RHS of the equation for A is zero, while the current conservation equation for the flows under considerations is obeyed identically. If a solution to the equation  1    (2.258) R A;R ;R þ R1 A;z ;z ¼ 0 is known in the cylindrical coordinates x1 ¼ z, x2 ¼ R, x3 ¼ c, all parameters of the flow, the particles of which are traveling along the circles around the z-axis, can be found by means of algebraic operations and differentiation as follows: h  2    i vc ¼ R1 A; r ¼ R1 Rj;R ;R þ Rj;z ;z (2.259) 2j ¼ R1 A ; Thus, the problem is reduced to the linear partial differential equation (2.258), which is analyzed in Chapter 4. Now let us consider the system (2.257) in Cartesian coordinates x1 ¼ x, 2 x ¼ y, x3 ¼ z and construct a solution in the form (Danilov, 1968a)

122

Exact Solutions to the Beam Equations

U ¼ shC; u ¼ chC; j ¼ u  ℋ; P ¼ PðCÞ; A ¼ AðCÞ;

ℋ ¼ ℋðCÞ;

(2.260)

where C ¼ C(f), with f being an arbitrary harmonic function f, xx þ f, yy ¼ 0. By substituting Eqs. (2.260) into Eqs. (2.257), we obtain ð ð ð ℋ ¼ chC þ a shC df ; P ¼ shC þ a chC df ; A ¼ a chC df ; ð j ¼ a shC df ; s ¼ aC0 ðrf Þ2 ; a ¼ const: (2.261) Equations (2.261) at C ¼ f appear as uz ¼ sh f ; A ¼ ash f ;

u ¼ ch f ; ℋ ¼ ð1 þ aÞch f ; j ¼ ach f ; s ¼ aðrf Þ2 :

P ¼ ð1 þ aÞsh f ;

(2.262)

From Eqs. (2.262) at a ¼  1 we obtain the expressions describing the non-vortex monoenergetic flow (Lomax, 1958): uz ¼ sh f ; j ¼ ch f ;

u ¼ ch f ; ℋ ¼ P ¼ 0; s ¼ ðrf Þ2 :

A ¼ shf ; (2.263)

The solutions (2.261)–(2.263) describe the relativistic flows along the straight-line trajectories calculated with regard to the self-magnetic field. The problem is thus reduced to a 2D Laplace equation.