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Invariant critical sets of conserved quantities
Chaos, Solitons & Fractals 44 (2011) 693–701
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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Invariant critical sets of conserved quantities Petre Birtea ⇑, Dan Coma˘nescu Department of Mathematics, West University of Timisßoara, Bd. V. Pârvan No. 4, 300223 Timisßoara, Romania
a r t i c l e
i n f o
Article history: Received 18 May 2010 Accepted 5 June 2011 Available online 26 July 2011
a b s t r a c t For a dynamical system we will construct various invariant sets starting from its conserved quantities. We will give conditions under which certain solutions of a nonlinear system are also solutions for a simpler dynamical system, for example when they are solutions for a linear dynamical system. We will apply these results to the example of Toda lattice. 2011 Elsevier Ltd. All rights reserved.
1. Introduction A particle moving in the Newtonian gravitational field is known as the Kepler problem. The equations of motion are
€x ¼
x kxk
3
;
where x 2 R3 n f0g is the position vector. It is well known that the motions for the Kepler problem are planar. If we consider the plane Ox1x2, the equations of motion becomes
8 x_1 ¼ y1 > > > > > < x_2 ¼ y2 y_1 ¼ ðx2 þxx12 Þ3=2 > > 1 2 > > > : y_2 ¼ 2 x22 3=2 ðx þx Þ 1
2
These equations are Hamiltonian with the standard symplectic form on R4 and the Hamiltonian function H ¼ 2 1 1 y1 þ y22 pffiffiffiffiffiffiffiffiffi . From Kepler’s second law we have an2 2 2 x1 þx2
other conserved quantity given by A = x1y2 x2y1. For a > 0, consider the following conserved quantity K ¼ H þ a13 A. After a straightforward computation we will obtain that rK = 0 if and only if y1 ¼ a13 x2 , and y2 ¼ a13 x1 and k(x1, x2)k = a2. Equivalently, the set {rK = 0} is equal with
the
set
fðx1 ; x2 ; y1 ; y2 Þjðx1 ; x2 Þ ðy1 ; y2 Þ ¼ 0,
and
kðx1 ; x2 Þk ¼ a2 , and kðy1 ; y2 Þk ¼ 1a and sgnðy1 Þ ¼ sgnðx2 Þg ⇑ Corresponding author. E-mail address: [email protected] (P. Birtea). 0960-0779/$ - see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2011.06.009
which is invariant under the dynamics and is filled with solutions that represent uniform circular motions moving clockwise. These particular motions are also solutions for the linear Hamiltonian system
8 x_1 ¼ x2 > > > < x_2 ¼ x1 > y > _1 ¼ y2 > : y_2 ¼ y1 ; where the Hamiltonian function is A. Given the above analysis, we can raise at least two questions. Is it true that for a dynamical system that admits conserved quantities, the set of points where the gradients of these conserved quantities are zero, is an invariant set? When do two Hamiltonian systems have common solutions? In what follows, we will give an answer to these questions. For the first question, the answer is positive and it is given in section two where we will also discuss various generalizations of this answer. In section three, we will present the conditions under which we can give an answer to the second question. In section four we will illustrate these results for the example of Toda lattice. Detailed computations are presented in the Appendix. 2. Invariant sets Let f : Rn ! Rn be a Cq, q P 1 function which generates the differential equation
x_ ¼ f ðxÞ:
ð2:1Þ
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701
694
We suppose that Eq. (2.1) admits a Cq vectorial conserved quantity F : Rn ! Rk with k 6 n. We denote by F1, . . ., Fk the components of F. For s 2 {0, . . ., k} we introduce the sets:
MFðsÞ
n
¼ fx 2 R jrankrFðxÞ ¼ sg
ð2:2Þ
where rF(x) is the Jacobian matrix
0 @F 1 @x1
ðxÞ
rFðxÞ ¼ B @ @F k @x1
@F 1 @xn
Theorem 2.5. The sets N FðrÞ are invariant sets for the dynamics generated by the differential Eq. (2.1). Proof. Let i 2 {1, . . ., k}, l 2 {1, . . ., q} and a = (a1, . . ., al) 2 {1, . . ., n}l. We will prove by mathematical induction that
1
C A: ðxÞ
ðxÞ
ðxÞ
Corollary 2.4. The set of critical points of F is an invariant set of the dynamics generated by the differential Eq. (2.1).
ð2:3Þ
n
ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ@ b F i ðUt ðxÞÞ
b1 ;...;bl ¼1
For r 2 {1, . . ., q} we introduce the sets:
NFðrÞ
n X
@ a F i ðxÞ ¼
@F k @xn
þ Si;a ðt; xÞ;
a
ð2:8Þ
¼ fx 2 R j@ F i ðxÞ ¼ 0; 8i 2 f1; . . . ; kg;
8a 2 f1; . . . ; ngl ; 8l 6 rg
ð2:4Þ l
Fi where we note @ a F i ¼ @xa @ @x if a = (a1, . . ., al). a 1
l
Remark 2.1. We observe that
@F i MFð0Þ ¼ NFð1Þ ¼ x 2 Rn j ðxÞ ¼ 0; i 2 f1; . . . ; kg; j 2 f1 . . . ng : @xj The set
n
M FðsÞ
o
ð2:5Þ with s 2 {0, . . ., k} is a partition of Rn . A
critical point of F is a point in Rn at which the rank of the matrix rF(x) is less than the maximum rank. A critical value is the image under F of a critical point. The set of critical points of F is k1
MFc ¼ [ MFðsÞ : s¼0
ð2:6Þ
Using Sard’s Theorem (see [9]) we have that the set of crit ical values F M Fc is of k-dimensional measure zero provid-
where b = (b1, . . ., bl) and Si,a(t, x) is a sum with the property: ‘‘all the terms contain a factor of the form @ cFi(Ut(x)) with jcj < l’’. We will prove this result by mathematical induction with respect to l. Componentwise the relation (2.7) implies our result for l = 1. Let a0 = (a1, . . ., al, al+1) 2 {1, . . ., n}l+1 where a = (a1, . . ., al) 2 {1, . . ., n}l. Using the induction hypothesis we have: ! n X @ a0 b @ F i ðxÞ ¼ ðrUt Þa1 b1 ðxÞ . . . ðrUt Þal bl ðxÞ@ F i ðUt ðxÞÞ @xalþ1 b ;...;b ¼1 1
By a straightforward computation we obtain
!
n X
@ @xalþ1
ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ@ b F i ðUt ðxÞÞ
b1 ;...;bl ¼1 n X
¼
ing that q P n k + 1.
l
@ ðSi;a ðt; xÞÞ: þ @xalþ1
ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ
b1 ;...;bl ¼1
Remark 2.2. We also have the obvious inclusions N FðqÞ # N Fðq1Þ # # N Fð1Þ . Theorem 2.3. The sets M FðsÞ are invariant under the dynamics generated by the differential Eq. (2.1).
@ @ b F i ðUt ðxÞÞ @xalþ1
@ ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ @ b F i ðUt ðxÞÞ @xalþ1 ;;b ¼1 n X
þ b1
l
n X
¼
ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ
b1 ;;bl ;blþ1 ¼1 0
ðrUt Þalþ1 blþ1 ðxÞ@ b F i ðUt ðxÞÞ
Proof. Because F is a conserved quantity, we have
FðUt ðxÞÞ ¼ FðxÞ; where Ut : Rn ! Rn is the flow generated by (2.1). Differentiating, we have
rFðUt ðxÞÞrUt ðxÞ ¼ rFðxÞ:
ð2:7Þ
As rUt(x) is an invertible matrix for any x 2 Rn which is not an equilibrium point for (2.1), (see [1]), we have that
rankrFðUt ðxÞÞ ¼ rankrFðxÞ; which implies the stated result. h As a consequence we will obtain the following well known result which was applied for studying invariant sets of various mechanical systems, see for example [6].
@ ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ @ b F i ðUt ðxÞÞ: @xalþ1 ;;b ¼1 n X
þ b1
l
We will note that
Si;a0 ðt; xÞ ¼
n X b1 ;...;bl ¼1
@ ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ @xalþ1
@ b F i ðUt ðxÞÞ þ
@ ðSi;a ðt; xÞÞ: @xalþ1
All the terms of Si;a0 ðt; xÞ contain a factor of the form @ cFi(Ut(x)) with jcj < l + 1, which had to be proved. Let b = (b1, . . ., bl) 2 {1, . . ., n}l and rU1 t ðxÞ be the inverse matrix of rUt(x), where x is not an equilibrium point for (2.1). Consequently, we have
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701 n X
@ b F i ðUt ðxÞÞ ¼
Another particular case is when the function f in (3.1) and (3.2) verifies the equality
1 ðrUt Þ1 b1 a1 ðxÞ ðrUt Þbl al ðxÞ
a1 ;...;al ¼1 a
@ F i ðxÞ Si;a ðt; xÞ :
695
ð2:9Þ
Also, we will prove by mathematical induction that the sets N FðjÞ are invariant under the dynamics generated by the differential Eq. (2.1). For j = 1 we have N Fð1Þ ¼ M Fð0Þ , which is an invariant set (see Theorem 2.3). We suppose that for all j 6 l the sets N FðjÞ are invariant. Let x 2 N Fðlþ1Þ be arbitrary chosen. Using (2.9) for l + 1-order of derivation and the induction hypothesis, we deduce that Ut ðxÞ 2 N Fðlþ1Þ ; 8t. Summing up, N FðjÞ are invariant sets for all j 2 {1, . . ., q}. h
< f ðx; yÞ; y >¼ 0; where< , > is the Euclidian product on Rn . In this case F is a conserved quantity for (3.1) and G is a conserved quantity for (3.2). The above equality is verified for the case when
f ðx; yÞ ¼ PðxÞy; where P(x) is an antisymmetric matrix. This is true for all almost Poisson manifolds (see [8]). In this situation the differential Eqs. (3.1) and (3.2) become
x_ ¼ PðxÞrFðxÞ and
3. Finding solutions using simpler dynamics Let F; G : Rn ! R be Cq functions and the differential equations:
x_ ¼ f ðx; rFðxÞÞ
ð3:1Þ
and
x_ ¼ f ðx; rGðxÞÞ 2n
ð3:2Þ n
q
where f : R ! R is a C vectorial function. For x 2 Rn , we denote with UFt ðxÞ and UGt ðxÞ the solutions of (3.1) and (3.2) with the initial conditions UF0 ðxÞ ¼ x and UG0 ðxÞ ¼ x. We will introduce the following set
E1 ¼ fx 2 Rn jrFðxÞ ¼ rGðxÞg: Theorem 3.1. If F G is a conserved quantity for (3.1) and x 2 E1, then for all t we have
UFt ðxÞ ¼ UGt ðxÞ: Proof. For L = F G we have the equality E1 ¼ MLð0Þ . By Theorem (2.3) the set E1 is invariant under the dynamics of (3.1). For x 2 E1 we have rF UFt ðxÞ ¼ rG UFt ðxÞ for all t and consequently
d F U ðxÞ ¼ f ðUFt ðxÞ; rF UFt ðxÞ ¼ f ðUFt ðxÞ; rG UFt ðxÞ : dt t
ð3:6Þ
We observe that F is a conserved quantity for (3.6) if and only if G is a conserved quantity for (3.5). Corollary 3.3. If G is a conserved quantity for (3.5), then for the initial conditions in fx 2 Rn jrFðxÞ ¼ rGðxÞg we have UFt ðxÞ ¼ UGt ðxÞ. A particular case of Corollary 3.3 is when we have a symplectic manifold with G being a quadratic function. In this case, the solutions of the Hamiltonian vector field XF starting in fx 2 Rn jrFðxÞ ¼ rGðxÞg are also the solutions of the linear Hamiltonian system XG. Analogous results are valid for the more general case of the vector valued conserved quantities and when the right side of Eqs. (3.1) and (3.2) depends on higher order derivatives. Firstly, we will introduce some notations. Let F; G : Rn ! Rk be Cq vectorial functions with q P 1 and also k 6 n. If F1, . . ., Fk are the components of F we denote with
D1 FðxÞ ¼
@F 1 @F 1 @F 2 @F 2 ðxÞ; . . . ; ðxÞ; ðxÞ; . . . ; ðxÞ; . . . ; @x1 @xn @x1 @xn @F k @F k ðxÞ; . . . ; ðxÞ 2 Rkn : @x1 @xn
r
Next we will discuss a particular case of the result presented above. For this we will take f(x, y) = h(x) + g(y) and F 0. Thus we have the two dynamics
ð3:3Þ
and the perturbed dynamics
x_ ¼ hðxÞ þ gðrGðxÞÞ:
x_ ¼ PðxÞrGðxÞ:
and
The above equality shows that UFt ðxÞ is also a solution for (3.2). Given the uniqueness of the solutions, we obtain the desired equality. h
x_ ¼ hðxÞ
ð3:5Þ
Dr FðxÞ ¼ ð. . . ; @ a F i ðxÞ; . . .Þ 2 Rkn ; where r 2 {1, . . ., q}, i 2 {1, . . ., k}, a 2 {1, . . ., n}r (jaj = r) and the components appear in the lexicographical order for (i, a) in Nrþ1 . For a fix r 2 {1, . . ., q}, we will consider, as before, the two differential equations
x_ ¼ fr ðx; D1 FðxÞ; . . . ; Dr FðxÞÞ
ð3:7Þ
and
ð3:4Þ
We denote by Uht the flux for (3.3).
x_ ¼ fr ðx; D1 GðxÞ; . . . ; Dr GðxÞÞ:
ð3:8Þ
Corollary 3.2. If G is a conserved quantity for (3.3), then for all initial conditions x 2 fx 2 Rn jrGðxÞ ¼ 0g we have
Let x 2 Rn , we denote with UFt ðxÞ and UGt ðxÞ the solutions of (3.7) and (3.8) with initial conditions UF0 ðxÞ ¼ x and UG0 ðxÞ ¼ x. We will introduce the following set
Uht ðxÞ ¼ UGt ðxÞ:
Er ¼ fx 2 Rn j@ a FðxÞ ¼ @ a GðxÞ; 8jaj 6 rg:
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701
696
Theorem 3.4. If F G is a conserved quantity for (3.7) and x 2 Er then for all t we have
UFt ðxÞ ¼ UGt ðxÞ: Also the obvious extension of Corollary 3.2 takes place.
(i) The indices i1, . . ., ik, j1, j1 + 1, . . ., jl, jl + 1, which appear in the term (either explicitly, or implicitly through a factor Xj) are all different (modulo n); (ii) The number of these indices is m, i.e. k + 2l = m. Two terms differing only in the order of factors are not considered different, and therefore only one of them appears in the sum.
The Toda lattice describes the one-dimensional motions of a chain of particles with nearest neighbor interactions. For a chain of particles with the equal masses m, Morikazu Toda came up with the choice of the interaction potential
In [4], Flaschka has proved that the above functions are conserved quantities using a Lax formulation. This was generalized to arbitrary Lie algebras by Adler [2], Kostant [7] and Damianou, Fernandes [3]. The first three scalar conserved quantities, depending on the variables (X1, . . ., Xn, u1, . . ., un), are
VðrÞ ¼ er þ r 1:
I1 ¼
4. Invariant sets for Toda lattices
i 2 Z:
ð4:1Þ
This second order differential system is equivalent with the first order differential system
x_i ¼ ui mu_i ¼ eðxi xi1 Þ eðxiþ1 xi Þ ;
ð4:2Þ
i 2 Z;
ui ¼ 0;
k 2 R ;
x0 2 R;
y_i ¼ ui k u_i ¼ em ðeðyi yi1 Þ eðyiþ1 yi Þ Þ;
i2Z
ð4:3Þ
ek ðyiþ1 yi Þ X i :¼ ; e m
ð4:4Þ
then the equations of motion become
X_ i ¼ X i ðui uiþ1 Þ u_i ¼ X i1 X i ; i 2 Z:
ð4:5Þ
The following particular cases are interesting: 1. The case of a periodic lattice, X iþn ¼ X i 8i 2 Z, 2. The case of a non-periodic lattice with the boundary conditions X0 = 0 (correspond to formally setting y0 = 1) and Xn = 0 (correspond to formally setting yn+1 = 1). In cases we investigate the motions of the particles 1 to n ðn 2 N Þ.
In this case Hénon proves in [5] that the following expressions are scalar conserved quantities
X
ui1 . . . uik ðX j1 Þ . . . ðX jl Þ;
I3 ¼
X
Xj
16j6n
X
ui1 ui2 ui3
16i1
ð4:8Þ ui X j ;
ðX 0 ¼ X n Þ:
16i;j6n;j–i;j–i1ðmodnÞ
ð4:9Þ We will introduce the vectorial conserved quantities,
I. The case n odd For this case we obtain as invariant sets only subsets of the set of equilibrium points or the empty set. 1 2 12 12 13 More precisely, M Ið0Þ ¼ MIð0Þ ¼ M Ið0Þ ¼ M Ið1Þ ¼ MIð0Þ ¼ I23 I123 I123 Mð0Þ ¼ M ð0Þ ¼ M ð1Þ ¼ ;. The following are subsets of the set of equilibrium points:
13 ¼ fðX; . . . ; X; 0; . . . ; 0ÞjX 2 Rg; M Ið1Þ n1 2 n1 2 23 ¼ u ;...; u ; u; . . . ; u ju 2 R ; M Ið1Þ 2 2 123 ¼ fðX; . . . ; X; u; . . . ; uÞjX; u 2 Rg: M Ið2Þ
II. The case n even 1 2 12 12 13 23 We have, M Ið0Þ ¼ MIð0Þ ¼ M Ið0Þ ¼ MIð1Þ ¼ M Ið0Þ ¼ MIð0Þ ¼ I123 I123 Mð0Þ ¼ M ð1Þ ¼ ;. As nontrivial invariant sets we have the following: 3 M Ið0Þ ¼ fðX 1 ; X 2 ; . . .; X 1 ; X 2 ; u1 ; u2 ; . . .; u1 ; u2 ÞjX 1 þ X 2
¼ u1 u2 ; u1 þ u2 ¼ 0g; 13 M Ið1Þ ¼ fðX 1 ; X 2 ; .. . ; X 1 ; X 2 ; u1 ; u2 ; .. . ; u1 ;u2 Þju1 þ u2 ¼ 0g; 23 ¼ fðX 1 ; X 2 ; .. . ; X 1 ; X 2 ; u1 ; u2 ; .. . ; u1 ;u2 ÞjX 1 þ X 2 M Ið1Þ o n ¼ ðu1 þ u2 Þ2 þ u1 u2 ; 4 123 M Ið2Þ ¼ fðX 1 ; X 2 ; . . . ;X 1 ;X 2 ;u1 ; u2 ; . . . ; u1 ; u2 ÞjX 1 ; X 2 ; u1 ; u2 2 Rg:
123 We obtain M Ið2Þ as the largest invariant set and the restricted dynamics is the dynamics of two particles
4.1. The case of a periodic lattice
Im ¼
ui1 ui2
16i1
3 M Ið0Þ ¼ fð0; . . . ; 0; 0; . . . ; 0Þg;
Let us define
(
ð4:7Þ X
ð4:10Þ
i 2 Z:
Let an equilibrium of the Toda lattice and yi = xi x0 ki the displacement of the i particle from its equilibrium position. The system in the variables yi and ui is
(
I2 ¼
X
I12 ¼ ðI1 ; I2 Þ; I13 ¼ ðI1 ; I3 Þ; I23 ¼ ðI2 ; I3 Þ; I123 ¼ ðI1 ; I2 ; I3 Þ:
where ui is the velocity of the particle i. An equilibrium of the Toda lattice has the form
xi ¼ x0 þ ki;
ui
16i6n
The system of motion reads explicitly
mx€i ¼ eðxi xi1 Þ eðxiþ1 xi Þ ;
X
ð4:6Þ
where m 2 {1, . . ., n} and the summation are extended to all terms which satisfy the following conditions:
8 _ > > > X 1 ¼ X 1 ðu1 u2 Þ > <_ X 2 ¼ X 2 ðu2 u1 Þ > u_ 1 ¼ X 2 X 1 > > > :_ u2 ¼ X 1 X 2
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701 23 On the invariant set M Ið1Þ we have the above dynamics subject to X 1 þ X 2 ¼ 4n ðu1 þ u2 Þ2 þ u1 u2 . On the invariant set 13 M Ið1Þ we have the above dynamics subject to u1 + u2 = 0 3 and on the invariant set MIð0Þ we have the above dynamics subject to u1 + u2 = 0 and X1 + X2 = u1u2. 3 23 For the sets MIð0Þ and M Ið1Þ the variables Xi have to take also negative values which from a mathematical point of view is correct and can be the solutions for the system (4.5). As the mechanical system is given by (4.3) and we do the change of variables (4.4), the variables Xi have to be strictly positive in order to have a physical meaning. Consequently, from a mechanical point of view only the 13 123 sets MIð1Þ and M Ið2Þ are meaningful. The computations can be found in the Appendix.
It is known that if we have the matrices
1 X1 0 0 0 C u2 X 2 C C C and C 1 un1 X n1 A 0 1 un 1 X 1 0 0 0 C 0 X 2 C C C; C 0 0 X n1 A 0 0 0
u1 B 1 B B L ¼ B B @ 0 0 0 0 B 0 B B B ¼ B B @ 0 0
k 2 f1; . . . ; ng:
F2 ¼
Xi þ
i¼1
F3 ¼
n1 X
u2i 2
123 M Fð2Þ
n1
23 M Fð1Þ
i¼1
123 M Fð2Þ
ð4:12Þ
n 1X u3 : 3 i¼1 i
We will introduce the vectorial conserved quantities,
F12 ¼ ðF 1 ; F 2 Þ;
F13 ¼ ðF 1 ; F 3 Þ;
F23 ¼ ðF 2 ; F 3 Þ;
F123 ¼ ðF 1 ; F 2 ; F 3 Þ:
n
8 9 < = ðX;0; .. . ;X; 0; X ;u1 ; u2 ; .. . ; u1 ;u2 Þju1 þ u2 ¼ 0; X ¼ u1 u2 ; : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ;
ð4:13Þ
As before we will distinguish two cases. I. The case n odd In this case we obtain as invariant sets only subsets of the set of equilibrium points or the empty set. 1 2 12 12 13 More precisely, M Fð0Þ ¼ MFð0Þ ¼ M Fð0Þ ¼ M Fð1Þ ¼ M Fð0Þ ¼ 23 123 123 MFð0Þ ¼ MFð0Þ ¼ MFð1Þ ¼ ;. The following are subsets of the set of equilibrium points:
n
8 9 < = ¼ ðX;0; .. . ; X; 0; X ;u1 ;u2 ; . .. ; u1 ; u2 Þju1 þ u2 ¼ 0 ; : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ; n
8 9 < = ¼ ðX;0; .. . ; X; 0; X ;u1 ;u2 ; . .. ; u1 ; u2 ÞjX ¼ u1 u2 ; |fflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflffl ffl } |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} : ; n1
X i ðui þ uiþ1 Þ þ
n
n1
8 9 < = ¼ ð0; . . . ; 0; u1 ; u2 ; . . . ; u1 Þju1 ; u2 2 R : : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} ;
n1
n X
ui i¼1 n X
n
n1
8 9 < = [ ð0; . . . ; 0; 0; u2 ; . . . ; 0; u2 ; 0Þju2 2 R ; : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ;
13 M Fð1Þ
For k 2 {1, 2, 3} we have
F1 ¼
23 M Fð1Þ
n
8 9 < = ¼ ðX;0; .. . ;X; 0; X ;u1 ; u2 ; . .. ; u1 ;u2 ÞjX; u1 ;u2 2 R : : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ; n1
ð4:11Þ
1 trðLk Þ; k
n
n1
n1
where [B, L] = BL LB. Using the Flaschka theorem (see [4]) we have the following scalar conserved quantities depending on the variables (X1, . . ., Xn1, u1, . . ., un)
Fk ¼
8 9 < = ¼ ð0; . . . ; 0; 0; . . . ; 0Þ ; : |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} ;
8 9 < = ¼ ð0; . . . ; 0; u1 ; 0; . . . ; u1 ; 0; u1 Þju1 2 R : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ;
3 M Fð0Þ ¼
then the system (4.5), in this case, has the Lax form
L_ ¼ ½B; L;
¼
13 M Fð1Þ
II. The case n even 1 2 12 12 13 23 We have, M Fð0Þ ¼ MFð0Þ ¼ MFð0Þ ¼ MFð1Þ ¼ M Fð0Þ ¼ MFð0Þ ¼ F123 F123 Mð0Þ ¼ Mð1Þ ¼ ;. As nontrivial invariant sets we have the following:
4.2. The case of non-periodic lattice
0
3 M Fð0Þ
697
n
123 We obtain M Fð2Þ as the largest invariant set and the restricted dynamics is given by
8 _ > > < X ¼ Xðu1 u2 Þ u_ 1 ¼ X > > : u_ 2 ¼ X 23 On the invariant set M Fð1Þ we have the above dynamics sub13 ject to X = u1u2. On the invariant set M Fð1Þ we have the above dynamics subject to u1 + u2 = 0 and on the invariant 3 set M Fð0Þ we have the above dynamics subject to u1 + u2 = 0 and X = u1u2. 123 For the set M Fð2Þ the variables Xi with i even are all equal with zero which from a mathematical point of view is correct. As before, the mechanical system is given by (4.3) and we do the change of variables (4.4). Consequently, the variables Xi have to be strictly positive in order to have a physical meaning. The computations can be found in the Appendix.
Acknowledgments Petre Birtea has been supported by CNCSIS UEFISCSU, project number PN II - IDEI 1081/2008.
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701
698
8 > < U uk ukþ1 ¼ 0 V uk ðU uk Þ ðY X k1 X k Þ ¼ V uq ðU uq Þ > : ðY X q1 X q Þ:
Appendix A A.1. The computations for the case of periodic lattice
ð5:7Þ
We will make the notations
U¼
X
ui ;
X
V¼
16i6n
ui1 ui2 ;
Y¼
16i1
X
Xj:
ð5:1Þ
16j6n
We observe that
2V ¼ U 2
n X
u2i :
ð5:2Þ
i¼1
13 MIð1Þ ¼ fðX; . . . ; X; 0; . . . ; 0ÞjX 2 Rg:
With these notations we have the following,
rI1 ¼ ð0; . . . ; 0; 1; . . . ; 1Þ 0
1
rI2 ¼ @1; . . . ; 1; U u1 ; . . . ; U uk ; . . . ; U un A |fflfflffl{zfflfflffl} nþ1
0
|fflfflffl{zfflfflffl} nþk
Adding the first n equations we obtain U = 0, u1 + u2 = u2 + u3 = = un1 + un = un + u1 and consequently ui ¼ ð1Þiþ1 u; u 2 R. The last n equations become X1 + X2 = X2 + X3 = = Xn1 + Xn = Xn + X1. The case n 2 2N þ 1. For this case note that u1 = u2 = = un = 0 and X 1 ¼ . . . ¼ X n ¼ X 2 R. In this case we have
|fflfflffl{zfflfflffl}
ð5:3Þ
The case n 2 2N. It is easy to see that X1 = X3 = = Xn1, X2 = X4 = = Xn. In this case we have
ð5:4Þ
13 MIð1Þ ¼ fðX 1 ; X 2 ; . . . ; X 1 ; X 2 ; u1 ; u2 ; . . . ; u1 ; u2 Þju1 þ u2 ¼ 0g:
2n
23 A point belongs to the set M Ið1Þ if and only if
rI3 ¼ B @. . . ; ðU uk ukþ1 Þ; . . . ;
detðAkq Þ ¼ 0; detðBkq Þ ¼ 0; detðC kq Þ ¼ 0; 8k; q 2 f1; . . . ; ng;
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} k
ð5:8Þ
1
C V uk ðU uk Þ ðY X k1 X k Þ; . . .A: |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
where
Akq ¼
nþk
3 M Ið0Þ .
The study of 3 The elements of MIð0Þ are the solutions of the algebraic system
ð5:5Þ
Bkq ¼
U uk ukþ1 ¼ 0; V uk ðU uk Þ ðY X k1 X k Þ ¼ 0; 8k 2 f1; . . . ; ng:
C kq ¼
1
1
ðU uk ukþ1 Þ ðU uq uqþ1 Þ
1 U uq ðU uk ukþ1 Þ V uq ðU uq Þ ðY X q1 X q Þ
U uq
U uk
V uk ðU uk Þ ðY X k1 X k Þ V uq ðU uq Þ ðY X q1 X q Þ
:
ð5:6Þ Adding the first n equations, we obtain U = 0 and u1 + u2 = u2 + u3 = = un1 + un = un + u1. We deduce the following results: The case n 2 2N þ 1. In this situation we have u1 = u2 = = un = 0. Adding the last n equations of (5.6) we obtain Y = 0 and X1 + X2 = X2 + X3 = = Xn1 + Xn = Xn + X1. It implies that X1 = = Xn = 0 and consequently
The case n 2 2N. For this situation ui ¼ ð1Þiþ1 u; u 2 R. In this case, using (5.2), we have V ¼ n2 u2 . The last n equations of (5.6) become
n 2
1 u2
8k 2 f1; . . . ; ng:
Making the addition we have Y ¼ n2 u2 and consequently X1 + X2 = X2 + X3 = = Xn1 + Xn = Xn + X1 = u2 which implies X1 = X3 = = Xn1, and X2 = X4 = = X2n. In this case we have 3 MIð0Þ ¼ fðX 1 ; X 2 ; . . . ; X 1 ; X 2 ; u1 ; u2 ; . . . ; u1 ; u2 ÞjX 1 þ X 2
¼ u1 u2 ; u1 þ u2 ¼ 0g: I
ðn 1Þu
1
Bkq ¼
ðn 2Þu
ðn2Þðn1Þ 2 u 2
!
ðY X q X q1 Þ
:
Using the condition that det(Bkq) = 0, we obtain
3 MIð0Þ ¼ fð0; . . . ; 0; 0; . . . ; 0Þg:
Y þ X k1 þ X k ¼
Using the expression of Akq we deduce that u1 + u2 = u2 + u3 = = un1 + un = un + u1. The case n 2 2N þ 1. In this case we have u1 = u2 = = un = u, and U = nu, and V ¼ ðn1Þn u2 and 2
Y X q X q1 ¼
8q 2 f1; . . . ; ng:
Adding these relations we have
ðn 1Þn 2 u ; 2 n1 2 u : X¼ 2
Y¼
and X 1 ¼ X 2 ¼ ¼ X n ¼ X
and
With this relation we have
C kq ¼
ðn 1Þu ðn 2Þðn 1Þu2
: ðn 2Þðn 1Þu2 ðn 1Þu
We observe that the equality det(Ckq) = 0 is verified. In conclusion we have
I
ij ij The study of M ð0Þ ; M ð1Þ with (i, j) 2 {(1, 2), (1, 3), (2, 3)}. 13 A point ðX 1 ; . . . ; X n ; u1 ; . . . ; un Þ 2 M Ið1Þ if and only if we have, for all k, q 2 {1, . . ., n},
ðn 2Þðn 1Þ 2 u ; 2
23 MIð1Þ ¼
n1 2 n1 2 u ;...; u ; u; . . . ; u ju 2 R : 2 2
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701
The case n 2 2N. In this case u1 = u3 = = un1, u2 = u4 = = un and we have U ¼ 2n ðu1 þ u2 Þ. Using (5.2) we obtain 2 2 V ¼ nðn2Þ u1 þ u22 þ n4 u1 u2 . From the relation det(Bkq) = 0, 8 we have
Y X q1 X q ¼ V ðU uq Þ
n2 U þ uq : n
ð5:9Þ
Consequently, X1 + X2 = X3 + X4 = = Xn1 + Xn and X2 + X3 = X4 + X5 = = Xn + X1 which further implies that X 1 ¼ X 3 ¼ . . . ¼ X n1 ; X 2 ¼ X 4 ¼ . . . ¼ X n and Y ¼ n2 ðX 1 þ X 2 Þ. By substitution into (5.9) we obtain X 1 þ X 2 ¼ n4 ðu1 þ u2 Þ2 þ u1 u2 . We have
B Akqr ¼ @
0
1
1
C U ur A
ðU uk ukþ1 Þ ðU uq uqþ1 Þ 0 B Bkqr ¼ @
0
1
1
U uq
ðU uk ukþ1 Þ 0
1
B C kqr ¼ @ U uk Vk
1 U uq Vq
1
Vr
A.2. The computations for the case of non-periodic lattice If we consider the variables (X1, . . ., Xn1, u1, . . ., un) we obtain
0
1
1
rF 1 ¼ @0; . . . ; 0; 1; . . . ; 1A
C U ur A
Vq 1
1
1
(i) detB111 = detB112 = 0 (by calculus). (ii) det Bkqr = det Bkrq. (iii) If q1 q2 2 2N and r 1 r 2 2 2N then detBk1 q1 r1 ¼ det Bk2 q2 r2 .
123 MIð2Þ ¼ fðX 1 ; X 2 ; . . . ; X 1 ; X 2 ; u1 ; u2 ; . . . ; u1 ; u2 ÞjX 1 ; X 2 ; u1 ; u2 2 Rg:
123 The study of M Ið2Þ . We introduce, for k, q, r 2 {1, . . ., n}, the matrices
0
If q = k + 1 and r = k + 2 we obtain Vk = Vk+2, which implies that X1 + X2 = X3 + X4 = = Xn1 + Xn and X2 + X3 = X4 + X5 = = Xn + X1. Consequently, we have X1 = X3 = = Xn1 and X2 = X4 = = Xn. We observe that if s t 2 2Z, then us = ut and Vs = Vt, which implies that det(Ckqr) = 0 " k, q, r 2 {1, . . ., n}. For the matrices Bkqr we have the following properties:
Using these properties we deduce that det Bkqr = 0, "k, q, r 2 {1, . . ., n} and we obtain
23 M Ið1Þ ¼ fðX 1 ; X 2 ; . . . ; X 1 ; X 2 ; u1 ; u2 ; . . . ; u1 ; u2 ÞjX 1 þ X 2 o n ¼ ðu1 þ u2 Þ2 þ u1 u2 : 4
0
0
Vr
n
n1
1
rF 2 ¼ @1; . . . ; 1; u1 ; . . . ; un A
1
ð5:11Þ
|fflfflfflffl{zfflfflfflffl}
C U ur A;
n1
rF 3 ¼ u1 þ u2 ; . . . ; un1 þ un ; X 1 þ u21 ; X 2 þ X 1
þ u22 ; . . . ; X n2 þ X n1 þ u2n1 ; X n1 þ u2n :
where Vs = V us(U us) (Y Xs1 Xs). We observe that rankðrI123 Þ ¼ 2 () detðAkqr Þ ¼ detðBkqr Þ ¼ detðC kqr Þ ¼ 0 8k; q; r 2 f1; . . . ; ng. The equations det (Akqr) = 0 "k, q, r 2 {1, . . ., n} give us
u1 þ u2 ¼ u2 þ u3 ¼ ¼ un1 þ un ¼ un þ u1 : The case n 2 2N þ 1. We have u:¼u1 = = un. With this notation we obtain U = nu, and V ¼ nðn1Þ u2 and V k ¼ 2 2 nðn1Þ u ðY X X Þ. It is easy to see that det(Ckqr) = k1 k 2 0. We have the equivalences
0
0 0 B 0 detðBkqr Þ ¼ 0 () det @ 1 Vk Vq Vr
1
1 C ðn 1Þu A ¼ 0 Vr
() V r ¼ V q () X r1 þ X r ¼ X q1 þ X q : Because n 2 2N þ 1, we obtain X1 = X2 = = Xn. In this case we have 123 M Ið2Þ ¼ fðX; . . . ; X; u; . . . ; uÞjX; u 2 Rg:
8 > < u1 þ u2 ¼ u2 þ u3 ¼ ¼ un1 þ un ¼ 0 X 1 þ u21 ¼ X 2 þ X 1 þ u22 ¼ ¼ X n2 þ X n1 þ u2n1 > : ¼ X n1 þ u2n ¼ 0: The case n 2 2N þ 1. We obtain 3 M Fð0Þ
8 9 < = ¼ ð0; . . . ; 0; 0; . . . ; 0Þ : : |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} ; n
n1
The case n 2 2N. In this situation u1 = u3 = = un1 = u, u2 = u4 = = un = u and Xk = u2 if k 2 2N þ 1 and Xk = 0 if k 2 2N. We have 8 9 < = F3 Mð0Þ ¼ ðX; 0; . . . ; X; 0; X ; u1 ; u2 ; . . . ; u1 ; u2 Þju1 þ u2 ¼ 0; X ¼ u1 u2 : : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ; F
0 uk uq Vq Vk
0
1
C uk ur A ¼ 0: Vr Vk
ð5:12Þ
3 The study of M Fð0Þ . 3 The elements of M Fð0Þ are the solutions of the system
n
n1
The case n 2 2N. We have u1 = u3 = = un1, u2 = u4 = = un. By calculus we obtain
1 B detðC kqr Þ ¼ 0 () det @ U uk Vk
ð5:10Þ
|fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl}
Vr
0
699
F
ij ij The study of M ð0Þ ; M ð1Þ with (i, j) 2 {(1, 2), (1, 3), (2,3)}. 13 The elements of M Fð1Þ are the solutions of the system
8 > < u1 þ u2 ¼ u2 þ u3 ¼ ¼ un1 þ un ¼ 0 X 1 þ u21 ¼ X 2 þ X 1 þ u22 ¼ ¼ X n2 þ X n1 þ u2n1 > : ¼ X n1 þ u2n :
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701
700
123 The study of M Fð2Þ .
The case n 2 2N þ 1. We obtain that 13 MFð1Þ
9 8 = < ¼ ð0; . . . ; 0; 0; . . . ; 0Þ : : |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} ;
123 we have For a point of the set M Fð2Þ
detðAkqr Þ ¼ 0;
n
n1
The case n 2 2N. In this situation u1 = u3 = = un1 = u, u2 = u4 = = un = u and Xk = X if k 2 2N þ 1 and Xk = 0 if k 2 2N. We have 13 MFð1Þ
8 9 < = ¼ ðX; 0; . . . ; X; 0; X ; u1 ; u2 ; . . . ; u1 ; u2 Þju1 þ u2 ¼ 0 : : |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ;
For a point of the set
detðAkq Þ ¼ 0;
23 M Fð1Þ
where
0 B Akqr ¼ @
detðC kq Þ ¼ 0;
8k; q ð5:13Þ
Akq ¼
1
1
uk þ ukþ1
uq þ uqþ1
;
B Bkqr ¼ @
1
uq
uk þ ukþ1
X q1 þ X q þ u2q
;
!
uq
Using the expression of Akq we deduce that u1 + u2 = u2 + u3 = = un1 + un and consequently uk = uq if k q 2 2Z. The matrices Bkq have the form
1
u1
u1 þ u2
X q1 þ X q þ u21
if q 2 2N þ 1
1
u2
u1 þ u2
X q1 þ X q þ u22
n
The case n 2 2N. We deduce that X1 = X3 = = Xn1 = X, X2 = X4 = = Xn2 = 0 and u1u2 = X. In this case we have 23 MFð1Þ
8 9 < = ¼ ðX; 0; X; . . . 0; X ; u1 ; u2 ; . . . ; u1 ; u2 ÞjX ¼ u1 u2 : : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ; n1
ur u2q
X r1 þ X r þ
C A; ð5:19Þ 1
1
uq
X k1 þ X k þ
u2r
q; r 2 f1; . . . ; ng 1
u2k
1
1
X q1 þ X q þ
ur
X q1 þ X q þ
u2q
X r1 þ X r þ
k; q; r 2 f1; . . . ; ng
u2r
C A;
ð5:20Þ
Using the expression of Akqr we deduce that u1 + u2 = u2 + u3 = = un1 + un and consequently uk = uq if k q 2 2Z. For k 2 {1, . . ., n 1}, we have 0 B det Bk;kþ1;k ¼ det @
0
0
1
1
1
ukþ1 uk
uk
C A¼0
uk þ ukþ1 X kþ1 X k1 þ u2kþ1 u2k X k1 þ X k þ u2k
and we deduce that Xk+1 = Xk1. The case n 2 2N þ 1. It is easy to see that X1 = X2 = = Xn1 = 0, u1 = u3 = = un and u2 = u4 = = un1. All the conditions (5.17) are verified and the set is
8 <
9 = ¼ ð0; . . . ; 0; u1 ; u2 ; . . . ; u1 Þju1 ; u2 2 R : : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} ; n
n1
9 8 = < ¼ ð0; . . . ; 0; u1 ; 0; . . . ; u1 ; 0; u1 Þju1 2 R ; : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} n n1 8 9 < = [ ð0; . . . ; 0; 0; u2 ; . . . ; 0; u2 ; 0Þju2 2 R : : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ; n1
uk
ð5:18Þ
if q 2 2N:
We have det Bkq = 0 if and only if X1 = X1 + X2 = . . . = Xn2 + Xn1 = Xn1 = u1u2. The case n 2 2N þ 1. It is easy to see that X1 = X2 = = Xn1 = 0 and u1u2 = 0. We observe that 23 MFð1Þ
B C kqr ¼ @
123 MFð2Þ
and
Bkq ¼
1
; k; q 2 f1; . . . ; ng: ð5:16Þ
0
k 2 f1; . . . ; n 1g ð5:15Þ
X k1 þ X k þ u2k X q1 þ X q þ u2q
Bkq ¼
uk þ ukþ1
C ur A; 2 X r1 þ X r þ ur
r 2 f1; . . . ; ng 1
!
uq
and q 2 f1; . . . ; ng C kq ¼
1 uq þ uqþ1
0
k; q 2 f1; . . . ; n 1g
1
uk
1 uk þ ukþ1
1
1
k 2 f1; . . . ; n 1g;
ð5:14Þ Bkq ¼
0
0
where
0
k; q 2 f1; . . . ; n 1g;
we have
detðBkq Þ ¼ 0;
8k; q; r
detðC kqr Þ ¼ 0;
ð5:17Þ
n
n1
detðBkqr Þ ¼ 0;
n
The case n 2 2N. It is easy to see that X1 = X3 = = Xn1 = X, X2 = X4 = = Xn2 = 0, u1 = u3 = = un1 and u2 = u4 = = un. All the conditions (5.17) are verified and the set is
123 MFð2Þ
8 <
9 = ¼ ðX; 0; . . . ; X; 0; X ; u1 ; u2 ; . . . ; u1 ; u2 ÞjX; u1 ; u2 2 R : : |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ; n1
n
References [1] Abraham R, Marsden JE, Ratiu T. Manifolds, tensor analysis, and applications. Applied Mathematical Sciences, Second ed., vol. 75. New-York: Springer-Verlag; 1978. [2] Adler M. On a trace functional for pseudo-differential operators and the symplectic structure of Korteveg–Devries type equations. Invent Math 1979;50:219–48. [3] Damianou PA, Fernandes Rui Loja. From the Toda lattice to the Volterra lattice and back. Rep Math Phys 2002;50:361–78. [4] Flaschka H. The Toda lattice. I. Existence of integrals. Phys Rev B 1974;9:1924–5.
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701 [5] Hénon M. Integrals of the Toda lattice. Phys Rev B 1974;9:1921–3. [6] Irtegov VD, Titorenko TN. The invariant manifolds of systems with first integrals. J Appl Math Mech 2009;73:379–84. [7] Kostant B. The solution to a generalized Toda lattice and representation theory. Adv Math 1979;34:195–338.
701
[8] Ortega J-P, Planas-Bielsa V. Dynamics on Leibniz manifolds. J Geom Phys 2004;52:1–27. [9] Sard A. The measure of the critical values of differentiable maps. Bullet Amer Math Soc 1942;48:883–90.
Contents lists available at ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Invariant critical sets of conserved quantities Petre Birtea ⇑, Dan Coma˘nescu Department of Mathematics, West University of Timisßoara, Bd. V. Pârvan No. 4, 300223 Timisßoara, Romania
a r t i c l e
i n f o
Article history: Received 18 May 2010 Accepted 5 June 2011 Available online 26 July 2011
a b s t r a c t For a dynamical system we will construct various invariant sets starting from its conserved quantities. We will give conditions under which certain solutions of a nonlinear system are also solutions for a simpler dynamical system, for example when they are solutions for a linear dynamical system. We will apply these results to the example of Toda lattice. 2011 Elsevier Ltd. All rights reserved.
1. Introduction A particle moving in the Newtonian gravitational field is known as the Kepler problem. The equations of motion are
€x ¼
x kxk
3
;
where x 2 R3 n f0g is the position vector. It is well known that the motions for the Kepler problem are planar. If we consider the plane Ox1x2, the equations of motion becomes
8 x_1 ¼ y1 > > > > > < x_2 ¼ y2 y_1 ¼ ðx2 þxx12 Þ3=2 > > 1 2 > > > : y_2 ¼ 2 x22 3=2 ðx þx Þ 1
2
These equations are Hamiltonian with the standard symplectic form on R4 and the Hamiltonian function H ¼ 2 1 1 y1 þ y22 pffiffiffiffiffiffiffiffiffi . From Kepler’s second law we have an2 2 2 x1 þx2
other conserved quantity given by A = x1y2 x2y1. For a > 0, consider the following conserved quantity K ¼ H þ a13 A. After a straightforward computation we will obtain that rK = 0 if and only if y1 ¼ a13 x2 , and y2 ¼ a13 x1 and k(x1, x2)k = a2. Equivalently, the set {rK = 0} is equal with
the
set
fðx1 ; x2 ; y1 ; y2 Þjðx1 ; x2 Þ ðy1 ; y2 Þ ¼ 0,
and
kðx1 ; x2 Þk ¼ a2 , and kðy1 ; y2 Þk ¼ 1a and sgnðy1 Þ ¼ sgnðx2 Þg ⇑ Corresponding author. E-mail address: [email protected] (P. Birtea). 0960-0779/$ - see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2011.06.009
which is invariant under the dynamics and is filled with solutions that represent uniform circular motions moving clockwise. These particular motions are also solutions for the linear Hamiltonian system
8 x_1 ¼ x2 > > > < x_2 ¼ x1 > y > _1 ¼ y2 > : y_2 ¼ y1 ; where the Hamiltonian function is A. Given the above analysis, we can raise at least two questions. Is it true that for a dynamical system that admits conserved quantities, the set of points where the gradients of these conserved quantities are zero, is an invariant set? When do two Hamiltonian systems have common solutions? In what follows, we will give an answer to these questions. For the first question, the answer is positive and it is given in section two where we will also discuss various generalizations of this answer. In section three, we will present the conditions under which we can give an answer to the second question. In section four we will illustrate these results for the example of Toda lattice. Detailed computations are presented in the Appendix. 2. Invariant sets Let f : Rn ! Rn be a Cq, q P 1 function which generates the differential equation
x_ ¼ f ðxÞ:
ð2:1Þ
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701
694
We suppose that Eq. (2.1) admits a Cq vectorial conserved quantity F : Rn ! Rk with k 6 n. We denote by F1, . . ., Fk the components of F. For s 2 {0, . . ., k} we introduce the sets:
MFðsÞ
n
¼ fx 2 R jrankrFðxÞ ¼ sg
ð2:2Þ
where rF(x) is the Jacobian matrix
0 @F 1 @x1
ðxÞ
rFðxÞ ¼ B @ @F k @x1
@F 1 @xn
Theorem 2.5. The sets N FðrÞ are invariant sets for the dynamics generated by the differential Eq. (2.1). Proof. Let i 2 {1, . . ., k}, l 2 {1, . . ., q} and a = (a1, . . ., al) 2 {1, . . ., n}l. We will prove by mathematical induction that
1
C A: ðxÞ
ðxÞ
ðxÞ
Corollary 2.4. The set of critical points of F is an invariant set of the dynamics generated by the differential Eq. (2.1).
ð2:3Þ
n
ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ@ b F i ðUt ðxÞÞ
b1 ;...;bl ¼1
For r 2 {1, . . ., q} we introduce the sets:
NFðrÞ
n X
@ a F i ðxÞ ¼
@F k @xn
þ Si;a ðt; xÞ;
a
ð2:8Þ
¼ fx 2 R j@ F i ðxÞ ¼ 0; 8i 2 f1; . . . ; kg;
8a 2 f1; . . . ; ngl ; 8l 6 rg
ð2:4Þ l
Fi where we note @ a F i ¼ @xa @ @x if a = (a1, . . ., al). a 1
l
Remark 2.1. We observe that
@F i MFð0Þ ¼ NFð1Þ ¼ x 2 Rn j ðxÞ ¼ 0; i 2 f1; . . . ; kg; j 2 f1 . . . ng : @xj The set
n
M FðsÞ
o
ð2:5Þ with s 2 {0, . . ., k} is a partition of Rn . A
critical point of F is a point in Rn at which the rank of the matrix rF(x) is less than the maximum rank. A critical value is the image under F of a critical point. The set of critical points of F is k1
MFc ¼ [ MFðsÞ : s¼0
ð2:6Þ
Using Sard’s Theorem (see [9]) we have that the set of crit ical values F M Fc is of k-dimensional measure zero provid-
where b = (b1, . . ., bl) and Si,a(t, x) is a sum with the property: ‘‘all the terms contain a factor of the form @ cFi(Ut(x)) with jcj < l’’. We will prove this result by mathematical induction with respect to l. Componentwise the relation (2.7) implies our result for l = 1. Let a0 = (a1, . . ., al, al+1) 2 {1, . . ., n}l+1 where a = (a1, . . ., al) 2 {1, . . ., n}l. Using the induction hypothesis we have: ! n X @ a0 b @ F i ðxÞ ¼ ðrUt Þa1 b1 ðxÞ . . . ðrUt Þal bl ðxÞ@ F i ðUt ðxÞÞ @xalþ1 b ;...;b ¼1 1
By a straightforward computation we obtain
!
n X
@ @xalþ1
ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ@ b F i ðUt ðxÞÞ
b1 ;...;bl ¼1 n X
¼
ing that q P n k + 1.
l
@ ðSi;a ðt; xÞÞ: þ @xalþ1
ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ
b1 ;...;bl ¼1
Remark 2.2. We also have the obvious inclusions N FðqÞ # N Fðq1Þ # # N Fð1Þ . Theorem 2.3. The sets M FðsÞ are invariant under the dynamics generated by the differential Eq. (2.1).
@ @ b F i ðUt ðxÞÞ @xalþ1
@ ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ @ b F i ðUt ðxÞÞ @xalþ1 ;;b ¼1 n X
þ b1
l
n X
¼
ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ
b1 ;;bl ;blþ1 ¼1 0
ðrUt Þalþ1 blþ1 ðxÞ@ b F i ðUt ðxÞÞ
Proof. Because F is a conserved quantity, we have
FðUt ðxÞÞ ¼ FðxÞ; where Ut : Rn ! Rn is the flow generated by (2.1). Differentiating, we have
rFðUt ðxÞÞrUt ðxÞ ¼ rFðxÞ:
ð2:7Þ
As rUt(x) is an invertible matrix for any x 2 Rn which is not an equilibrium point for (2.1), (see [1]), we have that
rankrFðUt ðxÞÞ ¼ rankrFðxÞ; which implies the stated result. h As a consequence we will obtain the following well known result which was applied for studying invariant sets of various mechanical systems, see for example [6].
@ ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ @ b F i ðUt ðxÞÞ: @xalþ1 ;;b ¼1 n X
þ b1
l
We will note that
Si;a0 ðt; xÞ ¼
n X b1 ;...;bl ¼1
@ ðrUt Þa1 b1 ðxÞ ðrUt Þal bl ðxÞ @xalþ1
@ b F i ðUt ðxÞÞ þ
@ ðSi;a ðt; xÞÞ: @xalþ1
All the terms of Si;a0 ðt; xÞ contain a factor of the form @ cFi(Ut(x)) with jcj < l + 1, which had to be proved. Let b = (b1, . . ., bl) 2 {1, . . ., n}l and rU1 t ðxÞ be the inverse matrix of rUt(x), where x is not an equilibrium point for (2.1). Consequently, we have
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701 n X
@ b F i ðUt ðxÞÞ ¼
Another particular case is when the function f in (3.1) and (3.2) verifies the equality
1 ðrUt Þ1 b1 a1 ðxÞ ðrUt Þbl al ðxÞ
a1 ;...;al ¼1 a
@ F i ðxÞ Si;a ðt; xÞ :
695
ð2:9Þ
Also, we will prove by mathematical induction that the sets N FðjÞ are invariant under the dynamics generated by the differential Eq. (2.1). For j = 1 we have N Fð1Þ ¼ M Fð0Þ , which is an invariant set (see Theorem 2.3). We suppose that for all j 6 l the sets N FðjÞ are invariant. Let x 2 N Fðlþ1Þ be arbitrary chosen. Using (2.9) for l + 1-order of derivation and the induction hypothesis, we deduce that Ut ðxÞ 2 N Fðlþ1Þ ; 8t. Summing up, N FðjÞ are invariant sets for all j 2 {1, . . ., q}. h
< f ðx; yÞ; y >¼ 0; where< , > is the Euclidian product on Rn . In this case F is a conserved quantity for (3.1) and G is a conserved quantity for (3.2). The above equality is verified for the case when
f ðx; yÞ ¼ PðxÞy; where P(x) is an antisymmetric matrix. This is true for all almost Poisson manifolds (see [8]). In this situation the differential Eqs. (3.1) and (3.2) become
x_ ¼ PðxÞrFðxÞ and
3. Finding solutions using simpler dynamics Let F; G : Rn ! R be Cq functions and the differential equations:
x_ ¼ f ðx; rFðxÞÞ
ð3:1Þ
and
x_ ¼ f ðx; rGðxÞÞ 2n
ð3:2Þ n
q
where f : R ! R is a C vectorial function. For x 2 Rn , we denote with UFt ðxÞ and UGt ðxÞ the solutions of (3.1) and (3.2) with the initial conditions UF0 ðxÞ ¼ x and UG0 ðxÞ ¼ x. We will introduce the following set
E1 ¼ fx 2 Rn jrFðxÞ ¼ rGðxÞg: Theorem 3.1. If F G is a conserved quantity for (3.1) and x 2 E1, then for all t we have
UFt ðxÞ ¼ UGt ðxÞ: Proof. For L = F G we have the equality E1 ¼ MLð0Þ . By Theorem (2.3) the set E1 is invariant under the dynamics of (3.1). For x 2 E1 we have rF UFt ðxÞ ¼ rG UFt ðxÞ for all t and consequently
d F U ðxÞ ¼ f ðUFt ðxÞ; rF UFt ðxÞ ¼ f ðUFt ðxÞ; rG UFt ðxÞ : dt t
ð3:6Þ
We observe that F is a conserved quantity for (3.6) if and only if G is a conserved quantity for (3.5). Corollary 3.3. If G is a conserved quantity for (3.5), then for the initial conditions in fx 2 Rn jrFðxÞ ¼ rGðxÞg we have UFt ðxÞ ¼ UGt ðxÞ. A particular case of Corollary 3.3 is when we have a symplectic manifold with G being a quadratic function. In this case, the solutions of the Hamiltonian vector field XF starting in fx 2 Rn jrFðxÞ ¼ rGðxÞg are also the solutions of the linear Hamiltonian system XG. Analogous results are valid for the more general case of the vector valued conserved quantities and when the right side of Eqs. (3.1) and (3.2) depends on higher order derivatives. Firstly, we will introduce some notations. Let F; G : Rn ! Rk be Cq vectorial functions with q P 1 and also k 6 n. If F1, . . ., Fk are the components of F we denote with
D1 FðxÞ ¼
@F 1 @F 1 @F 2 @F 2 ðxÞ; . . . ; ðxÞ; ðxÞ; . . . ; ðxÞ; . . . ; @x1 @xn @x1 @xn @F k @F k ðxÞ; . . . ; ðxÞ 2 Rkn : @x1 @xn
r
Next we will discuss a particular case of the result presented above. For this we will take f(x, y) = h(x) + g(y) and F 0. Thus we have the two dynamics
ð3:3Þ
and the perturbed dynamics
x_ ¼ hðxÞ þ gðrGðxÞÞ:
x_ ¼ PðxÞrGðxÞ:
and
The above equality shows that UFt ðxÞ is also a solution for (3.2). Given the uniqueness of the solutions, we obtain the desired equality. h
x_ ¼ hðxÞ
ð3:5Þ
Dr FðxÞ ¼ ð. . . ; @ a F i ðxÞ; . . .Þ 2 Rkn ; where r 2 {1, . . ., q}, i 2 {1, . . ., k}, a 2 {1, . . ., n}r (jaj = r) and the components appear in the lexicographical order for (i, a) in Nrþ1 . For a fix r 2 {1, . . ., q}, we will consider, as before, the two differential equations
x_ ¼ fr ðx; D1 FðxÞ; . . . ; Dr FðxÞÞ
ð3:7Þ
and
ð3:4Þ
We denote by Uht the flux for (3.3).
x_ ¼ fr ðx; D1 GðxÞ; . . . ; Dr GðxÞÞ:
ð3:8Þ
Corollary 3.2. If G is a conserved quantity for (3.3), then for all initial conditions x 2 fx 2 Rn jrGðxÞ ¼ 0g we have
Let x 2 Rn , we denote with UFt ðxÞ and UGt ðxÞ the solutions of (3.7) and (3.8) with initial conditions UF0 ðxÞ ¼ x and UG0 ðxÞ ¼ x. We will introduce the following set
Uht ðxÞ ¼ UGt ðxÞ:
Er ¼ fx 2 Rn j@ a FðxÞ ¼ @ a GðxÞ; 8jaj 6 rg:
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701
696
Theorem 3.4. If F G is a conserved quantity for (3.7) and x 2 Er then for all t we have
UFt ðxÞ ¼ UGt ðxÞ: Also the obvious extension of Corollary 3.2 takes place.
(i) The indices i1, . . ., ik, j1, j1 + 1, . . ., jl, jl + 1, which appear in the term (either explicitly, or implicitly through a factor Xj) are all different (modulo n); (ii) The number of these indices is m, i.e. k + 2l = m. Two terms differing only in the order of factors are not considered different, and therefore only one of them appears in the sum.
The Toda lattice describes the one-dimensional motions of a chain of particles with nearest neighbor interactions. For a chain of particles with the equal masses m, Morikazu Toda came up with the choice of the interaction potential
In [4], Flaschka has proved that the above functions are conserved quantities using a Lax formulation. This was generalized to arbitrary Lie algebras by Adler [2], Kostant [7] and Damianou, Fernandes [3]. The first three scalar conserved quantities, depending on the variables (X1, . . ., Xn, u1, . . ., un), are
VðrÞ ¼ er þ r 1:
I1 ¼
4. Invariant sets for Toda lattices
i 2 Z:
ð4:1Þ
This second order differential system is equivalent with the first order differential system
x_i ¼ ui mu_i ¼ eðxi xi1 Þ eðxiþ1 xi Þ ;
ð4:2Þ
i 2 Z;
ui ¼ 0;
k 2 R ;
x0 2 R;
y_i ¼ ui k u_i ¼ em ðeðyi yi1 Þ eðyiþ1 yi Þ Þ;
i2Z
ð4:3Þ
ek ðyiþ1 yi Þ X i :¼ ; e m
ð4:4Þ
then the equations of motion become
X_ i ¼ X i ðui uiþ1 Þ u_i ¼ X i1 X i ; i 2 Z:
ð4:5Þ
The following particular cases are interesting: 1. The case of a periodic lattice, X iþn ¼ X i 8i 2 Z, 2. The case of a non-periodic lattice with the boundary conditions X0 = 0 (correspond to formally setting y0 = 1) and Xn = 0 (correspond to formally setting yn+1 = 1). In cases we investigate the motions of the particles 1 to n ðn 2 N Þ.
In this case Hénon proves in [5] that the following expressions are scalar conserved quantities
X
ui1 . . . uik ðX j1 Þ . . . ðX jl Þ;
I3 ¼
X
Xj
16j6n
X
ui1 ui2 ui3
16i1
ð4:8Þ ui X j ;
ðX 0 ¼ X n Þ:
16i;j6n;j–i;j–i1ðmodnÞ
ð4:9Þ We will introduce the vectorial conserved quantities,
I. The case n odd For this case we obtain as invariant sets only subsets of the set of equilibrium points or the empty set. 1 2 12 12 13 More precisely, M Ið0Þ ¼ MIð0Þ ¼ M Ið0Þ ¼ M Ið1Þ ¼ MIð0Þ ¼ I23 I123 I123 Mð0Þ ¼ M ð0Þ ¼ M ð1Þ ¼ ;. The following are subsets of the set of equilibrium points:
13 ¼ fðX; . . . ; X; 0; . . . ; 0ÞjX 2 Rg; M Ið1Þ n1 2 n1 2 23 ¼ u ;...; u ; u; . . . ; u ju 2 R ; M Ið1Þ 2 2 123 ¼ fðX; . . . ; X; u; . . . ; uÞjX; u 2 Rg: M Ið2Þ
II. The case n even 1 2 12 12 13 23 We have, M Ið0Þ ¼ MIð0Þ ¼ M Ið0Þ ¼ MIð1Þ ¼ M Ið0Þ ¼ MIð0Þ ¼ I123 I123 Mð0Þ ¼ M ð1Þ ¼ ;. As nontrivial invariant sets we have the following: 3 M Ið0Þ ¼ fðX 1 ; X 2 ; . . .; X 1 ; X 2 ; u1 ; u2 ; . . .; u1 ; u2 ÞjX 1 þ X 2
¼ u1 u2 ; u1 þ u2 ¼ 0g; 13 M Ið1Þ ¼ fðX 1 ; X 2 ; .. . ; X 1 ; X 2 ; u1 ; u2 ; .. . ; u1 ;u2 Þju1 þ u2 ¼ 0g; 23 ¼ fðX 1 ; X 2 ; .. . ; X 1 ; X 2 ; u1 ; u2 ; .. . ; u1 ;u2 ÞjX 1 þ X 2 M Ið1Þ o n ¼ ðu1 þ u2 Þ2 þ u1 u2 ; 4 123 M Ið2Þ ¼ fðX 1 ; X 2 ; . . . ;X 1 ;X 2 ;u1 ; u2 ; . . . ; u1 ; u2 ÞjX 1 ; X 2 ; u1 ; u2 2 Rg:
123 We obtain M Ið2Þ as the largest invariant set and the restricted dynamics is the dynamics of two particles
4.1. The case of a periodic lattice
Im ¼
ui1 ui2
16i1
3 M Ið0Þ ¼ fð0; . . . ; 0; 0; . . . ; 0Þg;
Let us define
(
ð4:7Þ X
ð4:10Þ
i 2 Z:
Let an equilibrium of the Toda lattice and yi = xi x0 ki the displacement of the i particle from its equilibrium position. The system in the variables yi and ui is
(
I2 ¼
X
I12 ¼ ðI1 ; I2 Þ; I13 ¼ ðI1 ; I3 Þ; I23 ¼ ðI2 ; I3 Þ; I123 ¼ ðI1 ; I2 ; I3 Þ:
where ui is the velocity of the particle i. An equilibrium of the Toda lattice has the form
xi ¼ x0 þ ki;
ui
16i6n
The system of motion reads explicitly
mx€i ¼ eðxi xi1 Þ eðxiþ1 xi Þ ;
X
ð4:6Þ
where m 2 {1, . . ., n} and the summation are extended to all terms which satisfy the following conditions:
8 _ > > > X 1 ¼ X 1 ðu1 u2 Þ > <_ X 2 ¼ X 2 ðu2 u1 Þ > u_ 1 ¼ X 2 X 1 > > > :_ u2 ¼ X 1 X 2
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701 23 On the invariant set M Ið1Þ we have the above dynamics subject to X 1 þ X 2 ¼ 4n ðu1 þ u2 Þ2 þ u1 u2 . On the invariant set 13 M Ið1Þ we have the above dynamics subject to u1 + u2 = 0 3 and on the invariant set MIð0Þ we have the above dynamics subject to u1 + u2 = 0 and X1 + X2 = u1u2. 3 23 For the sets MIð0Þ and M Ið1Þ the variables Xi have to take also negative values which from a mathematical point of view is correct and can be the solutions for the system (4.5). As the mechanical system is given by (4.3) and we do the change of variables (4.4), the variables Xi have to be strictly positive in order to have a physical meaning. Consequently, from a mechanical point of view only the 13 123 sets MIð1Þ and M Ið2Þ are meaningful. The computations can be found in the Appendix.
It is known that if we have the matrices
1 X1 0 0 0 C u2 X 2 C C C and C 1 un1 X n1 A 0 1 un 1 X 1 0 0 0 C 0 X 2 C C C; C 0 0 X n1 A 0 0 0
u1 B 1 B B L ¼ B B @ 0 0 0 0 B 0 B B B ¼ B B @ 0 0
k 2 f1; . . . ; ng:
F2 ¼
Xi þ
i¼1
F3 ¼
n1 X
u2i 2
123 M Fð2Þ
n1
23 M Fð1Þ
i¼1
123 M Fð2Þ
ð4:12Þ
n 1X u3 : 3 i¼1 i
We will introduce the vectorial conserved quantities,
F12 ¼ ðF 1 ; F 2 Þ;
F13 ¼ ðF 1 ; F 3 Þ;
F23 ¼ ðF 2 ; F 3 Þ;
F123 ¼ ðF 1 ; F 2 ; F 3 Þ:
n
8 9 < = ðX;0; .. . ;X; 0; X ;u1 ; u2 ; .. . ; u1 ;u2 Þju1 þ u2 ¼ 0; X ¼ u1 u2 ; : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ;
ð4:13Þ
As before we will distinguish two cases. I. The case n odd In this case we obtain as invariant sets only subsets of the set of equilibrium points or the empty set. 1 2 12 12 13 More precisely, M Fð0Þ ¼ MFð0Þ ¼ M Fð0Þ ¼ M Fð1Þ ¼ M Fð0Þ ¼ 23 123 123 MFð0Þ ¼ MFð0Þ ¼ MFð1Þ ¼ ;. The following are subsets of the set of equilibrium points:
n
8 9 < = ¼ ðX;0; .. . ; X; 0; X ;u1 ;u2 ; . .. ; u1 ; u2 Þju1 þ u2 ¼ 0 ; : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ; n
8 9 < = ¼ ðX;0; .. . ; X; 0; X ;u1 ;u2 ; . .. ; u1 ; u2 ÞjX ¼ u1 u2 ; |fflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflffl ffl } |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} : ; n1
X i ðui þ uiþ1 Þ þ
n
n1
8 9 < = ¼ ð0; . . . ; 0; u1 ; u2 ; . . . ; u1 Þju1 ; u2 2 R : : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} ;
n1
n X
ui i¼1 n X
n
n1
8 9 < = [ ð0; . . . ; 0; 0; u2 ; . . . ; 0; u2 ; 0Þju2 2 R ; : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ;
13 M Fð1Þ
For k 2 {1, 2, 3} we have
F1 ¼
23 M Fð1Þ
n
8 9 < = ¼ ðX;0; .. . ;X; 0; X ;u1 ; u2 ; . .. ; u1 ;u2 ÞjX; u1 ;u2 2 R : : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ; n1
ð4:11Þ
1 trðLk Þ; k
n
n1
n1
where [B, L] = BL LB. Using the Flaschka theorem (see [4]) we have the following scalar conserved quantities depending on the variables (X1, . . ., Xn1, u1, . . ., un)
Fk ¼
8 9 < = ¼ ð0; . . . ; 0; 0; . . . ; 0Þ ; : |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} ;
8 9 < = ¼ ð0; . . . ; 0; u1 ; 0; . . . ; u1 ; 0; u1 Þju1 2 R : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ;
3 M Fð0Þ ¼
then the system (4.5), in this case, has the Lax form
L_ ¼ ½B; L;
¼
13 M Fð1Þ
II. The case n even 1 2 12 12 13 23 We have, M Fð0Þ ¼ MFð0Þ ¼ MFð0Þ ¼ MFð1Þ ¼ M Fð0Þ ¼ MFð0Þ ¼ F123 F123 Mð0Þ ¼ Mð1Þ ¼ ;. As nontrivial invariant sets we have the following:
4.2. The case of non-periodic lattice
0
3 M Fð0Þ
697
n
123 We obtain M Fð2Þ as the largest invariant set and the restricted dynamics is given by
8 _ > > < X ¼ Xðu1 u2 Þ u_ 1 ¼ X > > : u_ 2 ¼ X 23 On the invariant set M Fð1Þ we have the above dynamics sub13 ject to X = u1u2. On the invariant set M Fð1Þ we have the above dynamics subject to u1 + u2 = 0 and on the invariant 3 set M Fð0Þ we have the above dynamics subject to u1 + u2 = 0 and X = u1u2. 123 For the set M Fð2Þ the variables Xi with i even are all equal with zero which from a mathematical point of view is correct. As before, the mechanical system is given by (4.3) and we do the change of variables (4.4). Consequently, the variables Xi have to be strictly positive in order to have a physical meaning. The computations can be found in the Appendix.
Acknowledgments Petre Birtea has been supported by CNCSIS UEFISCSU, project number PN II - IDEI 1081/2008.
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701
698
8 > < U uk ukþ1 ¼ 0 V uk ðU uk Þ ðY X k1 X k Þ ¼ V uq ðU uq Þ > : ðY X q1 X q Þ:
Appendix A A.1. The computations for the case of periodic lattice
ð5:7Þ
We will make the notations
U¼
X
ui ;
X
V¼
16i6n
ui1 ui2 ;
Y¼
16i1
X
Xj:
ð5:1Þ
16j6n
We observe that
2V ¼ U 2
n X
u2i :
ð5:2Þ
i¼1
13 MIð1Þ ¼ fðX; . . . ; X; 0; . . . ; 0ÞjX 2 Rg:
With these notations we have the following,
rI1 ¼ ð0; . . . ; 0; 1; . . . ; 1Þ 0
1
rI2 ¼ @1; . . . ; 1; U u1 ; . . . ; U uk ; . . . ; U un A |fflfflffl{zfflfflffl} nþ1
0
|fflfflffl{zfflfflffl} nþk
Adding the first n equations we obtain U = 0, u1 + u2 = u2 + u3 = = un1 + un = un + u1 and consequently ui ¼ ð1Þiþ1 u; u 2 R. The last n equations become X1 + X2 = X2 + X3 = = Xn1 + Xn = Xn + X1. The case n 2 2N þ 1. For this case note that u1 = u2 = = un = 0 and X 1 ¼ . . . ¼ X n ¼ X 2 R. In this case we have
|fflfflffl{zfflfflffl}
ð5:3Þ
The case n 2 2N. It is easy to see that X1 = X3 = = Xn1, X2 = X4 = = Xn. In this case we have
ð5:4Þ
13 MIð1Þ ¼ fðX 1 ; X 2 ; . . . ; X 1 ; X 2 ; u1 ; u2 ; . . . ; u1 ; u2 Þju1 þ u2 ¼ 0g:
2n
23 A point belongs to the set M Ið1Þ if and only if
rI3 ¼ B @. . . ; ðU uk ukþ1 Þ; . . . ;
detðAkq Þ ¼ 0; detðBkq Þ ¼ 0; detðC kq Þ ¼ 0; 8k; q 2 f1; . . . ; ng;
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} k
ð5:8Þ
1
C V uk ðU uk Þ ðY X k1 X k Þ; . . .A: |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
where
Akq ¼
nþk
3 M Ið0Þ .
The study of 3 The elements of MIð0Þ are the solutions of the algebraic system
ð5:5Þ
Bkq ¼
U uk ukþ1 ¼ 0; V uk ðU uk Þ ðY X k1 X k Þ ¼ 0; 8k 2 f1; . . . ; ng:
C kq ¼
1
1
ðU uk ukþ1 Þ ðU uq uqþ1 Þ
1 U uq ðU uk ukþ1 Þ V uq ðU uq Þ ðY X q1 X q Þ
U uq
U uk
V uk ðU uk Þ ðY X k1 X k Þ V uq ðU uq Þ ðY X q1 X q Þ
:
ð5:6Þ Adding the first n equations, we obtain U = 0 and u1 + u2 = u2 + u3 = = un1 + un = un + u1. We deduce the following results: The case n 2 2N þ 1. In this situation we have u1 = u2 = = un = 0. Adding the last n equations of (5.6) we obtain Y = 0 and X1 + X2 = X2 + X3 = = Xn1 + Xn = Xn + X1. It implies that X1 = = Xn = 0 and consequently
The case n 2 2N. For this situation ui ¼ ð1Þiþ1 u; u 2 R. In this case, using (5.2), we have V ¼ n2 u2 . The last n equations of (5.6) become
n 2
1 u2
8k 2 f1; . . . ; ng:
Making the addition we have Y ¼ n2 u2 and consequently X1 + X2 = X2 + X3 = = Xn1 + Xn = Xn + X1 = u2 which implies X1 = X3 = = Xn1, and X2 = X4 = = X2n. In this case we have 3 MIð0Þ ¼ fðX 1 ; X 2 ; . . . ; X 1 ; X 2 ; u1 ; u2 ; . . . ; u1 ; u2 ÞjX 1 þ X 2
¼ u1 u2 ; u1 þ u2 ¼ 0g: I
ðn 1Þu
1
Bkq ¼
ðn 2Þu
ðn2Þðn1Þ 2 u 2
!
ðY X q X q1 Þ
:
Using the condition that det(Bkq) = 0, we obtain
3 MIð0Þ ¼ fð0; . . . ; 0; 0; . . . ; 0Þg:
Y þ X k1 þ X k ¼
Using the expression of Akq we deduce that u1 + u2 = u2 + u3 = = un1 + un = un + u1. The case n 2 2N þ 1. In this case we have u1 = u2 = = un = u, and U = nu, and V ¼ ðn1Þn u2 and 2
Y X q X q1 ¼
8q 2 f1; . . . ; ng:
Adding these relations we have
ðn 1Þn 2 u ; 2 n1 2 u : X¼ 2
Y¼
and X 1 ¼ X 2 ¼ ¼ X n ¼ X
and
With this relation we have
C kq ¼
ðn 1Þu ðn 2Þðn 1Þu2
: ðn 2Þðn 1Þu2 ðn 1Þu
We observe that the equality det(Ckq) = 0 is verified. In conclusion we have
I
ij ij The study of M ð0Þ ; M ð1Þ with (i, j) 2 {(1, 2), (1, 3), (2, 3)}. 13 A point ðX 1 ; . . . ; X n ; u1 ; . . . ; un Þ 2 M Ið1Þ if and only if we have, for all k, q 2 {1, . . ., n},
ðn 2Þðn 1Þ 2 u ; 2
23 MIð1Þ ¼
n1 2 n1 2 u ;...; u ; u; . . . ; u ju 2 R : 2 2
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701
The case n 2 2N. In this case u1 = u3 = = un1, u2 = u4 = = un and we have U ¼ 2n ðu1 þ u2 Þ. Using (5.2) we obtain 2 2 V ¼ nðn2Þ u1 þ u22 þ n4 u1 u2 . From the relation det(Bkq) = 0, 8 we have
Y X q1 X q ¼ V ðU uq Þ
n2 U þ uq : n
ð5:9Þ
Consequently, X1 + X2 = X3 + X4 = = Xn1 + Xn and X2 + X3 = X4 + X5 = = Xn + X1 which further implies that X 1 ¼ X 3 ¼ . . . ¼ X n1 ; X 2 ¼ X 4 ¼ . . . ¼ X n and Y ¼ n2 ðX 1 þ X 2 Þ. By substitution into (5.9) we obtain X 1 þ X 2 ¼ n4 ðu1 þ u2 Þ2 þ u1 u2 . We have
B Akqr ¼ @
0
1
1
C U ur A
ðU uk ukþ1 Þ ðU uq uqþ1 Þ 0 B Bkqr ¼ @
0
1
1
U uq
ðU uk ukþ1 Þ 0
1
B C kqr ¼ @ U uk Vk
1 U uq Vq
1
Vr
A.2. The computations for the case of non-periodic lattice If we consider the variables (X1, . . ., Xn1, u1, . . ., un) we obtain
0
1
1
rF 1 ¼ @0; . . . ; 0; 1; . . . ; 1A
C U ur A
Vq 1
1
1
(i) detB111 = detB112 = 0 (by calculus). (ii) det Bkqr = det Bkrq. (iii) If q1 q2 2 2N and r 1 r 2 2 2N then detBk1 q1 r1 ¼ det Bk2 q2 r2 .
123 MIð2Þ ¼ fðX 1 ; X 2 ; . . . ; X 1 ; X 2 ; u1 ; u2 ; . . . ; u1 ; u2 ÞjX 1 ; X 2 ; u1 ; u2 2 Rg:
123 The study of M Ið2Þ . We introduce, for k, q, r 2 {1, . . ., n}, the matrices
0
If q = k + 1 and r = k + 2 we obtain Vk = Vk+2, which implies that X1 + X2 = X3 + X4 = = Xn1 + Xn and X2 + X3 = X4 + X5 = = Xn + X1. Consequently, we have X1 = X3 = = Xn1 and X2 = X4 = = Xn. We observe that if s t 2 2Z, then us = ut and Vs = Vt, which implies that det(Ckqr) = 0 " k, q, r 2 {1, . . ., n}. For the matrices Bkqr we have the following properties:
Using these properties we deduce that det Bkqr = 0, "k, q, r 2 {1, . . ., n} and we obtain
23 M Ið1Þ ¼ fðX 1 ; X 2 ; . . . ; X 1 ; X 2 ; u1 ; u2 ; . . . ; u1 ; u2 ÞjX 1 þ X 2 o n ¼ ðu1 þ u2 Þ2 þ u1 u2 : 4
0
0
Vr
n
n1
1
rF 2 ¼ @1; . . . ; 1; u1 ; . . . ; un A
1
ð5:11Þ
|fflfflfflffl{zfflfflfflffl}
C U ur A;
n1
rF 3 ¼ u1 þ u2 ; . . . ; un1 þ un ; X 1 þ u21 ; X 2 þ X 1
þ u22 ; . . . ; X n2 þ X n1 þ u2n1 ; X n1 þ u2n :
where Vs = V us(U us) (Y Xs1 Xs). We observe that rankðrI123 Þ ¼ 2 () detðAkqr Þ ¼ detðBkqr Þ ¼ detðC kqr Þ ¼ 0 8k; q; r 2 f1; . . . ; ng. The equations det (Akqr) = 0 "k, q, r 2 {1, . . ., n} give us
u1 þ u2 ¼ u2 þ u3 ¼ ¼ un1 þ un ¼ un þ u1 : The case n 2 2N þ 1. We have u:¼u1 = = un. With this notation we obtain U = nu, and V ¼ nðn1Þ u2 and V k ¼ 2 2 nðn1Þ u ðY X X Þ. It is easy to see that det(Ckqr) = k1 k 2 0. We have the equivalences
0
0 0 B 0 detðBkqr Þ ¼ 0 () det @ 1 Vk Vq Vr
1
1 C ðn 1Þu A ¼ 0 Vr
() V r ¼ V q () X r1 þ X r ¼ X q1 þ X q : Because n 2 2N þ 1, we obtain X1 = X2 = = Xn. In this case we have 123 M Ið2Þ ¼ fðX; . . . ; X; u; . . . ; uÞjX; u 2 Rg:
8 > < u1 þ u2 ¼ u2 þ u3 ¼ ¼ un1 þ un ¼ 0 X 1 þ u21 ¼ X 2 þ X 1 þ u22 ¼ ¼ X n2 þ X n1 þ u2n1 > : ¼ X n1 þ u2n ¼ 0: The case n 2 2N þ 1. We obtain 3 M Fð0Þ
8 9 < = ¼ ð0; . . . ; 0; 0; . . . ; 0Þ : : |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} ; n
n1
The case n 2 2N. In this situation u1 = u3 = = un1 = u, u2 = u4 = = un = u and Xk = u2 if k 2 2N þ 1 and Xk = 0 if k 2 2N. We have 8 9 < = F3 Mð0Þ ¼ ðX; 0; . . . ; X; 0; X ; u1 ; u2 ; . . . ; u1 ; u2 Þju1 þ u2 ¼ 0; X ¼ u1 u2 : : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ; F
0 uk uq Vq Vk
0
1
C uk ur A ¼ 0: Vr Vk
ð5:12Þ
3 The study of M Fð0Þ . 3 The elements of M Fð0Þ are the solutions of the system
n
n1
The case n 2 2N. We have u1 = u3 = = un1, u2 = u4 = = un. By calculus we obtain
1 B detðC kqr Þ ¼ 0 () det @ U uk Vk
ð5:10Þ
|fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl}
Vr
0
699
F
ij ij The study of M ð0Þ ; M ð1Þ with (i, j) 2 {(1, 2), (1, 3), (2,3)}. 13 The elements of M Fð1Þ are the solutions of the system
8 > < u1 þ u2 ¼ u2 þ u3 ¼ ¼ un1 þ un ¼ 0 X 1 þ u21 ¼ X 2 þ X 1 þ u22 ¼ ¼ X n2 þ X n1 þ u2n1 > : ¼ X n1 þ u2n :
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701
700
123 The study of M Fð2Þ .
The case n 2 2N þ 1. We obtain that 13 MFð1Þ
9 8 = < ¼ ð0; . . . ; 0; 0; . . . ; 0Þ : : |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} ;
123 we have For a point of the set M Fð2Þ
detðAkqr Þ ¼ 0;
n
n1
The case n 2 2N. In this situation u1 = u3 = = un1 = u, u2 = u4 = = un = u and Xk = X if k 2 2N þ 1 and Xk = 0 if k 2 2N. We have 13 MFð1Þ
8 9 < = ¼ ðX; 0; . . . ; X; 0; X ; u1 ; u2 ; . . . ; u1 ; u2 Þju1 þ u2 ¼ 0 : : |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ;
For a point of the set
detðAkq Þ ¼ 0;
23 M Fð1Þ
where
0 B Akqr ¼ @
detðC kq Þ ¼ 0;
8k; q ð5:13Þ
Akq ¼
1
1
uk þ ukþ1
uq þ uqþ1
;
B Bkqr ¼ @
1
uq
uk þ ukþ1
X q1 þ X q þ u2q
;
!
uq
Using the expression of Akq we deduce that u1 + u2 = u2 + u3 = = un1 + un and consequently uk = uq if k q 2 2Z. The matrices Bkq have the form
1
u1
u1 þ u2
X q1 þ X q þ u21
if q 2 2N þ 1
1
u2
u1 þ u2
X q1 þ X q þ u22
n
The case n 2 2N. We deduce that X1 = X3 = = Xn1 = X, X2 = X4 = = Xn2 = 0 and u1u2 = X. In this case we have 23 MFð1Þ
8 9 < = ¼ ðX; 0; X; . . . 0; X ; u1 ; u2 ; . . . ; u1 ; u2 ÞjX ¼ u1 u2 : : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ; n1
ur u2q
X r1 þ X r þ
C A; ð5:19Þ 1
1
uq
X k1 þ X k þ
u2r
q; r 2 f1; . . . ; ng 1
u2k
1
1
X q1 þ X q þ
ur
X q1 þ X q þ
u2q
X r1 þ X r þ
k; q; r 2 f1; . . . ; ng
u2r
C A;
ð5:20Þ
Using the expression of Akqr we deduce that u1 + u2 = u2 + u3 = = un1 + un and consequently uk = uq if k q 2 2Z. For k 2 {1, . . ., n 1}, we have 0 B det Bk;kþ1;k ¼ det @
0
0
1
1
1
ukþ1 uk
uk
C A¼0
uk þ ukþ1 X kþ1 X k1 þ u2kþ1 u2k X k1 þ X k þ u2k
and we deduce that Xk+1 = Xk1. The case n 2 2N þ 1. It is easy to see that X1 = X2 = = Xn1 = 0, u1 = u3 = = un and u2 = u4 = = un1. All the conditions (5.17) are verified and the set is
8 <
9 = ¼ ð0; . . . ; 0; u1 ; u2 ; . . . ; u1 Þju1 ; u2 2 R : : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} ; n
n1
9 8 = < ¼ ð0; . . . ; 0; u1 ; 0; . . . ; u1 ; 0; u1 Þju1 2 R ; : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} n n1 8 9 < = [ ð0; . . . ; 0; 0; u2 ; . . . ; 0; u2 ; 0Þju2 2 R : : |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ; n1
uk
ð5:18Þ
if q 2 2N:
We have det Bkq = 0 if and only if X1 = X1 + X2 = . . . = Xn2 + Xn1 = Xn1 = u1u2. The case n 2 2N þ 1. It is easy to see that X1 = X2 = = Xn1 = 0 and u1u2 = 0. We observe that 23 MFð1Þ
B C kqr ¼ @
123 MFð2Þ
and
Bkq ¼
1
; k; q 2 f1; . . . ; ng: ð5:16Þ
0
k 2 f1; . . . ; n 1g ð5:15Þ
X k1 þ X k þ u2k X q1 þ X q þ u2q
Bkq ¼
uk þ ukþ1
C ur A; 2 X r1 þ X r þ ur
r 2 f1; . . . ; ng 1
!
uq
and q 2 f1; . . . ; ng C kq ¼
1 uq þ uqþ1
0
k; q 2 f1; . . . ; n 1g
1
uk
1 uk þ ukþ1
1
1
k 2 f1; . . . ; n 1g;
ð5:14Þ Bkq ¼
0
0
where
0
k; q 2 f1; . . . ; n 1g;
we have
detðBkq Þ ¼ 0;
8k; q; r
detðC kqr Þ ¼ 0;
ð5:17Þ
n
n1
detðBkqr Þ ¼ 0;
n
The case n 2 2N. It is easy to see that X1 = X3 = = Xn1 = X, X2 = X4 = = Xn2 = 0, u1 = u3 = = un1 and u2 = u4 = = un. All the conditions (5.17) are verified and the set is
123 MFð2Þ
8 <
9 = ¼ ðX; 0; . . . ; X; 0; X ; u1 ; u2 ; . . . ; u1 ; u2 ÞjX; u1 ; u2 2 R : : |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ; n1
n
References [1] Abraham R, Marsden JE, Ratiu T. Manifolds, tensor analysis, and applications. Applied Mathematical Sciences, Second ed., vol. 75. New-York: Springer-Verlag; 1978. [2] Adler M. On a trace functional for pseudo-differential operators and the symplectic structure of Korteveg–Devries type equations. Invent Math 1979;50:219–48. [3] Damianou PA, Fernandes Rui Loja. From the Toda lattice to the Volterra lattice and back. Rep Math Phys 2002;50:361–78. [4] Flaschka H. The Toda lattice. I. Existence of integrals. Phys Rev B 1974;9:1924–5.
P. Birtea, D. Coma˘nescu / Chaos, Solitons & Fractals 44 (2011) 693–701 [5] Hénon M. Integrals of the Toda lattice. Phys Rev B 1974;9:1921–3. [6] Irtegov VD, Titorenko TN. The invariant manifolds of systems with first integrals. J Appl Math Mech 2009;73:379–84. [7] Kostant B. The solution to a generalized Toda lattice and representation theory. Adv Math 1979;34:195–338.
701
[8] Ortega J-P, Planas-Bielsa V. Dynamics on Leibniz manifolds. J Geom Phys 2004;52:1–27. [9] Sard A. The measure of the critical values of differentiable maps. Bullet Amer Math Soc 1942;48:883–90.