- Email: [email protected]
Characteristic solutions of scalar newton equations in one space dimension
Int. J. Non-Linear Mechanics, Vol. 33, No. 6, pp. 1013—1026, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0020—7462/98 $19.00#0.00
PII: S0020–7462(97)00064-4
CHARACTERISTIC SOLUTIONS OF SCALAR NEWTON EQUATIONS IN ONE SPACE DIMENSION P. W. Doyle Department of Mathematics, University of Hawaii, Honolulu, HI 96822, U.S.A. (Received 1 July 1997) Abstract—Newton equations are dynamical systems on the space of fields. The solutions of a given equation which are curves of characteristic fields for its force are planar and have constant angular momentum. Separable solutions are characteristic with angular momentum equal to zero. A Newton equation is separable if and only if its characteristic equation is homogeneous. Separable equations correspond to invariants of homogeneous ordinary differential equations, and those associated with a given homogenous equation correspond to its generalized dilation symmetries. A Newton equation is compatible with the characteristic condition if and only if its characteristic equation is linear. Such equations correspond to invariants of linear ordinary differential equations. Those associated with a given linear equation correspond to the central force problems on its solution space. Regardless of compatibility, any Newton equation with a plane of characteristic fields has non-separable characteristic solutions. ( 1998 Elsevier Science Ltd. All rights reserved Keywords: separation of variables, characteristic equation, dilation symmetry, central force
1. INTRODUCTION
An autonomous Newton equation in one space dimension is a partial differential equation u "F(x, u, u , . . . , u ) (1) tt 1 k for a scalar function u of two variables x and t, where u denotes the second derivative with tt respect to t and u denotes the jth derivative with respect to x. We assume that F is smooth j (C=). A solution is a smooth function u(x, t) defined on a product of intervals in x and t with L2u(x, t)"F(x, u(x, t), L u(x, t), . . . , Lk u(x, t)), x t x at each point of its domain. In physical terms, the acceleration of the field x>u(x, t) at time t is due to a force depending on the spatial variable x and the spatial derivatives of the field. Thus, we interpret the differential operator F[u]"F(x, u, u , . . . , u ), 1 k as a field of force vectors on the space of fields u(x) and equation (1) as a dynamical system on this infinite-dimensional space. The force field is independent of time, so we consider solutions defined for values of t in some interval containing t"0. The relevant initial value problem requires solution of equation (1) subject to specified position and velocity u(x, 0)"v (x), u (x, 0)"v (x). 1 t 2 In general, we cannot claim existence or uniqueness of such a solution. A nonzero solution of equation (1) has separable form u(x, t)"v(x)w(t),
(2)
if and only if it satisfies the separation condition (u /u) "0. t x Non-zero characteristic solutions satisfy the characteristic condition (u /u) "0. tt x Contributed by W. F. Ames. 1013
(3)
(4)
1014
P. W. Doyle
Note that equation (3) implies equation (4), so separable solutions are characteristic. We find that characteristic solutions are those with planar form u(x, t)"v (x)w (t)#v (x)w (t) (5) 1 1 2 2 and constant angular momentum. We show that the Newton equation (1) is compatible with condition (3) if and only if its characteristic equation (F[u]/u) "0 (6) x is homogeneous. This geometric characterization of the separability criterion of Kalnins and Miller [1] is analogous to Doyle’s description of separable evolution equations [2]. Separable Newton equations correspond to invariants of homogeneous ordinary differential equations. The separable equations associated with a given homogeneous equation correspond to its generalized dilation symmetries, and their separable solutions are radial curves in its solution space. Equation (1) is compatible with condition (4) if and only if its characteristic equation is linear. In this case F stabilizes the solution space of equation (6), and the partial differential equation (1) restricts to a finite-dimensional dynamical system. This reduction is an example of the generalized method of separation of variables developed by Galaktionov and Posashkov [3, 4]. See refs [5, 6] for a clear geometrical description and for additional references. Equations compatible with the characteristic condition correspond to invariants of linear ordinary differential equations. The Newton equations associated with a given linear equation correspond to autonomous central force problems on its solution space, and their characteristic solutions are the central force trajectories. Regardless of compatibility, the restriction of a Newton equation to any linear subspace of characteristic fields is a central force problem, and its trajectories are characteristic solutions. We describe a general class of equations which are incompatible with the characteristic condition yet have non-separable characteristic solutions. The form of equation (1) is invariant under arbitrary smooth transformation of x, affine transformation of t, and dilation of u. Conditions (3) and (4), equation (6), and the functional forms (2) and (5) are also invariant under this transformation group. All functions in this study are assumed to be smooth.
2. EX A MP LES
The equation u "(u2#u2)~3@2u , tt x xx describes the motion of fields subject to force F[u]"(u2#u2)~3@2u . x xx
(7)
(8)
The function u(x, t)"w (t) cos x#w (t) sin x (9) 1 2 is a solution if and only if w and w satisfy the Kepler system 1 2 w A w 1 "!(w2#w2)~3@2 1 . (10) 1 2 w w 2 2 Given constants a, b, c, d with a2#b2O0, there is a unique function (9) which satisfies (7) with u(x, 0)"a cos x#b sin x, u (x, 0)"c cos x#d sin x. t Thus, there is a four-parameter family of solutions (9). Solutions corresponding to elliptical orbits of equation (10) are periodic in t. For example, the travelling waves
A B
A B
u(x, t)"c~2@3 cos(x!x !ct), 0
Characteristic solutions of scalar Newton equations
1015
correspond to circular orbits
A B
A B
w cos 1 (t)"c~2@3 (ct!t ). 0 w sin 2
Separable solutions u(x, t)"w(t) cos(x!x ), (11) 0 correspond to radial orbits. The function (11) satisfies (7) if and only if wA"!w~2, so there is a three-parameter family of such solutions. For any function (9), we have F[u(x, t)]"j(t)u(x, t) with j(t)"!(w (t)2#w (t)2)~3@2. 1 2 If u satisfies equation (7) then u (x, t)"j(t)u(x, t), tt so w and w satisfy the linear equation 1 2 wA"j(t) w.
(12)
(13)
Hence, (14) (w w@ !w w@ )@"0 2 1 1 2 which verifies the fact that each trajectory of equation (10) has constant angular momentum. As a second example, consider the operator F[u]"2u2u !uu2 . We have F[u(x)]" x xx ju(x) for some constant j if and only if u(x)"f #f x#f x2, in which case j has value 0 1 2 (15) *[u]"4f f !f2. 1 0 2 Fix independent quadratic polynomials v and v . Suppose that the x-quadratic function 1 2 u(x, t)"f (t)#f (t) x#f (t) x2, (16) 0 1 2 satisfies u "2u2u !uu2 x tt xx
(17)
with u(x, 0)"v (x), u (x, 0)"v (x). (18) 1 t 2 Then u satisfies the family of ordinary differential equations (12) with j(t)"4f (t)f (t)! 0 2 f (t)2. From equation (18) we have 1 u(x, t)"v (x)w (t)#v (x)w (t), (19) 1 1 2 2 where w and w are the solutions of equation (13) with 1 2 (20) w (0)"1, w@ (0)"0, w (0)"0, w@ (0)"1. 2 1 2 1 The function (19) satisfies equations (17) and (18) if and only if w and w have values (20) 1 2 and w A w 1 "(*[v ]w2#*[v , v ]w w #*[v ]w2) 1 , 2 1 1 2 1 2 2 1 w w 2 2 where
A B
A B
*[v , v ]"*[v #v ]!*[v ]!*[v ]. 1 2 1 2 1 2 Thus, the unique x-quadratic solution of the initial value problem (17, 18) is a curve in the plane of fields spanned by v and v . It has constant angular momentum because equation 1 2 (13) implies equation (14). Equation (17) also has a unique x-quadratic solution with given dependent initial values. The solution with u(x, 0)"f #f x#f x2, 0 1 2 is the separable function
u (x, 0)"c(f #f x#f x2) t 0 1 2
u(x, t)"(f #f x#f x2)w(t), 0 1 2
(21)
1016
P. W. Doyle
where wA"(4f f !f2)w3 1 0 2
(22)
and w(0)"1,
w@(0)"c.
The solution with u(x, 0)"0,
u (x, 0)"f #f x#f x2 t 0 1 2 is the function (21), where w satisfies equation (22) and w(0)"0,
w@(0)"1.
We will find that the four-parameter family thus obtained accounts for all separable solutions of equation (17). Equation (17) has a six-parameter family of solutions (16).
3. CHA RAC TER I ST IC SOL U TI ONS
A characteristic field of an operator F is a function u(x), not identically equal to zero, with F[u(x)]"ju(x) for some constant j, its characteristic value. A solution u(x, t) of equation (1) is characteristic if x>u(x, t) is a characteristic field of F for each t. A solution not identically equal to zero for any t is characteristic if and only if F[u(x, t)]"j(t)u(x, t), i.e. u (x, t)"j(t)u(x, t), (23) tt for some function j. A non-zero function defined on a product of intervals satisfies equation (23) for some j if and only if it satisfies equation (4). The solutions constructed in the previous section are characteristic. Solutions of equation (23) are functions (5), where w and w are independent solutions of 1 2 wA"j(t) w (24) with v and v arbitrary. Hence a solution of equation (1) is characteristic if and only if it has 1 2 form (5) (but is not identically equal to zero for any t), where w and w are independent 1 2 solutions of some equation (24), in which case j(t) is its characteristic value. Note that separable solutions are characteristic. ¹heorem 1 A solution of a Newton equation is characteristic if and only if it has form (5) with w w@ !w w@ constant and non-zero, in which case its characteristic value is 2 1 1 2 wA w@ !wA w@ 2 1. j" 1 2 (25) w w@ !w w@ 2 1 1 2 Proof If w and w are independent solutions of equation (24) then w w@ !w w@ is constant 1 2 1 2 2 1 and non-zero. Conversely, if w w@ !w w@ is constant and non-zero then w and w are 1 2 2 1 1 2 independent solutions of (24) with j given by equation (25). h If v and v are multiples of some field v then the solution (5) is separable. Otherwise it is 1 2 a curve in the plane spanned by v and v , with constant nonzero angular momentum 1 2 w w@ !w w@ . This property is independent of basis v , v . The momentum value is not 1 2 2 1 1 2 defined. Separable solutions are radial curves, so they are characteristic with angular momentum equal to zero.
Characteristic solutions of scalar Newton equations
1017
We assume L FO0, k*1. Equation (6) is a (k#1)th-order ordinary differential equauk tion u #(u/L F[u])(L #u L #) ) )#u L )(F[u]/u)"0 k`1 uk x 1 u k uk~1 and its solutions are the non-zero characteristic fields of F. A non-zero solution of equation (1) is characteristic if and only if it is a curve in the solution space of equation (6). For example, the characteristic equation for F[u]"!(u2#u2)~3@2 u x
(26)
is u #u"0. (27) xx Curves in its solution space are functions (9). Radial curves are functions (11). The forces (8) and (26) are identical on solutions of equation (27), so the solutions (9, 10) of equation (7) also satisfy (28) u #(u2#u2)~3@2 u"0. x tt This accounts for all characteristic solutions of equation (28). Of course, solutions of equation (27) satisfy the characteristic equation u "(1#3u(u#u )/(u2#u2))u u /u x x xx xxx xx of equation (7). The characteristic equation of equation (17) is
(29)
u "0, (30) xxx curves in its solution space are functions (16), and radial curves are functions (21). Function (2) represents dilation of a field v by a variable factor w, suggesting that the geometric nature of separability is projective. Hence, we emphasize non-zero separable solutions. In Doyle’s study of evolution equations [2], for which characteristic and separable solutions are identical, characteristic fields are by definition nowhere equal to zero. This restriction is artificial in the context of Newton equations. Note that zeros of a non-separable function (5) may evolve as t varies.
4 . S EPA R AB I LI TY
Equation (1) is (regularly) separable [1] if it has a solution (2) with u(x , 0)"c , . . . , u (x , 0)"c , u (x , 0)"d, 0 0 k 0 k t 0 for each relevant set of values x and c O0, c , . . . , c , d. This requirement represents 0 0 1 k compatibility of equations (1) and (3). Kalnins and Miller [1] derive additive separability conditions for partial differential equations of general form, i.e. necessary and sufficient conditions for the existence of a complete family of additively separable solutions in a given coordinate system. See ref. [7] for application to non-linear Laplace equations. Doyle [2] describes compatibility of the separation condition (3) and evolution equation u "F[u], (31) t as integrability of the contact structure on their intersection, and finds that equation (31) is separable if and only if equation (6) has local homogeneous form u "u(x, u /u, . . . , u /u) u. (32) k`1 1 k See also ref. [8]. We now prove the analogous result for Newton equations. See ref. [9] or ref. [10] for discussion of symmetry and differential equations. ¹heorem 2 A Newton equation is separable if and only if its characteristic equation is locally homogeneous.
1018
P. W. Doyle
Proof For u'0 or u(0, condition (3) is given in the variable uN "lnDuD by uN "0, xt
(33)
equation (1) is given by uN #uN 2"FM [uN ] t tt with FM [uN ]"F[u]/u, and equation (6) is given by
(34)
FM [uN ] "0. x According to ref. [1], equations (33) and (34) are compatible if and only if
(35)
L 6 ((L #uN L 6 #) ) )#uN L 6 )FM [uN ]/L 6 FM [uN ])"0. u x 1 u k uk~1 uk This condition holds if and only if L 6 is a symmetry of equation (35), i.e. if and only if uL is u u a symmetry of equation (6). h As in Theorem 8, it can be shown that dilation invariance of the characteristic equation is equivalent to integrability of the contact structure on the intersection of equations (1) and (3) and their differential consequences. Equation (7) is separable because its characteristic equation (29) is homogeneous. Equations (28) and (17) are separable because their characteristic equations (27) and (30) are linear (hence homogeneous). A separable kth-order Newton equation has a (k#2)-parameter family of separable solutions. Above we describe the three-parameter family for equation (28) and the fourparameter family for equation (17). Note that a first-order Newton equation (k"1) is actually a second-order differential equation. The following description of separable solutions explains the role of dilation invariance of the characteristic equation. ¹heorem 3 Fix a non-zero characteristic field v of a separable equation u "F[u]. (36) tt For any constant c and any point x in the domain of v there is a separable solution u on 0 a neighborhood of x with 0 u(x, 0)"v(x), u (x, 0)"cv(x). (37) t If u is a separable solution on a neighborhood of x with values (37) then there are positive 0 constants e and e such that u(x, t)"v(x)w(t) for Dx!x D(e and DtD(e , where 1 2 0 1 2 wA"¼(w) and w(0)"1,
w@(0)"c,
(38)
where ¼(w)"F[v(x)w]/v(x). This accounts for all non-zero separable solutions of equation (36). Proof Locally, v(x)w is characteristic for all values of w sufficiently close to w"1 because the characteristic equation (36) is homogeneous. Hence F[v(x)w]/v(x) is locally independent of x near w"1. The function u(x, t)"v(x)w(t) satisfies equations (36) and (37) if and only if w satisfies equation (38) and wA"F[v(x)w]/v(x). This implies local existence and uniqueness as claimed. A non-zero separable solution u is a curve of characteristic fields. In particular, its initial value u(x, 0)"v(x) is characteristic. We have u(x, t)"v(x)w(t) for some function w. Hence u (x, 0)"w@(0)v(x). We have already t encountered this solution. h
Characteristic solutions of scalar Newton equations
1019
We now formulate Theorem 2 in terms of invariants. A first integral or invariant of an nth-order ordinary differential equation has generic order n!1. ¹heorem 4 The Newton equation with force F is separable if and only if F/u is an invariant of a locally homogeneous ordinary differential equation. Proof Equation (6) is the unique equation for which F/u is invariant. h Thus, separable kth-order Newton equations correspond (locally) to invariants of homogeneous (k#1)th-order ordinary differential equations. The Newton equation with force F is associated with a given homogeneous ordinary differential equation if F/u is invariant. The separable solutions of each associated equation are radial curves in the homogeneous solution space, as described in Theorem 3. Consider the homogeneous equation u "u u /u. xxx x xx Its solutions are the solutions of all equations
(39)
u /u"i (40) xx with i constant. Its invariants are functional combinations of phase, frequency, and amplitude. Phase and frequency are homogeneous. Amplitude is inhomogeneous. Phase depends on x. Frequency and amplitude are combinations of the x-independent invariants f "u /u and f "uu !u2 . The translation invariant equations associated with equax 1 xx 2 xx tion (39) are (41) u "g(u /u, uu !u2)u, x tt xx xx where g is arbitrary. These are the only translation invariant second-order Newton equations which propagate arbitrary exponential data by dilation. We obtain the free linear wave equation u "u (42) tt xx from g(f , f )"f , and the non-linear wave equation 1 2 1 (43) u "(1#u2)u !uu2 x tt xx from g(f , f )"f #f . 1 2 1 2 The evolution equation (31) generates the formal flow of F. We now show that a Newton equation is separable if and only if its force flow stabilizes the space of characteristic fields. See ref. [11] or ref. [10] for discussion of generalized symmetry. ¹heorem 5 The Newton equation with force F is separable if and only if FL is a symmetry of its u characteristic equation. Proof We have (F[u]L #F[u] L #F[u] L #) ) ))(F[u]/u) u x ux xx uxx x "(F[u]L #(uF[u]/u) L #(uF[u]/u) L #) ) ))(F[u]/u) xx uxx x u x ux ,(F[u]/u)(uL #u L #u L #) ) ))(F[u]/u) mod (F[u]/u) "0, xx uxx x x u x ux so FL is a symmetry of equation (6) if and only if uL is a symmetry. h u u Suppose that F/u is an invariant of the homogeneous equation (32), so that FL is u a symmetry. The solution space of (32) is a locally conic subset of the space of fields. The value of FL on a solution u(x) is the vector lu(x)L , where l is the constant F[u(x)]/u(x). u u Thus FL is radial on each solution of (32), and is tangent to the solution cone. We regard it u
1020
P. W. Doyle
as a generator of dilations with variable scale. By contrast, the vector field uL generates u dilation of the entire space of fields, with constant scale l"1. Conversely, if a kth order vector field FL is radial on the solutions of (32) then F/u is invariant. The generalized u dilation symmetries of a homogeneous ordinary differential equation are the vector fields FL with F/u invariant. u ¹heorem 6 The Newton equations associated with a locally homogeneous ordinary differential equation correspond to its generalized dilation symmetries. Finally we characterize trivial separability. A Newton equation is homogeneous if it has the form u "g(x, u /u, . . . , u /u)u, tt 1 k e.g., if it is linear. Any such equation is separable, and its separation mechanism transparent: a function (2) is a solution if and only if v and w satisfy the ordinary differential equations g(x, v /v, . . . , v /v)"l, wA/w"l 1 k for some constant l. Note that the characteristic value of any separable solution is independent of t. The generic invariant of a homogeneous ordinary differential equation is inhomogeneous, so generic separable Newton equations are inhomogeneous. For example, the separable equation (41) is homogeneous if and only if Lg/Lf "0. In particular, equation 2 (43) is inhomogeneous. The separable equations (7), (17), and (28) are also inhomogeneous. The proof of the following characterization of homogeneity is similar to its analogue for evolution equations [2]. ¹heorem 7 A separable Newton equation is locally homogeneous if and only if the characteristic value of each separable solution is constant. Separability implies that FL is tangent to the solution space of (6). This space is foliated u by the solution spaces of equations F[u]/u"l.
(44)
The Newton equation is homogeneous if and only if its force is tangent to the characteristic value foliation, i.e., if and only if FL is a symmetry of each equation (44). u 5 . COM PATI BIL ITY WIT H T HE CH AR A CT ER IS T I C CO NDI T IO N
The non-zero characteristic solutions of equation (1) are the solutions of equations (1) and (6). These are the solutions of equations (1) and (4). Equations (1) and (4) are compatible if they share a solution u(x, t) with u(x , 0)"c , . . . , u (x , 0)"c , u (x , 0)"d , . . . , u (x , 0)"d , 0 0 k 0 k t 0 0 k,t 0 k for each relevant set of values x and c O0, c , . . . , c , d , . . . , d . The natural context for 0 0 1 k 0 k a geometric analysis of compatibility is the bundle J of (k#2)th-order jets of smooth maps R2"R. The variables x, t, and u with i#j)k#2 are coordinates on J, where i,j u denotes the derivative Li Lju. We also use notation u "u and u "u . See ref. [12] i,t i,1 i,tt i,2 x t i,j for elementary facts regarding jet bundles, contact structure, and prolongation. We write equation (6) as u "((x, u, u , . . . , u ). k`1 1 k Solutions u(x, t) also satisfy the conditions
(45)
u "(L #u L #) ) )#u L #([u]L ) ([u] k`2 x 1 u k uk~1 uk
(46)
u "¶ (x, u, u , . . . , u , u , u , . . . , u ), k`1,t 1 k t 1,t k,t
(47)
and
Characteristic solutions of scalar Newton equations
1021
where ¶ [u]"(u L #) ) )#u L ) ([u]. t u k,t uk Differentiating equation (4) successively with respect to x, we find that solutions of equations (1) and (4) satisfy u "u F[u]/u, i"0, . . . , k. i,tt i Differentiating with respect to t, we find that solutions satisfy u "¹j(u F[u]/u), i#j)k, i,j`2 i
(48)
where k k ¹"L # + u L # + (u F[u]/u)L . i ui,t t i,t ui i/0 i/0 Relations (45—48) define a (2k#4)-dimensional submanifold R of J, parametrized by variables x, t, u, u , . . . , u , u , u , . . . , u . The pullback of the contact structure to R 1 k t 1,t k,t includes the independent 1-forms h "du !u dx!u dt, i"0, . . . , k!1, i i i`1 i,t h "du !([u] dx!u dt, k k k,t / "du !u dx!(u F[u]/u) dt, i"0, . . . , k!1, i i,t i`1,t i / "du !¶ [u] dx!(u F[u]/u) dt. k k,t k The 1-forms dx, dt, h, and / are also independent, so contact integrals in R are transverse to the projection (x, t, u, . . . )>(x, t). Locally, contact surfaces in R are prolonged joint solutions of equations (1) and (4). In fact, it suffices to use the two-codimensional subsystem h, /. ¸emma The surface u"u(x, t), u "u (x, t), i"1, . . . , k, i i u "u (x, t), i"0, . . . , k, i,t i,t is an integral of h, / if and only if u(x, t) is a solution of equations (1) and (4), and u (x, t)"Li u(x, t), i"1, . . . , k, x i u (x, t)"Li L u(x, t), i"0, . . . , k. x t i,t ¹heorem 8 A Newton equation is compatible with the characteristic condition if and only if its characteristic equation is locally linear. Proof By the lemma, equations (1) and (4) are compatible if and only if there is an integral surface of h, / through each point of R. The independent vector fields ¹ and k~1 k~1 X"L # + u L #([u] L # + u L #¶ [u] L i k x i`1 u u i`1,t ui,t uk,t i/0 i/0 are annulled by h, /, and [X, ¹]"((F/u!¹¶ )L , uk,t so h, / is integrable if and only if (F/u"¹¶, i.e. k k (F/u"(F/u) + u L (# + u u L L (. i,t j,t ui uj i ui i/0 i,j/0
1022
P. W. Doyle
The conditions L L ("0 and +u L ("( hold if and only if ( has local form ui uj i ui ([u]"a (x)u #) ) )#a (x)u. h k k 0 For example, the equations u #g(u2#u2)u"0 x tt
(49)
and u "g(u , 2uu !u2)u, (50) tt xx xx x are compatible with equation (4). Their characteristic equations are equations (27) and (30). Note that equation (28) has form (49) and equation (17) has form (50). In general, a kth-order Newton equation compatible with the characteristic condition has a (2k#2)-parameter family of characteristic solutions. Any Newton equation compatible with the characteristic condition is separable, by Theorem 2, and its force operator is a symmetry of its characteristic equation, by Theorem 5. We have the following analogue of Theorem 4. ¹heorem 9 The Newton equation with force F is compatible with the characteristic condition if and only if F/u is an invariant of a locally linear ordinary differential equation. Thus, kth-order Newton equations compatible with the characteristic condition correspond to invariants of linear (k#1)th-order ordinary differential equations. The Newton equations associated with a given linear equation correspond to its generalized dilation symmetries, by Theorem 6. The characteristic solutions of each associated equation are planar curves in the linear solution space, as described in Theorem 10. For example, fix arbitrary functions v (x) and v (x) with v v@ !v v@ O0. The equation 2 1 1 2 1 2 u #a(x)u #b(x)u"0 (51) xx x with coefficients vA v !vA v v@ vA !v@ vA 2 1, 2 1, a" 1 2 b" 1 2 v v@ !v v@ v v@ !v v@ 2 1 1 2 2 1 1 2 is the unique linear second-order ordinary differential equation for which v and v are 1 2 solutions. The linear operators v (x)u !v@ (x)u v (x)u !v@ (x)u 2 x 2 1 x 1 f (x, u, u )" , f (x, u, u )" , 1 x 2 x v (x)v@ (x)!v (x)v@ (x) v (x)v@ (x)!v (x)v@ (x) 2 1 1 1 2 2 2 1 are independent invariants. The general invariant is a functional combination of f and f , 1 2 so the associated Newton equations are u "g(f (x, u, u ), f (x, u, u ))u. (52) tt 1 x 2 x A curve u(x, t)"v (x)w (t)#v (x)w (t) in the solution plane of equation (51) satisfies 1 1 2 2 equation (52) if and only if
A B
A B
w A w 1 "g(w , w ) 1 . (53) 1 2 w w 2 2 Thus, we obtain the four-parameter family of characteristic solutions of equation (52). For equation (27) we have v "cos x, v "sin x, and 1 2 f "u cos x!u sin x, f "u sin x#u cos x. 1 x 2 x The combination f2#f2 yields the translation invariant equations (49). The function (9) 2 1 satisfies equation (49) if and only if
A B
A B
w A w 1 "!g(w2#w2) 1 . 1 2 w w 2 2
(54)
Characteristic solutions of scalar Newton equations
1023
6 . RE DUC TI ON TO CE NTRA L F ORCE DY NAM I CS
An autonomous finite-dimensional dynamical system is a second-order ordinary differential equation fA"f (f) for unknown functions f , . . . , f of a variable t. The force f : )"Rn, 1 n defined on an open domain of Rn, is central if it has form f (f)"g(f)f for some scalar function g. The form of the central force problem fA"g(f)f
(55)
is invariant under linear tranformation of Rn. Central forces depending only on the radius DfD are conservative. The converse holds on any spherically symmetric domain. The physically important case n"3 is discussed in texts on classical mechanics, e.g. Arnold [13]. We do not assume that central forces are symmetric. An n-dimensional central force problem has a 2n-parameter family of solutions f(t). For independent vectors v 3) and v 3Rn the unique solution of equation (55) with 1 2 f(0)"v , f@(0)"v 1 2 is f(t)"w (t)v #w (t)v , 1 1 2 2
(56)
where
A B
A B
w A w 1 "g(w v #w v ) 1 1 1 2 2 w w 2 2
and w (0)"1, w@ (0)"0, 1 1 For v3) and c3R the solution with f (0)"v,
w (0)"0, w@ (0)"1. 2 2 f@(0)"cv
is f(t)"w(t)v,
(57)
wA"g(wv)w
(58)
where
and w(0)"1, w@(0)"c. If 03) then for v3Rn the solution with f(0)"0, f@(0)"v is the function (57), where w satisfies equation (58) and w(0)"0, w@(0)"1. The coefficients w and w of any solution (56) satisfy equation (24) with j(t)"g(f(t)). Thus 1 2 each trajectory of a central force problem is planar or radial, and has constant angular momentum. These properties are independent of the choice of linear coordinates for Rn. We now describe the relationship between central force problems, Newton equations compatible with the characteristic condition, and generalized dilation symmetries of linear ordinary differential equations. The solutions of a linear equation u
"a (x)u #) ) )#a (x)u, (59) k`1 k k 0 form a (k#1)-dimensional vector space Z. For independent solutions v (x), . . . , v (x) the 0 k matrix function »(x)"(v(i) (x)) ) ) j 0 i,j k
1024
P. W. Doyle
is invertible, and the linear differential operators f , . . . , f defined by 0 k f[u]"(u, u , . . . , u )»(x)~t, (60) 1 k are invariants with values c , . . . , c on the solution u(x)"+c v (x). The general invariant is 0 k i i a functional combination, so the Newton equations associated with equation (59) are u "g(f [u], . . . , f [u]) u. (61) tt 0 k The force of equation (61) stabilizes its space Z of characteristic fields, so equation (61) restricts to a finite-dimensional dynamical system on Z. Using the invariants f as linear coordinates we find that the restriction has form (55). Characteristic solutions of any Newton equation compatible with the characteristic condition are thus identified with trajectories of a central force problem. ¹heorem 10 Let f , . . . , f be the invariants due to a solution basis v , . . . , v for a linear ordinary 0 k 0 k differential equation. The characteristic solutions of u "g(f[u])u, tt are the functions k u(x, t)" + f (t)v (x) i i i/0 with fA"g(f)f. The construction of explicit solutions of non-linear partial differential equations using invariant finite-dimensional linear function spaces is discussed by Galaktionov and Posashkov [3, 4] and Svirshchevskii [5, 6]. A linear prototype is found in the invariant spaces of quasipolynomial solutions of the one-dimensional diffusion equation. Theorem 10 exemplifies their technique. An operator F stabilizes the solution space of a given linear ordinary differential equation if and only if FL is a symmetry. For example, the force field g(f)uL for equation (61) is u u a generalized dilation symmetry of equation (59). It is radial on Z, reflecting the fact that the reduced equation is a central force problem. Its alternate form g(f)(f v (x)#) ) )#f v (x))L , 0 0 k k u is obtained from the identity u"+ f [u]v (x). See refs [14] or [5] for a complete description i i of the symmetries of linear ordinary differential equations in terms of invariants. The reduction for k"1 is given in the previous section: the restriction of equation (52) to the solution plane of equation (51) is the central force problem (53). In general, independent functions v , . . . , v form a basis for a unique (k#1)th-order ordinary differential equation. 0 k The dual invariants are given by equation (60). The associated equations are given by equation (61), and their central force reductions are described in Theorem 10. We illustrate Theorem 10 with an example in the case k"2. The operators f "u!xu #x2u /2, f "u !xu , f "u /2 0 x xx 1 x xx 2 xx are invariants of equation (30). The combination (15) yields equation (17). Its characteristic solutions are functions (16), where f(t) is any trajectory of the three-dimensional central force problem fA"(4f f !f2)f. 1 0 2 7 . E QUA TI ONS WI TH NON -S EPA R AB LE CH AR ACT ER I STI C S OLU TIO NS
Compatibility with the characteristic condition is unnecessary for the existence of non-separable characteristic solutions, just as compatibility with the separation condition is
Characteristic solutions of scalar Newton equations
1025
unnecessary for the existence of separable solutions. An operator F stabilizes any linear subspace Z of its characteristic fields. Moreover, the restriction of F to Z is radial. The corresponding Newton equation reduces to a central force problem, and its generic trajectories are non-radial if dim Z'1. ¹heorem 11 The restriction of a Newton equation to any vector space of characteristic fields is a central force problem. Its trajectories are characteristic solutions, generically non-separable if the space is at least two dimensional. Hence any equation with a (homogeneous) line of characteristic fields has separable solutions, and any equation with a plane of characteristic fields has non-separable characteristic solutions. Equation (7) is not compatible with the characteristic condition. Its restriction to the plane of characteristic fields defined by equation (27) is the system (10). In fact, the solutions of equation (27) are characteristic fields for any equation (62) u "g(u2#u2)u . x xx tt The restriction is the conservative central force problem (54). We obtain a four-parameter family of solutions (9). If g'0 then equation (62) is hyperbolic and the central force of equation (54) is attractive. Bound orbits of equation (54) with non-zero angular momentum are bounded non-separable solutions of equation (62) defined for all t. Circular orbits are travelling wave solutions. The separable equation (1) defined by an invariant F/u of the homogeneous equation ((a (x)u #) ) )#a (x)u)/u) "0 (63) k k 0 x (a O0) is not compatible with the characteristic condition. Solutions of equation (63) are k the non-zero solutions of all equations a (x)u #) ) )#a (x)u"iu (64) k k 0 with i constant. The restriction of equation (1) to the solution space of equation (64) yields a 2k-parameter family of characteristic solutions. The (2k#1)-parameter family thus obtained accounts for all non-zero characteristic solutions of equation (1). They are separable if k"1, and generically non-separable if k*2. Equation (39) has form (63), so the characteristic solutions of equation (41) are those which satisfy equation (40) for some i. We obtain the five-parameter family u(x, t)"a#bt#cx#dxt, u(x, t)"a cosh ux cosh ut#b cosh ux sinh ut#c sinh ux cosh ut#d sinh ux sinh ut, u(x, t)"a cos ux cos ut#b cos ux sin ut#c sin ux cos ut#d sin ux sin ut, of characteristic solutions of the linear wave equation (42). The characteristic solutions of equation (43) are functions u(x, t)"w (t)#w (t)x, 1 2
A B
A B
w A w 1 "!w2 1 , 2 w w 2 2
functions u(x, t)"w (t) cosh ux#w (t) sinh ux 1 2 with
A B
A B
w A w 1 "u2(1#w2!w2) 1 2 w 1 w 2 2
(65)
u(x, t)"w (t) cos ux#w (t) sin ux 1 2
(66)
and
1026
P. W. Doyle
with
A B
A B
w A w 1 "!u2(1#w2#w2) 1 . 2 w 1 w 2 2 The effective potential for the radial component of the conservative system (67) is
(67)
»(r)"u2(r2/2#r4/4)#¸2/2r2, where the arbitrary constant ¸ represents angular momentum. All solutions of the radial equation rA#»@(r)"0 are bounded. Hence the solutions (66) of equation (43) are defined for all t and are bounded. They need not be periodic in t (Bertrand’s theorem [13]). The travelling wave solutions u(x, t)"Jc2!1 cos u(x!x !ct) 0 are circular orbits of equation (67), and the travelling waves u(x, t)"$Jc2!1 cosh u(x!x !ct), 0 are hyperbolic orbits of equation (65).
u(x, t)"J1!c2 sinh u(x!x !ct) 0
R EFE RE NC ES 1. E. G. Kalnins and W. Miller Jr., Differential-Sta¨ckel matrices. J. Math. Phys. 26, 1560—1565 (1985). 2. P. W. Doyle, Separation of variables for scalar evolution equations in one space dimension. J. Phys. A 29, 7581—7595 (1996). 3. V. A. Galaktionov and S. A. Posashkov, Exact solutions and invariant spaces for non-linear gradient diffusion equations. Comput. Maths Math. Phys. 34, 313—321 (1994). 4. V. A. Galaktionov, Invariant subspaces and new explicit solutions to evolution equations with quadratic non-linearities. Proc. Roy. Soc. Edin. 125 A, 225—246 (1995). 5. S. R. Svirshchevskii, Lie-Ba¨cklund symmetries of linear ODEs and generalized separation of variables in non-linear equations. Phys. ¸ett. A 199, 344—348 (1995). 6. S. R. Svirshchevskii, Invariant linear spaces and exact solutions of non-linear evolution equations. Non-linear Math. Phys. 3, 164—169 (1996). 7. W. Miller Jr. and L. A. Rubel, Functional separation of variables for Laplace equations in two dimensions. J. Phys. A 26, 1901—1913 (1993). 8. P. W. Doyle and P. J. Vassiliou, Separation of variables for the 1-dimensional non-linear diffusion equation. Int. J. Non-¸inear Mech. 33, 315—326 (1998). 9. G. W. Bluman and S. Kumei, Symmetries and Differential Equations. Springer, New York (1989). 10. P. J. Olver, Applications of ¸ie Groups to Differential Equations. Springer, New York (1993). 11. R. L. Anderson and N. H. Ibragimov, ¸ie-Ba( cklund ¹ransformations in Applications. SIAM, Philadelphia (1979). 12. P. J. Olver, Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995). 13. V. I. Arnold, Mathematical Methods of Classical Mechanics. Springer, New York (1978). 14. A. V. Samokhin, Symmetries of linear ordinary differential equations. Amer. Math. Soc. ¹ransl. 167, 193—206 (1995).
PII: S0020–7462(97)00064-4
CHARACTERISTIC SOLUTIONS OF SCALAR NEWTON EQUATIONS IN ONE SPACE DIMENSION P. W. Doyle Department of Mathematics, University of Hawaii, Honolulu, HI 96822, U.S.A. (Received 1 July 1997) Abstract—Newton equations are dynamical systems on the space of fields. The solutions of a given equation which are curves of characteristic fields for its force are planar and have constant angular momentum. Separable solutions are characteristic with angular momentum equal to zero. A Newton equation is separable if and only if its characteristic equation is homogeneous. Separable equations correspond to invariants of homogeneous ordinary differential equations, and those associated with a given homogenous equation correspond to its generalized dilation symmetries. A Newton equation is compatible with the characteristic condition if and only if its characteristic equation is linear. Such equations correspond to invariants of linear ordinary differential equations. Those associated with a given linear equation correspond to the central force problems on its solution space. Regardless of compatibility, any Newton equation with a plane of characteristic fields has non-separable characteristic solutions. ( 1998 Elsevier Science Ltd. All rights reserved Keywords: separation of variables, characteristic equation, dilation symmetry, central force
1. INTRODUCTION
An autonomous Newton equation in one space dimension is a partial differential equation u "F(x, u, u , . . . , u ) (1) tt 1 k for a scalar function u of two variables x and t, where u denotes the second derivative with tt respect to t and u denotes the jth derivative with respect to x. We assume that F is smooth j (C=). A solution is a smooth function u(x, t) defined on a product of intervals in x and t with L2u(x, t)"F(x, u(x, t), L u(x, t), . . . , Lk u(x, t)), x t x at each point of its domain. In physical terms, the acceleration of the field x>u(x, t) at time t is due to a force depending on the spatial variable x and the spatial derivatives of the field. Thus, we interpret the differential operator F[u]"F(x, u, u , . . . , u ), 1 k as a field of force vectors on the space of fields u(x) and equation (1) as a dynamical system on this infinite-dimensional space. The force field is independent of time, so we consider solutions defined for values of t in some interval containing t"0. The relevant initial value problem requires solution of equation (1) subject to specified position and velocity u(x, 0)"v (x), u (x, 0)"v (x). 1 t 2 In general, we cannot claim existence or uniqueness of such a solution. A nonzero solution of equation (1) has separable form u(x, t)"v(x)w(t),
(2)
if and only if it satisfies the separation condition (u /u) "0. t x Non-zero characteristic solutions satisfy the characteristic condition (u /u) "0. tt x Contributed by W. F. Ames. 1013
(3)
(4)
1014
P. W. Doyle
Note that equation (3) implies equation (4), so separable solutions are characteristic. We find that characteristic solutions are those with planar form u(x, t)"v (x)w (t)#v (x)w (t) (5) 1 1 2 2 and constant angular momentum. We show that the Newton equation (1) is compatible with condition (3) if and only if its characteristic equation (F[u]/u) "0 (6) x is homogeneous. This geometric characterization of the separability criterion of Kalnins and Miller [1] is analogous to Doyle’s description of separable evolution equations [2]. Separable Newton equations correspond to invariants of homogeneous ordinary differential equations. The separable equations associated with a given homogeneous equation correspond to its generalized dilation symmetries, and their separable solutions are radial curves in its solution space. Equation (1) is compatible with condition (4) if and only if its characteristic equation is linear. In this case F stabilizes the solution space of equation (6), and the partial differential equation (1) restricts to a finite-dimensional dynamical system. This reduction is an example of the generalized method of separation of variables developed by Galaktionov and Posashkov [3, 4]. See refs [5, 6] for a clear geometrical description and for additional references. Equations compatible with the characteristic condition correspond to invariants of linear ordinary differential equations. The Newton equations associated with a given linear equation correspond to autonomous central force problems on its solution space, and their characteristic solutions are the central force trajectories. Regardless of compatibility, the restriction of a Newton equation to any linear subspace of characteristic fields is a central force problem, and its trajectories are characteristic solutions. We describe a general class of equations which are incompatible with the characteristic condition yet have non-separable characteristic solutions. The form of equation (1) is invariant under arbitrary smooth transformation of x, affine transformation of t, and dilation of u. Conditions (3) and (4), equation (6), and the functional forms (2) and (5) are also invariant under this transformation group. All functions in this study are assumed to be smooth.
2. EX A MP LES
The equation u "(u2#u2)~3@2u , tt x xx describes the motion of fields subject to force F[u]"(u2#u2)~3@2u . x xx
(7)
(8)
The function u(x, t)"w (t) cos x#w (t) sin x (9) 1 2 is a solution if and only if w and w satisfy the Kepler system 1 2 w A w 1 "!(w2#w2)~3@2 1 . (10) 1 2 w w 2 2 Given constants a, b, c, d with a2#b2O0, there is a unique function (9) which satisfies (7) with u(x, 0)"a cos x#b sin x, u (x, 0)"c cos x#d sin x. t Thus, there is a four-parameter family of solutions (9). Solutions corresponding to elliptical orbits of equation (10) are periodic in t. For example, the travelling waves
A B
A B
u(x, t)"c~2@3 cos(x!x !ct), 0
Characteristic solutions of scalar Newton equations
1015
correspond to circular orbits
A B
A B
w cos 1 (t)"c~2@3 (ct!t ). 0 w sin 2
Separable solutions u(x, t)"w(t) cos(x!x ), (11) 0 correspond to radial orbits. The function (11) satisfies (7) if and only if wA"!w~2, so there is a three-parameter family of such solutions. For any function (9), we have F[u(x, t)]"j(t)u(x, t) with j(t)"!(w (t)2#w (t)2)~3@2. 1 2 If u satisfies equation (7) then u (x, t)"j(t)u(x, t), tt so w and w satisfy the linear equation 1 2 wA"j(t) w.
(12)
(13)
Hence, (14) (w w@ !w w@ )@"0 2 1 1 2 which verifies the fact that each trajectory of equation (10) has constant angular momentum. As a second example, consider the operator F[u]"2u2u !uu2 . We have F[u(x)]" x xx ju(x) for some constant j if and only if u(x)"f #f x#f x2, in which case j has value 0 1 2 (15) *[u]"4f f !f2. 1 0 2 Fix independent quadratic polynomials v and v . Suppose that the x-quadratic function 1 2 u(x, t)"f (t)#f (t) x#f (t) x2, (16) 0 1 2 satisfies u "2u2u !uu2 x tt xx
(17)
with u(x, 0)"v (x), u (x, 0)"v (x). (18) 1 t 2 Then u satisfies the family of ordinary differential equations (12) with j(t)"4f (t)f (t)! 0 2 f (t)2. From equation (18) we have 1 u(x, t)"v (x)w (t)#v (x)w (t), (19) 1 1 2 2 where w and w are the solutions of equation (13) with 1 2 (20) w (0)"1, w@ (0)"0, w (0)"0, w@ (0)"1. 2 1 2 1 The function (19) satisfies equations (17) and (18) if and only if w and w have values (20) 1 2 and w A w 1 "(*[v ]w2#*[v , v ]w w #*[v ]w2) 1 , 2 1 1 2 1 2 2 1 w w 2 2 where
A B
A B
*[v , v ]"*[v #v ]!*[v ]!*[v ]. 1 2 1 2 1 2 Thus, the unique x-quadratic solution of the initial value problem (17, 18) is a curve in the plane of fields spanned by v and v . It has constant angular momentum because equation 1 2 (13) implies equation (14). Equation (17) also has a unique x-quadratic solution with given dependent initial values. The solution with u(x, 0)"f #f x#f x2, 0 1 2 is the separable function
u (x, 0)"c(f #f x#f x2) t 0 1 2
u(x, t)"(f #f x#f x2)w(t), 0 1 2
(21)
1016
P. W. Doyle
where wA"(4f f !f2)w3 1 0 2
(22)
and w(0)"1,
w@(0)"c.
The solution with u(x, 0)"0,
u (x, 0)"f #f x#f x2 t 0 1 2 is the function (21), where w satisfies equation (22) and w(0)"0,
w@(0)"1.
We will find that the four-parameter family thus obtained accounts for all separable solutions of equation (17). Equation (17) has a six-parameter family of solutions (16).
3. CHA RAC TER I ST IC SOL U TI ONS
A characteristic field of an operator F is a function u(x), not identically equal to zero, with F[u(x)]"ju(x) for some constant j, its characteristic value. A solution u(x, t) of equation (1) is characteristic if x>u(x, t) is a characteristic field of F for each t. A solution not identically equal to zero for any t is characteristic if and only if F[u(x, t)]"j(t)u(x, t), i.e. u (x, t)"j(t)u(x, t), (23) tt for some function j. A non-zero function defined on a product of intervals satisfies equation (23) for some j if and only if it satisfies equation (4). The solutions constructed in the previous section are characteristic. Solutions of equation (23) are functions (5), where w and w are independent solutions of 1 2 wA"j(t) w (24) with v and v arbitrary. Hence a solution of equation (1) is characteristic if and only if it has 1 2 form (5) (but is not identically equal to zero for any t), where w and w are independent 1 2 solutions of some equation (24), in which case j(t) is its characteristic value. Note that separable solutions are characteristic. ¹heorem 1 A solution of a Newton equation is characteristic if and only if it has form (5) with w w@ !w w@ constant and non-zero, in which case its characteristic value is 2 1 1 2 wA w@ !wA w@ 2 1. j" 1 2 (25) w w@ !w w@ 2 1 1 2 Proof If w and w are independent solutions of equation (24) then w w@ !w w@ is constant 1 2 1 2 2 1 and non-zero. Conversely, if w w@ !w w@ is constant and non-zero then w and w are 1 2 2 1 1 2 independent solutions of (24) with j given by equation (25). h If v and v are multiples of some field v then the solution (5) is separable. Otherwise it is 1 2 a curve in the plane spanned by v and v , with constant nonzero angular momentum 1 2 w w@ !w w@ . This property is independent of basis v , v . The momentum value is not 1 2 2 1 1 2 defined. Separable solutions are radial curves, so they are characteristic with angular momentum equal to zero.
Characteristic solutions of scalar Newton equations
1017
We assume L FO0, k*1. Equation (6) is a (k#1)th-order ordinary differential equauk tion u #(u/L F[u])(L #u L #) ) )#u L )(F[u]/u)"0 k`1 uk x 1 u k uk~1 and its solutions are the non-zero characteristic fields of F. A non-zero solution of equation (1) is characteristic if and only if it is a curve in the solution space of equation (6). For example, the characteristic equation for F[u]"!(u2#u2)~3@2 u x
(26)
is u #u"0. (27) xx Curves in its solution space are functions (9). Radial curves are functions (11). The forces (8) and (26) are identical on solutions of equation (27), so the solutions (9, 10) of equation (7) also satisfy (28) u #(u2#u2)~3@2 u"0. x tt This accounts for all characteristic solutions of equation (28). Of course, solutions of equation (27) satisfy the characteristic equation u "(1#3u(u#u )/(u2#u2))u u /u x x xx xxx xx of equation (7). The characteristic equation of equation (17) is
(29)
u "0, (30) xxx curves in its solution space are functions (16), and radial curves are functions (21). Function (2) represents dilation of a field v by a variable factor w, suggesting that the geometric nature of separability is projective. Hence, we emphasize non-zero separable solutions. In Doyle’s study of evolution equations [2], for which characteristic and separable solutions are identical, characteristic fields are by definition nowhere equal to zero. This restriction is artificial in the context of Newton equations. Note that zeros of a non-separable function (5) may evolve as t varies.
4 . S EPA R AB I LI TY
Equation (1) is (regularly) separable [1] if it has a solution (2) with u(x , 0)"c , . . . , u (x , 0)"c , u (x , 0)"d, 0 0 k 0 k t 0 for each relevant set of values x and c O0, c , . . . , c , d. This requirement represents 0 0 1 k compatibility of equations (1) and (3). Kalnins and Miller [1] derive additive separability conditions for partial differential equations of general form, i.e. necessary and sufficient conditions for the existence of a complete family of additively separable solutions in a given coordinate system. See ref. [7] for application to non-linear Laplace equations. Doyle [2] describes compatibility of the separation condition (3) and evolution equation u "F[u], (31) t as integrability of the contact structure on their intersection, and finds that equation (31) is separable if and only if equation (6) has local homogeneous form u "u(x, u /u, . . . , u /u) u. (32) k`1 1 k See also ref. [8]. We now prove the analogous result for Newton equations. See ref. [9] or ref. [10] for discussion of symmetry and differential equations. ¹heorem 2 A Newton equation is separable if and only if its characteristic equation is locally homogeneous.
1018
P. W. Doyle
Proof For u'0 or u(0, condition (3) is given in the variable uN "lnDuD by uN "0, xt
(33)
equation (1) is given by uN #uN 2"FM [uN ] t tt with FM [uN ]"F[u]/u, and equation (6) is given by
(34)
FM [uN ] "0. x According to ref. [1], equations (33) and (34) are compatible if and only if
(35)
L 6 ((L #uN L 6 #) ) )#uN L 6 )FM [uN ]/L 6 FM [uN ])"0. u x 1 u k uk~1 uk This condition holds if and only if L 6 is a symmetry of equation (35), i.e. if and only if uL is u u a symmetry of equation (6). h As in Theorem 8, it can be shown that dilation invariance of the characteristic equation is equivalent to integrability of the contact structure on the intersection of equations (1) and (3) and their differential consequences. Equation (7) is separable because its characteristic equation (29) is homogeneous. Equations (28) and (17) are separable because their characteristic equations (27) and (30) are linear (hence homogeneous). A separable kth-order Newton equation has a (k#2)-parameter family of separable solutions. Above we describe the three-parameter family for equation (28) and the fourparameter family for equation (17). Note that a first-order Newton equation (k"1) is actually a second-order differential equation. The following description of separable solutions explains the role of dilation invariance of the characteristic equation. ¹heorem 3 Fix a non-zero characteristic field v of a separable equation u "F[u]. (36) tt For any constant c and any point x in the domain of v there is a separable solution u on 0 a neighborhood of x with 0 u(x, 0)"v(x), u (x, 0)"cv(x). (37) t If u is a separable solution on a neighborhood of x with values (37) then there are positive 0 constants e and e such that u(x, t)"v(x)w(t) for Dx!x D(e and DtD(e , where 1 2 0 1 2 wA"¼(w) and w(0)"1,
w@(0)"c,
(38)
where ¼(w)"F[v(x)w]/v(x). This accounts for all non-zero separable solutions of equation (36). Proof Locally, v(x)w is characteristic for all values of w sufficiently close to w"1 because the characteristic equation (36) is homogeneous. Hence F[v(x)w]/v(x) is locally independent of x near w"1. The function u(x, t)"v(x)w(t) satisfies equations (36) and (37) if and only if w satisfies equation (38) and wA"F[v(x)w]/v(x). This implies local existence and uniqueness as claimed. A non-zero separable solution u is a curve of characteristic fields. In particular, its initial value u(x, 0)"v(x) is characteristic. We have u(x, t)"v(x)w(t) for some function w. Hence u (x, 0)"w@(0)v(x). We have already t encountered this solution. h
Characteristic solutions of scalar Newton equations
1019
We now formulate Theorem 2 in terms of invariants. A first integral or invariant of an nth-order ordinary differential equation has generic order n!1. ¹heorem 4 The Newton equation with force F is separable if and only if F/u is an invariant of a locally homogeneous ordinary differential equation. Proof Equation (6) is the unique equation for which F/u is invariant. h Thus, separable kth-order Newton equations correspond (locally) to invariants of homogeneous (k#1)th-order ordinary differential equations. The Newton equation with force F is associated with a given homogeneous ordinary differential equation if F/u is invariant. The separable solutions of each associated equation are radial curves in the homogeneous solution space, as described in Theorem 3. Consider the homogeneous equation u "u u /u. xxx x xx Its solutions are the solutions of all equations
(39)
u /u"i (40) xx with i constant. Its invariants are functional combinations of phase, frequency, and amplitude. Phase and frequency are homogeneous. Amplitude is inhomogeneous. Phase depends on x. Frequency and amplitude are combinations of the x-independent invariants f "u /u and f "uu !u2 . The translation invariant equations associated with equax 1 xx 2 xx tion (39) are (41) u "g(u /u, uu !u2)u, x tt xx xx where g is arbitrary. These are the only translation invariant second-order Newton equations which propagate arbitrary exponential data by dilation. We obtain the free linear wave equation u "u (42) tt xx from g(f , f )"f , and the non-linear wave equation 1 2 1 (43) u "(1#u2)u !uu2 x tt xx from g(f , f )"f #f . 1 2 1 2 The evolution equation (31) generates the formal flow of F. We now show that a Newton equation is separable if and only if its force flow stabilizes the space of characteristic fields. See ref. [11] or ref. [10] for discussion of generalized symmetry. ¹heorem 5 The Newton equation with force F is separable if and only if FL is a symmetry of its u characteristic equation. Proof We have (F[u]L #F[u] L #F[u] L #) ) ))(F[u]/u) u x ux xx uxx x "(F[u]L #(uF[u]/u) L #(uF[u]/u) L #) ) ))(F[u]/u) xx uxx x u x ux ,(F[u]/u)(uL #u L #u L #) ) ))(F[u]/u) mod (F[u]/u) "0, xx uxx x x u x ux so FL is a symmetry of equation (6) if and only if uL is a symmetry. h u u Suppose that F/u is an invariant of the homogeneous equation (32), so that FL is u a symmetry. The solution space of (32) is a locally conic subset of the space of fields. The value of FL on a solution u(x) is the vector lu(x)L , where l is the constant F[u(x)]/u(x). u u Thus FL is radial on each solution of (32), and is tangent to the solution cone. We regard it u
1020
P. W. Doyle
as a generator of dilations with variable scale. By contrast, the vector field uL generates u dilation of the entire space of fields, with constant scale l"1. Conversely, if a kth order vector field FL is radial on the solutions of (32) then F/u is invariant. The generalized u dilation symmetries of a homogeneous ordinary differential equation are the vector fields FL with F/u invariant. u ¹heorem 6 The Newton equations associated with a locally homogeneous ordinary differential equation correspond to its generalized dilation symmetries. Finally we characterize trivial separability. A Newton equation is homogeneous if it has the form u "g(x, u /u, . . . , u /u)u, tt 1 k e.g., if it is linear. Any such equation is separable, and its separation mechanism transparent: a function (2) is a solution if and only if v and w satisfy the ordinary differential equations g(x, v /v, . . . , v /v)"l, wA/w"l 1 k for some constant l. Note that the characteristic value of any separable solution is independent of t. The generic invariant of a homogeneous ordinary differential equation is inhomogeneous, so generic separable Newton equations are inhomogeneous. For example, the separable equation (41) is homogeneous if and only if Lg/Lf "0. In particular, equation 2 (43) is inhomogeneous. The separable equations (7), (17), and (28) are also inhomogeneous. The proof of the following characterization of homogeneity is similar to its analogue for evolution equations [2]. ¹heorem 7 A separable Newton equation is locally homogeneous if and only if the characteristic value of each separable solution is constant. Separability implies that FL is tangent to the solution space of (6). This space is foliated u by the solution spaces of equations F[u]/u"l.
(44)
The Newton equation is homogeneous if and only if its force is tangent to the characteristic value foliation, i.e., if and only if FL is a symmetry of each equation (44). u 5 . COM PATI BIL ITY WIT H T HE CH AR A CT ER IS T I C CO NDI T IO N
The non-zero characteristic solutions of equation (1) are the solutions of equations (1) and (6). These are the solutions of equations (1) and (4). Equations (1) and (4) are compatible if they share a solution u(x, t) with u(x , 0)"c , . . . , u (x , 0)"c , u (x , 0)"d , . . . , u (x , 0)"d , 0 0 k 0 k t 0 0 k,t 0 k for each relevant set of values x and c O0, c , . . . , c , d , . . . , d . The natural context for 0 0 1 k 0 k a geometric analysis of compatibility is the bundle J of (k#2)th-order jets of smooth maps R2"R. The variables x, t, and u with i#j)k#2 are coordinates on J, where i,j u denotes the derivative Li Lju. We also use notation u "u and u "u . See ref. [12] i,t i,1 i,tt i,2 x t i,j for elementary facts regarding jet bundles, contact structure, and prolongation. We write equation (6) as u "((x, u, u , . . . , u ). k`1 1 k Solutions u(x, t) also satisfy the conditions
(45)
u "(L #u L #) ) )#u L #([u]L ) ([u] k`2 x 1 u k uk~1 uk
(46)
u "¶ (x, u, u , . . . , u , u , u , . . . , u ), k`1,t 1 k t 1,t k,t
(47)
and
Characteristic solutions of scalar Newton equations
1021
where ¶ [u]"(u L #) ) )#u L ) ([u]. t u k,t uk Differentiating equation (4) successively with respect to x, we find that solutions of equations (1) and (4) satisfy u "u F[u]/u, i"0, . . . , k. i,tt i Differentiating with respect to t, we find that solutions satisfy u "¹j(u F[u]/u), i#j)k, i,j`2 i
(48)
where k k ¹"L # + u L # + (u F[u]/u)L . i ui,t t i,t ui i/0 i/0 Relations (45—48) define a (2k#4)-dimensional submanifold R of J, parametrized by variables x, t, u, u , . . . , u , u , u , . . . , u . The pullback of the contact structure to R 1 k t 1,t k,t includes the independent 1-forms h "du !u dx!u dt, i"0, . . . , k!1, i i i`1 i,t h "du !([u] dx!u dt, k k k,t / "du !u dx!(u F[u]/u) dt, i"0, . . . , k!1, i i,t i`1,t i / "du !¶ [u] dx!(u F[u]/u) dt. k k,t k The 1-forms dx, dt, h, and / are also independent, so contact integrals in R are transverse to the projection (x, t, u, . . . )>(x, t). Locally, contact surfaces in R are prolonged joint solutions of equations (1) and (4). In fact, it suffices to use the two-codimensional subsystem h, /. ¸emma The surface u"u(x, t), u "u (x, t), i"1, . . . , k, i i u "u (x, t), i"0, . . . , k, i,t i,t is an integral of h, / if and only if u(x, t) is a solution of equations (1) and (4), and u (x, t)"Li u(x, t), i"1, . . . , k, x i u (x, t)"Li L u(x, t), i"0, . . . , k. x t i,t ¹heorem 8 A Newton equation is compatible with the characteristic condition if and only if its characteristic equation is locally linear. Proof By the lemma, equations (1) and (4) are compatible if and only if there is an integral surface of h, / through each point of R. The independent vector fields ¹ and k~1 k~1 X"L # + u L #([u] L # + u L #¶ [u] L i k x i`1 u u i`1,t ui,t uk,t i/0 i/0 are annulled by h, /, and [X, ¹]"((F/u!¹¶ )L , uk,t so h, / is integrable if and only if (F/u"¹¶, i.e. k k (F/u"(F/u) + u L (# + u u L L (. i,t j,t ui uj i ui i/0 i,j/0
1022
P. W. Doyle
The conditions L L ("0 and +u L ("( hold if and only if ( has local form ui uj i ui ([u]"a (x)u #) ) )#a (x)u. h k k 0 For example, the equations u #g(u2#u2)u"0 x tt
(49)
and u "g(u , 2uu !u2)u, (50) tt xx xx x are compatible with equation (4). Their characteristic equations are equations (27) and (30). Note that equation (28) has form (49) and equation (17) has form (50). In general, a kth-order Newton equation compatible with the characteristic condition has a (2k#2)-parameter family of characteristic solutions. Any Newton equation compatible with the characteristic condition is separable, by Theorem 2, and its force operator is a symmetry of its characteristic equation, by Theorem 5. We have the following analogue of Theorem 4. ¹heorem 9 The Newton equation with force F is compatible with the characteristic condition if and only if F/u is an invariant of a locally linear ordinary differential equation. Thus, kth-order Newton equations compatible with the characteristic condition correspond to invariants of linear (k#1)th-order ordinary differential equations. The Newton equations associated with a given linear equation correspond to its generalized dilation symmetries, by Theorem 6. The characteristic solutions of each associated equation are planar curves in the linear solution space, as described in Theorem 10. For example, fix arbitrary functions v (x) and v (x) with v v@ !v v@ O0. The equation 2 1 1 2 1 2 u #a(x)u #b(x)u"0 (51) xx x with coefficients vA v !vA v v@ vA !v@ vA 2 1, 2 1, a" 1 2 b" 1 2 v v@ !v v@ v v@ !v v@ 2 1 1 2 2 1 1 2 is the unique linear second-order ordinary differential equation for which v and v are 1 2 solutions. The linear operators v (x)u !v@ (x)u v (x)u !v@ (x)u 2 x 2 1 x 1 f (x, u, u )" , f (x, u, u )" , 1 x 2 x v (x)v@ (x)!v (x)v@ (x) v (x)v@ (x)!v (x)v@ (x) 2 1 1 1 2 2 2 1 are independent invariants. The general invariant is a functional combination of f and f , 1 2 so the associated Newton equations are u "g(f (x, u, u ), f (x, u, u ))u. (52) tt 1 x 2 x A curve u(x, t)"v (x)w (t)#v (x)w (t) in the solution plane of equation (51) satisfies 1 1 2 2 equation (52) if and only if
A B
A B
w A w 1 "g(w , w ) 1 . (53) 1 2 w w 2 2 Thus, we obtain the four-parameter family of characteristic solutions of equation (52). For equation (27) we have v "cos x, v "sin x, and 1 2 f "u cos x!u sin x, f "u sin x#u cos x. 1 x 2 x The combination f2#f2 yields the translation invariant equations (49). The function (9) 2 1 satisfies equation (49) if and only if
A B
A B
w A w 1 "!g(w2#w2) 1 . 1 2 w w 2 2
(54)
Characteristic solutions of scalar Newton equations
1023
6 . RE DUC TI ON TO CE NTRA L F ORCE DY NAM I CS
An autonomous finite-dimensional dynamical system is a second-order ordinary differential equation fA"f (f) for unknown functions f , . . . , f of a variable t. The force f : )"Rn, 1 n defined on an open domain of Rn, is central if it has form f (f)"g(f)f for some scalar function g. The form of the central force problem fA"g(f)f
(55)
is invariant under linear tranformation of Rn. Central forces depending only on the radius DfD are conservative. The converse holds on any spherically symmetric domain. The physically important case n"3 is discussed in texts on classical mechanics, e.g. Arnold [13]. We do not assume that central forces are symmetric. An n-dimensional central force problem has a 2n-parameter family of solutions f(t). For independent vectors v 3) and v 3Rn the unique solution of equation (55) with 1 2 f(0)"v , f@(0)"v 1 2 is f(t)"w (t)v #w (t)v , 1 1 2 2
(56)
where
A B
A B
w A w 1 "g(w v #w v ) 1 1 1 2 2 w w 2 2
and w (0)"1, w@ (0)"0, 1 1 For v3) and c3R the solution with f (0)"v,
w (0)"0, w@ (0)"1. 2 2 f@(0)"cv
is f(t)"w(t)v,
(57)
wA"g(wv)w
(58)
where
and w(0)"1, w@(0)"c. If 03) then for v3Rn the solution with f(0)"0, f@(0)"v is the function (57), where w satisfies equation (58) and w(0)"0, w@(0)"1. The coefficients w and w of any solution (56) satisfy equation (24) with j(t)"g(f(t)). Thus 1 2 each trajectory of a central force problem is planar or radial, and has constant angular momentum. These properties are independent of the choice of linear coordinates for Rn. We now describe the relationship between central force problems, Newton equations compatible with the characteristic condition, and generalized dilation symmetries of linear ordinary differential equations. The solutions of a linear equation u
"a (x)u #) ) )#a (x)u, (59) k`1 k k 0 form a (k#1)-dimensional vector space Z. For independent solutions v (x), . . . , v (x) the 0 k matrix function »(x)"(v(i) (x)) ) ) j 0 i,j k
1024
P. W. Doyle
is invertible, and the linear differential operators f , . . . , f defined by 0 k f[u]"(u, u , . . . , u )»(x)~t, (60) 1 k are invariants with values c , . . . , c on the solution u(x)"+c v (x). The general invariant is 0 k i i a functional combination, so the Newton equations associated with equation (59) are u "g(f [u], . . . , f [u]) u. (61) tt 0 k The force of equation (61) stabilizes its space Z of characteristic fields, so equation (61) restricts to a finite-dimensional dynamical system on Z. Using the invariants f as linear coordinates we find that the restriction has form (55). Characteristic solutions of any Newton equation compatible with the characteristic condition are thus identified with trajectories of a central force problem. ¹heorem 10 Let f , . . . , f be the invariants due to a solution basis v , . . . , v for a linear ordinary 0 k 0 k differential equation. The characteristic solutions of u "g(f[u])u, tt are the functions k u(x, t)" + f (t)v (x) i i i/0 with fA"g(f)f. The construction of explicit solutions of non-linear partial differential equations using invariant finite-dimensional linear function spaces is discussed by Galaktionov and Posashkov [3, 4] and Svirshchevskii [5, 6]. A linear prototype is found in the invariant spaces of quasipolynomial solutions of the one-dimensional diffusion equation. Theorem 10 exemplifies their technique. An operator F stabilizes the solution space of a given linear ordinary differential equation if and only if FL is a symmetry. For example, the force field g(f)uL for equation (61) is u u a generalized dilation symmetry of equation (59). It is radial on Z, reflecting the fact that the reduced equation is a central force problem. Its alternate form g(f)(f v (x)#) ) )#f v (x))L , 0 0 k k u is obtained from the identity u"+ f [u]v (x). See refs [14] or [5] for a complete description i i of the symmetries of linear ordinary differential equations in terms of invariants. The reduction for k"1 is given in the previous section: the restriction of equation (52) to the solution plane of equation (51) is the central force problem (53). In general, independent functions v , . . . , v form a basis for a unique (k#1)th-order ordinary differential equation. 0 k The dual invariants are given by equation (60). The associated equations are given by equation (61), and their central force reductions are described in Theorem 10. We illustrate Theorem 10 with an example in the case k"2. The operators f "u!xu #x2u /2, f "u !xu , f "u /2 0 x xx 1 x xx 2 xx are invariants of equation (30). The combination (15) yields equation (17). Its characteristic solutions are functions (16), where f(t) is any trajectory of the three-dimensional central force problem fA"(4f f !f2)f. 1 0 2 7 . E QUA TI ONS WI TH NON -S EPA R AB LE CH AR ACT ER I STI C S OLU TIO NS
Compatibility with the characteristic condition is unnecessary for the existence of non-separable characteristic solutions, just as compatibility with the separation condition is
Characteristic solutions of scalar Newton equations
1025
unnecessary for the existence of separable solutions. An operator F stabilizes any linear subspace Z of its characteristic fields. Moreover, the restriction of F to Z is radial. The corresponding Newton equation reduces to a central force problem, and its generic trajectories are non-radial if dim Z'1. ¹heorem 11 The restriction of a Newton equation to any vector space of characteristic fields is a central force problem. Its trajectories are characteristic solutions, generically non-separable if the space is at least two dimensional. Hence any equation with a (homogeneous) line of characteristic fields has separable solutions, and any equation with a plane of characteristic fields has non-separable characteristic solutions. Equation (7) is not compatible with the characteristic condition. Its restriction to the plane of characteristic fields defined by equation (27) is the system (10). In fact, the solutions of equation (27) are characteristic fields for any equation (62) u "g(u2#u2)u . x xx tt The restriction is the conservative central force problem (54). We obtain a four-parameter family of solutions (9). If g'0 then equation (62) is hyperbolic and the central force of equation (54) is attractive. Bound orbits of equation (54) with non-zero angular momentum are bounded non-separable solutions of equation (62) defined for all t. Circular orbits are travelling wave solutions. The separable equation (1) defined by an invariant F/u of the homogeneous equation ((a (x)u #) ) )#a (x)u)/u) "0 (63) k k 0 x (a O0) is not compatible with the characteristic condition. Solutions of equation (63) are k the non-zero solutions of all equations a (x)u #) ) )#a (x)u"iu (64) k k 0 with i constant. The restriction of equation (1) to the solution space of equation (64) yields a 2k-parameter family of characteristic solutions. The (2k#1)-parameter family thus obtained accounts for all non-zero characteristic solutions of equation (1). They are separable if k"1, and generically non-separable if k*2. Equation (39) has form (63), so the characteristic solutions of equation (41) are those which satisfy equation (40) for some i. We obtain the five-parameter family u(x, t)"a#bt#cx#dxt, u(x, t)"a cosh ux cosh ut#b cosh ux sinh ut#c sinh ux cosh ut#d sinh ux sinh ut, u(x, t)"a cos ux cos ut#b cos ux sin ut#c sin ux cos ut#d sin ux sin ut, of characteristic solutions of the linear wave equation (42). The characteristic solutions of equation (43) are functions u(x, t)"w (t)#w (t)x, 1 2
A B
A B
w A w 1 "!w2 1 , 2 w w 2 2
functions u(x, t)"w (t) cosh ux#w (t) sinh ux 1 2 with
A B
A B
w A w 1 "u2(1#w2!w2) 1 2 w 1 w 2 2
(65)
u(x, t)"w (t) cos ux#w (t) sin ux 1 2
(66)
and
1026
P. W. Doyle
with
A B
A B
w A w 1 "!u2(1#w2#w2) 1 . 2 w 1 w 2 2 The effective potential for the radial component of the conservative system (67) is
(67)
»(r)"u2(r2/2#r4/4)#¸2/2r2, where the arbitrary constant ¸ represents angular momentum. All solutions of the radial equation rA#»@(r)"0 are bounded. Hence the solutions (66) of equation (43) are defined for all t and are bounded. They need not be periodic in t (Bertrand’s theorem [13]). The travelling wave solutions u(x, t)"Jc2!1 cos u(x!x !ct) 0 are circular orbits of equation (67), and the travelling waves u(x, t)"$Jc2!1 cosh u(x!x !ct), 0 are hyperbolic orbits of equation (65).
u(x, t)"J1!c2 sinh u(x!x !ct) 0
R EFE RE NC ES 1. E. G. Kalnins and W. Miller Jr., Differential-Sta¨ckel matrices. J. Math. Phys. 26, 1560—1565 (1985). 2. P. W. Doyle, Separation of variables for scalar evolution equations in one space dimension. J. Phys. A 29, 7581—7595 (1996). 3. V. A. Galaktionov and S. A. Posashkov, Exact solutions and invariant spaces for non-linear gradient diffusion equations. Comput. Maths Math. Phys. 34, 313—321 (1994). 4. V. A. Galaktionov, Invariant subspaces and new explicit solutions to evolution equations with quadratic non-linearities. Proc. Roy. Soc. Edin. 125 A, 225—246 (1995). 5. S. R. Svirshchevskii, Lie-Ba¨cklund symmetries of linear ODEs and generalized separation of variables in non-linear equations. Phys. ¸ett. A 199, 344—348 (1995). 6. S. R. Svirshchevskii, Invariant linear spaces and exact solutions of non-linear evolution equations. Non-linear Math. Phys. 3, 164—169 (1996). 7. W. Miller Jr. and L. A. Rubel, Functional separation of variables for Laplace equations in two dimensions. J. Phys. A 26, 1901—1913 (1993). 8. P. W. Doyle and P. J. Vassiliou, Separation of variables for the 1-dimensional non-linear diffusion equation. Int. J. Non-¸inear Mech. 33, 315—326 (1998). 9. G. W. Bluman and S. Kumei, Symmetries and Differential Equations. Springer, New York (1989). 10. P. J. Olver, Applications of ¸ie Groups to Differential Equations. Springer, New York (1993). 11. R. L. Anderson and N. H. Ibragimov, ¸ie-Ba( cklund ¹ransformations in Applications. SIAM, Philadelphia (1979). 12. P. J. Olver, Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995). 13. V. I. Arnold, Mathematical Methods of Classical Mechanics. Springer, New York (1978). 14. A. V. Samokhin, Symmetries of linear ordinary differential equations. Amer. Math. Soc. ¹ransl. 167, 193—206 (1995).