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A non-integrability test for perturbed separable planar Hamiltonians
22 May 2000
Physics Letters A 270 Ž2000. 47–54 www.elsevier.nlrlocaterpla
A non-integrability test for perturbed separable planar Hamiltonians Efi Meletlidou Department of Physics, UniÕersity of Thessaloniki, 54006 Thessaloniki, Greece Received 6 September 1999; received in revised form 27 March 2000; accepted 30 March 2000 Communicated by A.P. Fordy
Abstract We prove the non-integrability of perturbed separable planar potentials by using the perturbed normal variational equations, with a reasoning analogous to Ziglin’s theorem. We apply the above criterion to a Hamiltonian that cannot be proved non-integrable by other known non-integrability criteria. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.20.q i; 05.45.q b; 45.10.q z Keywords: Hamiltonian systems; Non-integrability; Ziglin’s theorem
1. Introduction It is already known from the work of Poincare´ Žw1x, Ch. 4. that if a Hamiltonian H possesses a second integral of motion I, then the linear system of the variational equations around a periodic solution of H possesses an integral, linear in the variations, which depends periodically on time through = I, evaluated on the periodic solution. In many cases, however, this integral turns out to be a function of the corresponding integral, produced by the Hamiltonian, or merely a trivial constant. Ziglin w2;3x generalized this result of Poincare´ by showing that, if I is an independent integral of H
E-mail address: [email protected] ŽE. Meletlidou..
which may be dependent on H on the specific solution, then the variational equations always possess an independent non-trivial integral, homogeneous of some degree m G 1 in the variations, with coefficients that depend periodically on time through the mth order derivatives of I, evaluated on the periodic solution. By using this result, finally he proved an important non-integrability theorem, involving properties of the monodromy group of the normal variational equations in the complex time domain. By applying Ziglin’s theorem, Yoshida w4x proposed an algorithmic criterion for the case of planar motion of a point mass in a homogeneous potential of degree k / 0, "2. Important extensions of Ziglin’s theory have been recently made by Morales-Ruiz and Simo´ w5x and Morales-Ruiz and Ramis w6;7x. Applications of the above theorems in various systems were made in w8–19x. The geometrical sig-
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 2 5 4 - 1
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E. Meletlidour Physics Letters A 270 (2000) 47–54
nificance of Ziglin’s theorem has been explored in w20;21x. There are several other non-integrability theorems apart from Ziglin’s results. Poincare’s ´ Žw1x, Ch. 5. theorem on non-integrability, although showed the genericity of this property, is difficult to apply in specific systems. For two non-integrability criteria that are closely related to Poincare’s ´ theorem, see w22–24x. Another criterion for establishing whether a perturbed system is non-integrable Žand also chaotic. is the Poincare–Mel’nikov homoclinic integral w25x. ´ If this integral, evaluated along an unperturbed homoclinic loop, has simple zeros, it guarantees the existence of transverse homoclinic orbits in the perturbed system, which is an obstacle to integrability. Poincare’s ´ and Mel’nikov’s theorems depend on the real time dynamics and are of a perturbative nature. Another theorem for perturbed Hamiltonians, that deals with the complex time dynamics is Kozlov’s w26x theorem, which examines the branching of the unperturbed solution to the perturbed one. In this Letter we consider the problem of the planar motion of a particle in a potential that consists of a part that is separable in cartesian coordinates and a small coupling term. We assume that a family of vertically unstable straight-line periodic solutions that are also single valued functions in complex time exists along one of the axes. We also assume that it is preserved under the perturbation. It is proved that, if the product of the second derivative of the perturbing potential with respect to the vertical coordinate, evaluated on the straight-line solution, with an exponential function of time has at least one pole in the complex time domain with non-zero residue, then the perturbed system does not possess a holomorphic integral. In the last section the theory is applied to a polynomial potential whose non-integrability cannot be proved by Mel’nikov’s method, since the integrable part does not possess a homoclinic surface. Yoshida’s w27x theorem is also not applicable in this case, since the homogeneous parts of higher and lower degree are integrable. Finally, for the application of Poincare’s ´ or Kozlov’s theorems, one needs open neighbourhoods of bounded motion, i.e. the unperturbed part must possess invariant tori which is not the case in this system, where the unperturbed potential possesses no bounded motions other than an isolated family of straight line solutions.
2. Perturbed separable potentials We consider two degrees of freedom Hamiltonians of the following form H s 12 Ž p x2 q p y2 . q V01 Ž x . q V02 Ž y . q ´ V1 Ž x , y, ´ . .
Ž 1.
The above Hamiltonian describes planar motion of a particle in the potential V s V01 q V02 q ´ V1 and has a separable part H0 s 12 Ž p x2 q p y2 . q V01 Ž x . q V02 Ž y . .
Ž 2.
and a perturbation in the potential H1 s V1 Ž x , y, ´ . ,
Ž 3.
analytic with respect to ´ g R in an open neighbourhood of ´ s 0. The Hamiltonian H is also analytic with respect to its arguments h s Ž x, y, p x , p y .T g C 4 . We suppose that the Hamiltonian Ž2. has a family of straight line solutions ŽSLS. on the x-axis, that is we suppose that dV02
Ž 0. s 0 .
dy
Ž 4.
and also that at least one of these straight line solutions is periodic with some real period T. We are going to consider cases of Hamiltonians of the form Ž1. for which the T-periodic SLS on the x-axis of the separable part Ž2., is preserved for ´ / 0. This means that if the unperturbed SLS is given by the periodic function h 0 Ž t ., the perturbed SLS is given by the same function. We also impose the condition that this SLS with real period T is a single valued function of the complex time t. We will see later on that it is essential for our method that the SLS has poles. The necessary and sufficient conditions for the SLS to be preserved are
E V1 Ex
Ž x ,0, ´ . s 0 ,
E V1 Ey
Ž x ,0, ´ . s 0
Ž 5.
i.e. the perturbing force must be zero on the SLS. Finally we also suppose that d 2 V02 dy 2
Ž 0. / 0 .
Ž 6.
E. Meletlidour Physics Letters A 270 (2000) 47–54
The conditions Ž4. and Ž6. result to the existence of a non-degenerate extremum of the function V02 Ž y . at y s 0. Later on this extremum will be considered a maximum, so that the SLS will be unstable with respect to vertical perturbations. We write Hamilton’s equations in the form
h˙ a s V a b
EH
first order perturbation of h10 with respect to the initial conditions. Since = V1Ž x,0, ´ . s 0, there is no perturbation of the SLS from the unperturbed to the perturbed system, and therefore h10 Ž t . ' 0. By introducing the above expansion to the equations of motion Ž7., after some straightforward calculations, we obtain
Ž 7.
Eh b
a h˙ 00 sV ab
where
Vs
ž
O yI
I O
a h˙ 01 sV ab
/
is the matrix of the standard symplectic structure. O is the zero and I the unit 2 = 2 matrices. Let h Ž t . be a solution of the perturbed Hamiltonian with initial conditions that belong to a neighbourhood U g C 4 of the initial conditions of the unperturbed SLS. We expand this solution with respect to ´
h Ž t . s h 0 Ž t . q ´h1 Ž t . q O Ž ´ 2 .
a h˙ 11 sV ab
E H0 Eh b
, h 00
E 2 H0 Eh bEh g
g h 01 , h 00
E 2 H0 Eh bEh g
qV ab
ž
g h11 h 00
E 3 H0 Eh bEh gEh d
d h10 q h 00
E 2 H1 Eh bEh g
h 00
/
g h 01 .
Ž 10 .
Ž 8.
For real time t it is known Žw28x, p. 12. that the perturbed solutions are analytic with respect to their arguments, i.e. x 0 , p x 0 ,t and ´ . For complex time Poincare´ has proved Žw1x, Ch. 2; w29x, p. 297. that the perturbed solutions are analytic functions of ´ for < ´ < - ´ 0 and can be expanded in convergent power series of ´ with coefficients that depend on time for those paths on complex time for which the unperturbed solutions have no singularities. This does not exclude the possibility of h1 ,h 2 , . . . being multivalued functions of complex time. Anyway, our approach will depend on the multivaluedness of the perturbed solution. We also expand the solution with respect to small perturbations to the initial conditions around the SLS, which are governed by a small parameter d . This gives
h Ž t . s h 00 Ž t . q dh 01 Ž t . q . . . q´h10 Ž t . q ´dh11 Ž t . q . . . .
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Ž 9.
Notice that h 00 s Ž x,Ž t ., p x Ž t .0,0.T is the unperturbed SLS, h10 is the first order perturbation of the SLS, h 01 is the first order perturbation of h 00 with respect to the initial conditions and finally h11 is the
In the following, subscripts x and y in the components of the potential will denote partial differentiation. By considering the pertinent form of the Hamiltonian Ž1., Eqs. Ž10. attain the form x˙ 00 s p x 00 ,
y˙ 00 s p y 00 s 0 ,
p˙ x 00 s yV01 x ,
p˙ y 00 s yV02 y s 0 , x˙ 01 s p x 01 ,
y˙ 01 s p y 01 ,
p˙ y 01 s yV02 y y y 01 , x˙ 11 s p x11 ,
Ž 11 . p˙ x 01 s yV01 x x x 01 ,
Ž 12 .
y˙ 11 s p y11 ,
p˙ x11 s yV01 x x x 11 q a x 01 , p˙ y11 s yV02 y y y 11 q b y 01 ,
Ž 13 .
where hi j s Ž x i j , yi j , p x i j , p y i j .T, i, j g 0,14 , a s yV01 x x x x 10 y V1 x x and b s yV1 y y . In Eqs. Ž12., Ž13. the derivatives of the potential are evaluated on the SLS h 00 Ž t ., which satisfies Eqs. Ž11.. The variational equations of the perturbed SLS are decomposed to the tangent variational equations ŽTVE. and to the normal variational equations ŽNVE.. Since the continued orbit still remains a straight line,
E. Meletlidour Physics Letters A 270 (2000) 47–54
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the normal coordinates are y, p y . Let h˜ 1 s Ž y 1 , p y1 .T be the O Ž d . perturbation with respect to the initial conditions of the SLS for ´ / 0. They evolve in time as
Ý ´ i yi1Ž t .
and
is0 `
p y1 Ž t . s
`
D0 Ž t y t 0 . s exp A Ž t y t 0 . s
Ý
Ž t y t0 .
is0
i!
i
Ai
is the matrizant of Ž14., i.e. a fundamental solution such that for t s t 0 , D0 Ž0. s I. Its value D0 ŽT . at t s t 0 q T is the monodromy matrix of system Ž14.. It can be easily seen that A2 k s n k I and A2 kq1 s k n A, so that
`
y1Ž t . s
where
Ý ´ i p y i1Ž t . . is0
The right hand side infinite sums are convergent, as can be established if we apply Poincare’s ´ theorem to the NVE. From now on we define h˜ i j s Ž yi j , p y i j .T. The NVE of zeroth and first order with respect to ´ , as can be seen from Eqs. Ž12. and Ž13., are
h˜˙ 01 s Ah˜ 01
Ž 14 .
and
h˜˙ 11 s Ah˜ 11 q Bh˜ 01
Ž 15 .
respectively, where As
0
1 , 0
žn /
Bs
ž
0 b
0 0
D0 Ž t y t 0 . s Icosh Ž v Ž t y t 0 . . 1 q
v
Asinh Ž v Ž t y t 0 . .
Ž 18 .
where v s 'n . Note that if the unperturbed SLS is vertically stable, n - 0 while if it is unstable, n ) 0. For reasons that will become clear later we will consider the case of vertical instability and v will be considered real. Both A and D0 become diagonal by the same linear change of coordinates with a constant matrix Us 1 v
ž
1 . yv
/
Ž 19 .
The new coordinates j s Ž j 1 , j 2 .T are defined by h˜ 01 s Uj and the linear system Ž14. becomes
/
Ž 16 .
and n s yV02 y y Ž0. / 0 is a constant, while b Ž t . s yV1 y y Ž x Ž t .,0,0. is a known T-periodic function of time, evaluated on the unperturbed SLS.
j˙s A)j
Ž 20 .
where A) s Uy1AU s diagŽ v ,y v .. The solution of Ž20. is
j Ž t . s D0) Ž t y t 0 . j Ž t 0 .
Ž 21 .
D0) Ž t y t 0 . s diagŽe v Ž tyt 0 .,ey v Ž tyt 0 . ..
3. Solution of the NVE of the unperturbed and perturbed systems The matric A of the linear Eq. Ž14. is a constant matrix and system Ž14. is autonomous. This is not so in the general case of a periodic orbit of a general Hamiltonian system. For the present case, we know that the solution of the linear system Ž14. with constant coefficients will depend on time through t y t 0 , where t 0 is an arbitrary initial time and it is of the form
h˜ 01 Ž t . s D0 Ž t y t 0 . h˜ 01 Ž t 0 .
Ž 17 .
where Since system Ž14. is a linear Hamiltonian system, matrix A is a Hamiltonian matrix Žsee e.g. w30x, p. 34. and its eigenvalues appear in opposite pairs, while D0 is a symplectic matrix and its eigenvalues form a reciprocal pair, as can also be seen by the above diagonalization. The perturbed variational Eqs. Ž15. are of the form
h˜˙ 11 s Ah˜ 11 q Bh˜ 01 .
Ž 22 .
By transforming h˜ 11 by the linear transformation U, in the new variables z defined by h˜ 11 s Uz , the linear system Ž22. transforms to
z˙s A)z q B )j
Ž 23 .
E. Meletlidour Physics Letters A 270 (2000) 47–54
where B ) s Uy1 BU. The solution of this system is given by
z Ž t . s D0) Ž t y t 0 . z Ž t 0 . q
t
Ht D
)y1 0
Ž s y t0 .
0
=B )j Ž s y t 0 . ds .
Ž 24 .
Since j is the solution Ž21. of the homogeneous system Ž20., we find for z
z Ž t . s D0) Ž t y t 0 . z Ž t 0 . q D1) Ž t ,t 0 . j Ž t 0 . t
Ht D
)y1 0
Ž s y t0 . B )
0
= D0) Ž s y t 0 . ds .
Ž 26 .
After some straightforward calculations we find that
D1) Ž t ,t 0 . s
ž
a Ž t ,t 0 .
b Ž t ,t 0 .
c Ž t ,t 0 .
d Ž t ,t 0 .
/
Ž 27 .
Ý f k Ž t . z1k z 2myk .
Ž 29 .
The coefficients f k depend on time only through the SLS Ž x Ž t,t 0 ., p x Ž t,t 0 ... Thus Ž29. is a univalued periodic function with respect to time, with the same period T of the perturbed SLS, which coincides with the period of the unperturbed SLS. By expanding z and f k with respect to ´ , f k s Ý`ls0 ´ l f l k , for some s g N we obtain m
Fs´ s
where
b Ž t ,t 0 . s
m
Fs
ks0
D1) Ž t ,t 0 . s D0) Ž t y t 0 .
a Ž t ,t 0 . s
Note that the z’s remain analytic functions of ´ around ´ s 0, due to the fact that y 1Ž t ., p y1Ž t . are analytic functions of ´ around ´ s 0. Ziglin w2;3x has proved that the NVE of a solution of a Hamiltonian system with a second integral of motion independent of the Hamiltonian, possess an integral which is a homogeneous polynomial in the normal variations. Therefore, the NVE of the SLS of the perturbed Hamiltonian system have an integral of motion
Ž 25 .
where
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e v Ž tyt 0 . 2v
c Ž t ,t 0 . s y d Ž t ,t 0 . s y
f sk j 1k j 2myk
ks0 t
m
Ht b ds ,
q´ sq 1
0
Ý
f Ž sq1 . k j 1k j 2myk
ks0
e v Ž tyt 0 . 2v
Ý
t
Ht
b ey2 v Ž syt 0 . ds ,
m
q
0
eyv Ž tyt 0 . 2v eyv Ž tyt 0 . 2v
Ý
f sk Ž Ž m y k . j 1k j 2myky1z 2
ks0 t
Ht b e
2 v Ž syt 0 .
ds ,
q k j 1ky 1j 2mykz 1 . q O Ž ´ sq2 . .
0
t
Ht b ds .
Ž 28 .
0
All the above calculations are both valid for real and complex time. Notice that D1Ž t,t 0 . is no longer univalued, if the integrands have poles in complex time. This is a common property of solutions of linear systems in the complex domain.
Ziglin’s theorem guarantees that the integral of the NVE of the continued orbit of the perturbed system is non-trivial. This leads to the conclusion that there always exists an integer s, such that at least one coefficient f sk Ž t . is not identically zero. This property is essential for the conclusion of the proof. Dividing the above expression by ´ s, is equivalent to putting s s 0, i.e. m
4. Ziglin’s lemma for the existence of an integral of the NVE
Fs
m
Ý
f 0 k j 1k j 2myk q ´
ks0
Ý
f 1 k j 1k j 2myk
ks0 m
We transform the NVE in the basis where D0 Ž t y t 0 . becomes diagonal, i.e. in coordinates z s Ž z 1 , z 2 .T defined by
h˜ 1 s Uz s Uj q ´ Uz q O Ž ´ 2 . .
q
Ý
f 0 k Ž Ž m y k . j 1k j 2myky1z 2
ks0
qk j 1ky 1j 2mykz 1 . q O Ž ´ 2 . .
Ž 30 .
E. Meletlidour Physics Letters A 270 (2000) 47–54
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The above series is convergent with respect to ´ , since both the integral of the perturbed Hamiltonian and z are analytic functions of ´ . At zeroth order we obtain an integral of the unperturbed variational Eqs. Ž20. which is m
F0s
f 0 k j 1k j 2myk
Ý
Ž 31 .
ks0
and, at first order in ´ , the following integral of the joint system Ž20. and Ž23. m
F1 s
ks0 m
q
Ý
f 0 k Ž Ž m y k . j 1k j 2myky1z 2
ks0
qk j 1ky 1j 2my kz 1 . .
Ž 32 .
Let r ,1rr be the eigenvalues of the monodromy matrix D0 ŽT . of the unperturbed NVE Ž14.. This matrix is called non-resonant if the eigenvalues are not roots of unit. It is known w2;3x that, when the integral Ž31. is expressed in the variables that diagonalize the monodromy matrix, either there is a resonance condition in the eigenvalues, i.e. an integer n exists, such that r n s 1, or the integral, at t s t 0 , is of the form m m
t 0 j 12
F 0 Ž t0 . s f 0 Ž . 2
m
Ž t 0 . j 22 Ž t 0 .
Ž 33 .
for some even integer m. This is proved by replacing j Ž t 0 q T . s D0) ŽT . j Ž t 0 . in the equation F 0 Ž t 0 q T . s F 0 Ž t 0 . and using the fact that f 0 k are T-periodic functions of time. We are going to assume that the monodromy matrix of the unperturbed NVE D0 ŽT . is non-resonant, i.e. we are going to exclude the case where the eigenvalues e " v T are roots of unit. Since these roots are dense on the unit circle, this assumption cannot be checked for imaginary v , and for this reason we assumed that v is real, i.e. that the unperturbed SLS is unstable with respect to normal variations. Thus, the integral F 0 Ž t 0 . of the unperturbed NVE is in our case of the form Ž33.. Now we equate
F 1Ž t0 q T . s F 1Ž t0 .
m 2
which are the coefficients of all the monomials except j 1 q1j 2 y1, j 1 y1j 2 q1 , j 1 j 2 . The coefficient f 1 Ž t 0 . may be different from zero, but is eliminated from Ž34.. The remaining non-zero coefficients of the above monomials satisfy the equations m f m Ž d r q a ry1 . s 0 , 2 02 m f 1Ž m q1. r 2 q f 0 m c r y f 1Ž m q1. s 0 , 2 2 2 2 m f 1Ž m y1. ry2 q f 0 m bry1 y f 1Ž m y1. s 0 Ž 35 . 2 2 2 2 m 2
m 2
m 2
m 2
m 2
m 2
m 2
f 1 k j 1k j 2myk
Ý
and replace j Ž t 0 q T ., z Ž t 0 q T . by the solutions Ž21. and Ž25. for t s t 0 q T. Due to the fact that only f 0 is different from zero, the arrangement of the terms of the polynomials in Ž32., after substituting j and z by their solutions Ž21. and Ž25. respectively, leads to m m m f1 k Ž t0 . s 0 ; k / y 1 , , q 1 , 2 2 2
Ž 34 .
where the values of f 0 , f 1Ž q1. , f 1Ž y1. are considered for t s t 0 , since they are T-periodic functions. Notice also that a,b,c,d that enter in the equations are given by Ž28. with upper limit t s t 0 q T, since F 1 , and therefore j and z , are evaluated at this t. Since the integral of the NVE of the perturbed SLS is non-trivial, for suitable t 0 it holds that f 0 Ž t 0 . / 0. Therefore Eq. Ž35a. gives m 2
m 2
m 2
m 2
d r q a ry1 s 0 . It is easily seen from Ž28. that this equation is satisfied identically. In Eqs. Ž35b. and Ž35c. f 0 / 0, f 1Ž q1. , f 1Ž y1. are univalued functions of t 0 and, by solving with respect to c and b respectively, we obtain m 2
m 2
t 0qT
Ht
m 2
b Ž s y t 0 . e 2 v Ž syt 0 . ds s const.s c1 ,
0
t 0qT
Ht
b Ž s y t 0 . ey2 v Ž syt 0 . ds s const.s c 2
Ž 36 .
0
where the constants c1 , c 2 depend on f 0 Ž t 0 ., f 1Ž " 1.Ž t 0 . and also on v ,T. However, the integrals on the left-hand sides can be taken from t 0 to t 0 q T along any path in the complex time plane. If the composition of two such paths encircles a pole either m 2
m 2
E. Meletlidour Physics Letters A 270 (2000) 47–54
of the function b Ž t .e 2 v t or the function b Ž t .ey2 v t with nonzero residue, then either of these two integrals along the different paths attains different values. This contradicts the fact that, according to Ž36. these integrals must equal to constants, which is a necessary condition for the existence of an additional independent integral of motion of the perturbed Hamiltonian system. If therefore either of the functions b Ž t .e " 2 v t has a pole with non-zero residue, the perturbed Hamiltonian system does not possess a second integral of motion, analytic with respect to ´ and holomorphic with respect to x, y, p x , p y at least in an open neighbourhood of the SLS and independent of H. Note that, since functions e " v t are analytic, the poles of the integrands coincide with the poles of b . Thus we have proved the following
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non-integrability by applying Mel’nikov’s w25x method. Moreover, the parts of lower and higher degree in the potential are by themselves integrable, so neither Yoshida’s w27x theorem is applicable. The straight line solution of the unperturbed and perturbed Hamiltonians on the x-axis is Žsee e.g. w31x, p. 207. x Ž t y t 0 . s Ccn l Ž t y t 0 . ,k where l,t 0 are arbitrary constants, while k 2 s Ž l2 y 1.rŽ2 l2 . and C 2 s l2 y 1. This is a periodic solution with period T s 4 K Ž k .rl, where K is the complete elliptic integral of the first kind. We see that V02 y y s y1 so the SLS is vertically unstable and we evaluate
b Ž t . s yV1 y y Ž x Ž t y t 0 . ,0,0 . s y2Ccn l Ž t y t 0 . ,k .
Theorem 1. Consider the Hamiltonian H s 12 Ž p x2 q p y2 . q V01 Ž x . q V02 Ž y . q ´ V1 Ž x , y, ´ . where V0 1 ,V0 2 ,V1 are holomorphic functions of (x,y,p x ,p y ) g C 4 and analytic with respect to ´ g R , for which the following conditions are satisfied: (i) V0 2 y (0) s 0, (ii) = V1(x,0,´ ) s 0, (iii) V0 2 y y (0) is strictly negatiÕe. Then, if either of the functions V1 y y e 2 v t or V1 y y e y 2 v t where v s (yV0 2 y y )1r 2 , eÕaluated along the unperturbed straight-line solution, haÕe at least one pole in the complex time domain with non-zero residue, the Hamiltonian H is non-integrable, i.e. it does not haÕe an integral of motion, independent of H, holomorphic in an open neighbourhood of the straight line solution and analytic with respect to ´ g R , in an open interÕal of ´ around zero.
5. Application and conclusions We will apply the previous results to the system described by the Hamiltonian H s 12 Ž p x2 q p y2 . q 12 x 2 q 14 x 4 y 12 y 2 q ´ xy 2 .
Ž 37 . Note that, since the integrable part of H does not possess a homoclinic loop, one cannot prove its
Since this function has simple poles on the complex t-plane, then expŽ"v t . b Ž t . have also simple poles and the Hamiltonian H is non-integrable for ´ / 0 in an open neighbourhood around zero. In order to attain the above results, we have used the period-T monodromy matrix of the NVE of the unperturbed SLS. The form of the Hamiltonian has simplified the calculations considerably. The criterion falls, as has been expected, on the branching of the solutions of the perturbed NVE. On the other hand, we do not want the SLS to branch at all, since we assumed that the unperturbed SLS is preserved in the same form under the perturbation. We have not used the monodromy group of the NVE of the unperturbed SLS, since it is trivial, i.e. it is the unit matrix and first order perturbations do not guarantee that the perturbed monodromy group possesses a non-resonant element. In the case where the NVE of the unperturbed SLS have non-trivial monodromy group, the calculations would be, although straightforward, much more sophisticated. It is certainly a more general case that can be applied to a general class of perturbed integrable Hamiltonians but on the other hand its applicability is restrained by the need of knowing the solution of the unperturbed NVE which has no more constant coefficients but is a Hill equation. However, the present case cannot be treated through this general method, due to the triviality of the monodromy group.
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E. Meletlidour Physics Letters A 270 (2000) 47–54
Acknowledgements The author wishes to thank Drs. S. Ichtiaroglou and S. Tanabe for very fruitful discussions. This work was partially supported by a post-doctoral fellowship of the Greek Scholarship Foundation ŽI.K.Y..
References w1x H. Poincare, Nouvelles de la Mechanique ´ 1892, Les Methodes ´ ´ Celeste, vol. I, Gauthier-Villars, Paris. English translation: ´ Goroff D.L. ŽEd.., New Methods in Celestial Mechanics, American Institute of Physics, 1993. w2x S.L. Ziglin, Func. Anal. Appl. 16 Ž1983. 181. w3x S.L. Ziglin, Func. Anal. Appl. 17 Ž1983. 6. w4x H. Yoshida, Physica D 29 Ž1987. 128. w5x J.J. Moralez-Ruiz, C. Simo, ´ J. Diff. Eq. 107 Ž1994. 140. w6x J.J. Morales-Ruiz, C. Ramis, 1998a, Galoisian obstructions to integrability of Hamiltonian systems Žpreprint.. w7x J.J. Morales-Ruiz, C. Ramis, 1998b, Galoisian obstructions to integrability of Hamiltonian systems II Žpreprint.. w8x H. Ito, Kodai Math. J. 8 Ž1995. 120. w9x H. Ito, ZAMP 38 Ž1987. 459. w10x H. Ito, Bol. Soc. Bras. Mat. 21 Ž1990. 95. w11x S. Ichtiaroglou, J. Phys. A: Math. Gen. 22 Ž1989. 3461. w12x M.A. Almeida, I.C. Moreira, H. Yoshida, J. Phys. A: Math. Gen. 25 Ž1992. L227. w13x M. Irigoyen, C. Simo, ´ Celest. Mech. Dyn. Astr. 55 Ž1993. 281.
w14x J.-P. Franc¸oise, M. Irigoyen, Geom. Phys. 10 Ž1993. 231. w15x J.M. Ferrandiz, M.E. Sansaturio, Phys. Lett. A 207 Ž1995. ´ 180. w16x J.M. Ferrandiz, M.E. Sansaturio, I. Vigo, Phys. Lett. A 221 ´ Ž1996. 153. w17x M.E. Sansaturio, I. Vigo-Aguiar, J.M. Ferrandiz, J. Phys. A: ´ Math. Gen. 30 Ž1997. 5869. w18x M.E. Sansaturio, I. Vigo-Aguiar, J.M. Ferrandiz, in: B.A. ´ Steves, A.E. Roy ŽEds.., The Dynamics of Small Bodies in the Solar System, Kluwer, the Netherlands, 1999, pp. 295– 302. w19x M.I. Vigo-Aguiar, 1999, No integrabilidad del problema del satelite, Doctor’s Thesis, Universidad de Alicante. ´ w20x R.C. Churchill, D.L. Rod, J. Diff. Eq. 76 Ž1988. 91. w21x R.C. Churchill, D.L. Rod, SIAM J. Math. Annal. 2 Ž1991. 1790. w22x E. Meletlidou, S. Ichtiaroglou, Physica D 71 Ž1994. 261. w23x E. Meletlidou, S. Ichtiaroglou, J. Phys. A 27 Ž1994. 3919. w24x E. Meletlidou, S. Ichtiaroglou, Phys. Lett. A 188 Ž1994. 157. w25x V.K. Mel’nikov, Trans. Moscow Math. Soc. 12 Ž1963. 3. w26x V.V. Kozlov, J. Appl. Math. Mech. 42 Ž1979. 420. w27x H. Yoshida, Commun. Math. Phys. 116 Ž1988. 529. w28x G.D. Birkhoff, 1927, Dynamical Systems, Colloquium Publications, vol. IX, Am. Math. Soc., Providence, Rhode Island, Acta Math. 50. w29x V.V. Kozlov, 1996, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, Berlin. w30x K.R. Meyer, G.R. Hall, 1992, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer, New York. w31x H.T. Davis, 1962, Introduction to Nonlinear Differential and Integral Equations, Dover, New York.
Physics Letters A 270 Ž2000. 47–54 www.elsevier.nlrlocaterpla
A non-integrability test for perturbed separable planar Hamiltonians Efi Meletlidou Department of Physics, UniÕersity of Thessaloniki, 54006 Thessaloniki, Greece Received 6 September 1999; received in revised form 27 March 2000; accepted 30 March 2000 Communicated by A.P. Fordy
Abstract We prove the non-integrability of perturbed separable planar potentials by using the perturbed normal variational equations, with a reasoning analogous to Ziglin’s theorem. We apply the above criterion to a Hamiltonian that cannot be proved non-integrable by other known non-integrability criteria. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.20.q i; 05.45.q b; 45.10.q z Keywords: Hamiltonian systems; Non-integrability; Ziglin’s theorem
1. Introduction It is already known from the work of Poincare´ Žw1x, Ch. 4. that if a Hamiltonian H possesses a second integral of motion I, then the linear system of the variational equations around a periodic solution of H possesses an integral, linear in the variations, which depends periodically on time through = I, evaluated on the periodic solution. In many cases, however, this integral turns out to be a function of the corresponding integral, produced by the Hamiltonian, or merely a trivial constant. Ziglin w2;3x generalized this result of Poincare´ by showing that, if I is an independent integral of H
E-mail address: [email protected] ŽE. Meletlidou..
which may be dependent on H on the specific solution, then the variational equations always possess an independent non-trivial integral, homogeneous of some degree m G 1 in the variations, with coefficients that depend periodically on time through the mth order derivatives of I, evaluated on the periodic solution. By using this result, finally he proved an important non-integrability theorem, involving properties of the monodromy group of the normal variational equations in the complex time domain. By applying Ziglin’s theorem, Yoshida w4x proposed an algorithmic criterion for the case of planar motion of a point mass in a homogeneous potential of degree k / 0, "2. Important extensions of Ziglin’s theory have been recently made by Morales-Ruiz and Simo´ w5x and Morales-Ruiz and Ramis w6;7x. Applications of the above theorems in various systems were made in w8–19x. The geometrical sig-
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 2 5 4 - 1
48
E. Meletlidour Physics Letters A 270 (2000) 47–54
nificance of Ziglin’s theorem has been explored in w20;21x. There are several other non-integrability theorems apart from Ziglin’s results. Poincare’s ´ Žw1x, Ch. 5. theorem on non-integrability, although showed the genericity of this property, is difficult to apply in specific systems. For two non-integrability criteria that are closely related to Poincare’s ´ theorem, see w22–24x. Another criterion for establishing whether a perturbed system is non-integrable Žand also chaotic. is the Poincare–Mel’nikov homoclinic integral w25x. ´ If this integral, evaluated along an unperturbed homoclinic loop, has simple zeros, it guarantees the existence of transverse homoclinic orbits in the perturbed system, which is an obstacle to integrability. Poincare’s ´ and Mel’nikov’s theorems depend on the real time dynamics and are of a perturbative nature. Another theorem for perturbed Hamiltonians, that deals with the complex time dynamics is Kozlov’s w26x theorem, which examines the branching of the unperturbed solution to the perturbed one. In this Letter we consider the problem of the planar motion of a particle in a potential that consists of a part that is separable in cartesian coordinates and a small coupling term. We assume that a family of vertically unstable straight-line periodic solutions that are also single valued functions in complex time exists along one of the axes. We also assume that it is preserved under the perturbation. It is proved that, if the product of the second derivative of the perturbing potential with respect to the vertical coordinate, evaluated on the straight-line solution, with an exponential function of time has at least one pole in the complex time domain with non-zero residue, then the perturbed system does not possess a holomorphic integral. In the last section the theory is applied to a polynomial potential whose non-integrability cannot be proved by Mel’nikov’s method, since the integrable part does not possess a homoclinic surface. Yoshida’s w27x theorem is also not applicable in this case, since the homogeneous parts of higher and lower degree are integrable. Finally, for the application of Poincare’s ´ or Kozlov’s theorems, one needs open neighbourhoods of bounded motion, i.e. the unperturbed part must possess invariant tori which is not the case in this system, where the unperturbed potential possesses no bounded motions other than an isolated family of straight line solutions.
2. Perturbed separable potentials We consider two degrees of freedom Hamiltonians of the following form H s 12 Ž p x2 q p y2 . q V01 Ž x . q V02 Ž y . q ´ V1 Ž x , y, ´ . .
Ž 1.
The above Hamiltonian describes planar motion of a particle in the potential V s V01 q V02 q ´ V1 and has a separable part H0 s 12 Ž p x2 q p y2 . q V01 Ž x . q V02 Ž y . .
Ž 2.
and a perturbation in the potential H1 s V1 Ž x , y, ´ . ,
Ž 3.
analytic with respect to ´ g R in an open neighbourhood of ´ s 0. The Hamiltonian H is also analytic with respect to its arguments h s Ž x, y, p x , p y .T g C 4 . We suppose that the Hamiltonian Ž2. has a family of straight line solutions ŽSLS. on the x-axis, that is we suppose that dV02
Ž 0. s 0 .
dy
Ž 4.
and also that at least one of these straight line solutions is periodic with some real period T. We are going to consider cases of Hamiltonians of the form Ž1. for which the T-periodic SLS on the x-axis of the separable part Ž2., is preserved for ´ / 0. This means that if the unperturbed SLS is given by the periodic function h 0 Ž t ., the perturbed SLS is given by the same function. We also impose the condition that this SLS with real period T is a single valued function of the complex time t. We will see later on that it is essential for our method that the SLS has poles. The necessary and sufficient conditions for the SLS to be preserved are
E V1 Ex
Ž x ,0, ´ . s 0 ,
E V1 Ey
Ž x ,0, ´ . s 0
Ž 5.
i.e. the perturbing force must be zero on the SLS. Finally we also suppose that d 2 V02 dy 2
Ž 0. / 0 .
Ž 6.
E. Meletlidour Physics Letters A 270 (2000) 47–54
The conditions Ž4. and Ž6. result to the existence of a non-degenerate extremum of the function V02 Ž y . at y s 0. Later on this extremum will be considered a maximum, so that the SLS will be unstable with respect to vertical perturbations. We write Hamilton’s equations in the form
h˙ a s V a b
EH
first order perturbation of h10 with respect to the initial conditions. Since = V1Ž x,0, ´ . s 0, there is no perturbation of the SLS from the unperturbed to the perturbed system, and therefore h10 Ž t . ' 0. By introducing the above expansion to the equations of motion Ž7., after some straightforward calculations, we obtain
Ž 7.
Eh b
a h˙ 00 sV ab
where
Vs
ž
O yI
I O
a h˙ 01 sV ab
/
is the matrix of the standard symplectic structure. O is the zero and I the unit 2 = 2 matrices. Let h Ž t . be a solution of the perturbed Hamiltonian with initial conditions that belong to a neighbourhood U g C 4 of the initial conditions of the unperturbed SLS. We expand this solution with respect to ´
h Ž t . s h 0 Ž t . q ´h1 Ž t . q O Ž ´ 2 .
a h˙ 11 sV ab
E H0 Eh b
, h 00
E 2 H0 Eh bEh g
g h 01 , h 00
E 2 H0 Eh bEh g
qV ab
ž
g h11 h 00
E 3 H0 Eh bEh gEh d
d h10 q h 00
E 2 H1 Eh bEh g
h 00
/
g h 01 .
Ž 10 .
Ž 8.
For real time t it is known Žw28x, p. 12. that the perturbed solutions are analytic with respect to their arguments, i.e. x 0 , p x 0 ,t and ´ . For complex time Poincare´ has proved Žw1x, Ch. 2; w29x, p. 297. that the perturbed solutions are analytic functions of ´ for < ´ < - ´ 0 and can be expanded in convergent power series of ´ with coefficients that depend on time for those paths on complex time for which the unperturbed solutions have no singularities. This does not exclude the possibility of h1 ,h 2 , . . . being multivalued functions of complex time. Anyway, our approach will depend on the multivaluedness of the perturbed solution. We also expand the solution with respect to small perturbations to the initial conditions around the SLS, which are governed by a small parameter d . This gives
h Ž t . s h 00 Ž t . q dh 01 Ž t . q . . . q´h10 Ž t . q ´dh11 Ž t . q . . . .
49
Ž 9.
Notice that h 00 s Ž x,Ž t ., p x Ž t .0,0.T is the unperturbed SLS, h10 is the first order perturbation of the SLS, h 01 is the first order perturbation of h 00 with respect to the initial conditions and finally h11 is the
In the following, subscripts x and y in the components of the potential will denote partial differentiation. By considering the pertinent form of the Hamiltonian Ž1., Eqs. Ž10. attain the form x˙ 00 s p x 00 ,
y˙ 00 s p y 00 s 0 ,
p˙ x 00 s yV01 x ,
p˙ y 00 s yV02 y s 0 , x˙ 01 s p x 01 ,
y˙ 01 s p y 01 ,
p˙ y 01 s yV02 y y y 01 , x˙ 11 s p x11 ,
Ž 11 . p˙ x 01 s yV01 x x x 01 ,
Ž 12 .
y˙ 11 s p y11 ,
p˙ x11 s yV01 x x x 11 q a x 01 , p˙ y11 s yV02 y y y 11 q b y 01 ,
Ž 13 .
where hi j s Ž x i j , yi j , p x i j , p y i j .T, i, j g 0,14 , a s yV01 x x x x 10 y V1 x x and b s yV1 y y . In Eqs. Ž12., Ž13. the derivatives of the potential are evaluated on the SLS h 00 Ž t ., which satisfies Eqs. Ž11.. The variational equations of the perturbed SLS are decomposed to the tangent variational equations ŽTVE. and to the normal variational equations ŽNVE.. Since the continued orbit still remains a straight line,
E. Meletlidour Physics Letters A 270 (2000) 47–54
50
the normal coordinates are y, p y . Let h˜ 1 s Ž y 1 , p y1 .T be the O Ž d . perturbation with respect to the initial conditions of the SLS for ´ / 0. They evolve in time as
Ý ´ i yi1Ž t .
and
is0 `
p y1 Ž t . s
`
D0 Ž t y t 0 . s exp A Ž t y t 0 . s
Ý
Ž t y t0 .
is0
i!
i
Ai
is the matrizant of Ž14., i.e. a fundamental solution such that for t s t 0 , D0 Ž0. s I. Its value D0 ŽT . at t s t 0 q T is the monodromy matrix of system Ž14.. It can be easily seen that A2 k s n k I and A2 kq1 s k n A, so that
`
y1Ž t . s
where
Ý ´ i p y i1Ž t . . is0
The right hand side infinite sums are convergent, as can be established if we apply Poincare’s ´ theorem to the NVE. From now on we define h˜ i j s Ž yi j , p y i j .T. The NVE of zeroth and first order with respect to ´ , as can be seen from Eqs. Ž12. and Ž13., are
h˜˙ 01 s Ah˜ 01
Ž 14 .
and
h˜˙ 11 s Ah˜ 11 q Bh˜ 01
Ž 15 .
respectively, where As
0
1 , 0
žn /
Bs
ž
0 b
0 0
D0 Ž t y t 0 . s Icosh Ž v Ž t y t 0 . . 1 q
v
Asinh Ž v Ž t y t 0 . .
Ž 18 .
where v s 'n . Note that if the unperturbed SLS is vertically stable, n - 0 while if it is unstable, n ) 0. For reasons that will become clear later we will consider the case of vertical instability and v will be considered real. Both A and D0 become diagonal by the same linear change of coordinates with a constant matrix Us 1 v
ž
1 . yv
/
Ž 19 .
The new coordinates j s Ž j 1 , j 2 .T are defined by h˜ 01 s Uj and the linear system Ž14. becomes
/
Ž 16 .
and n s yV02 y y Ž0. / 0 is a constant, while b Ž t . s yV1 y y Ž x Ž t .,0,0. is a known T-periodic function of time, evaluated on the unperturbed SLS.
j˙s A)j
Ž 20 .
where A) s Uy1AU s diagŽ v ,y v .. The solution of Ž20. is
j Ž t . s D0) Ž t y t 0 . j Ž t 0 .
Ž 21 .
D0) Ž t y t 0 . s diagŽe v Ž tyt 0 .,ey v Ž tyt 0 . ..
3. Solution of the NVE of the unperturbed and perturbed systems The matric A of the linear Eq. Ž14. is a constant matrix and system Ž14. is autonomous. This is not so in the general case of a periodic orbit of a general Hamiltonian system. For the present case, we know that the solution of the linear system Ž14. with constant coefficients will depend on time through t y t 0 , where t 0 is an arbitrary initial time and it is of the form
h˜ 01 Ž t . s D0 Ž t y t 0 . h˜ 01 Ž t 0 .
Ž 17 .
where Since system Ž14. is a linear Hamiltonian system, matrix A is a Hamiltonian matrix Žsee e.g. w30x, p. 34. and its eigenvalues appear in opposite pairs, while D0 is a symplectic matrix and its eigenvalues form a reciprocal pair, as can also be seen by the above diagonalization. The perturbed variational Eqs. Ž15. are of the form
h˜˙ 11 s Ah˜ 11 q Bh˜ 01 .
Ž 22 .
By transforming h˜ 11 by the linear transformation U, in the new variables z defined by h˜ 11 s Uz , the linear system Ž22. transforms to
z˙s A)z q B )j
Ž 23 .
E. Meletlidour Physics Letters A 270 (2000) 47–54
where B ) s Uy1 BU. The solution of this system is given by
z Ž t . s D0) Ž t y t 0 . z Ž t 0 . q
t
Ht D
)y1 0
Ž s y t0 .
0
=B )j Ž s y t 0 . ds .
Ž 24 .
Since j is the solution Ž21. of the homogeneous system Ž20., we find for z
z Ž t . s D0) Ž t y t 0 . z Ž t 0 . q D1) Ž t ,t 0 . j Ž t 0 . t
Ht D
)y1 0
Ž s y t0 . B )
0
= D0) Ž s y t 0 . ds .
Ž 26 .
After some straightforward calculations we find that
D1) Ž t ,t 0 . s
ž
a Ž t ,t 0 .
b Ž t ,t 0 .
c Ž t ,t 0 .
d Ž t ,t 0 .
/
Ž 27 .
Ý f k Ž t . z1k z 2myk .
Ž 29 .
The coefficients f k depend on time only through the SLS Ž x Ž t,t 0 ., p x Ž t,t 0 ... Thus Ž29. is a univalued periodic function with respect to time, with the same period T of the perturbed SLS, which coincides with the period of the unperturbed SLS. By expanding z and f k with respect to ´ , f k s Ý`ls0 ´ l f l k , for some s g N we obtain m
Fs´ s
where
b Ž t ,t 0 . s
m
Fs
ks0
D1) Ž t ,t 0 . s D0) Ž t y t 0 .
a Ž t ,t 0 . s
Note that the z’s remain analytic functions of ´ around ´ s 0, due to the fact that y 1Ž t ., p y1Ž t . are analytic functions of ´ around ´ s 0. Ziglin w2;3x has proved that the NVE of a solution of a Hamiltonian system with a second integral of motion independent of the Hamiltonian, possess an integral which is a homogeneous polynomial in the normal variations. Therefore, the NVE of the SLS of the perturbed Hamiltonian system have an integral of motion
Ž 25 .
where
51
e v Ž tyt 0 . 2v
c Ž t ,t 0 . s y d Ž t ,t 0 . s y
f sk j 1k j 2myk
ks0 t
m
Ht b ds ,
q´ sq 1
0
Ý
f Ž sq1 . k j 1k j 2myk
ks0
e v Ž tyt 0 . 2v
Ý
t
Ht
b ey2 v Ž syt 0 . ds ,
m
q
0
eyv Ž tyt 0 . 2v eyv Ž tyt 0 . 2v
Ý
f sk Ž Ž m y k . j 1k j 2myky1z 2
ks0 t
Ht b e
2 v Ž syt 0 .
ds ,
q k j 1ky 1j 2mykz 1 . q O Ž ´ sq2 . .
0
t
Ht b ds .
Ž 28 .
0
All the above calculations are both valid for real and complex time. Notice that D1Ž t,t 0 . is no longer univalued, if the integrands have poles in complex time. This is a common property of solutions of linear systems in the complex domain.
Ziglin’s theorem guarantees that the integral of the NVE of the continued orbit of the perturbed system is non-trivial. This leads to the conclusion that there always exists an integer s, such that at least one coefficient f sk Ž t . is not identically zero. This property is essential for the conclusion of the proof. Dividing the above expression by ´ s, is equivalent to putting s s 0, i.e. m
4. Ziglin’s lemma for the existence of an integral of the NVE
Fs
m
Ý
f 0 k j 1k j 2myk q ´
ks0
Ý
f 1 k j 1k j 2myk
ks0 m
We transform the NVE in the basis where D0 Ž t y t 0 . becomes diagonal, i.e. in coordinates z s Ž z 1 , z 2 .T defined by
h˜ 1 s Uz s Uj q ´ Uz q O Ž ´ 2 . .
q
Ý
f 0 k Ž Ž m y k . j 1k j 2myky1z 2
ks0
qk j 1ky 1j 2mykz 1 . q O Ž ´ 2 . .
Ž 30 .
E. Meletlidour Physics Letters A 270 (2000) 47–54
52
The above series is convergent with respect to ´ , since both the integral of the perturbed Hamiltonian and z are analytic functions of ´ . At zeroth order we obtain an integral of the unperturbed variational Eqs. Ž20. which is m
F0s
f 0 k j 1k j 2myk
Ý
Ž 31 .
ks0
and, at first order in ´ , the following integral of the joint system Ž20. and Ž23. m
F1 s
ks0 m
q
Ý
f 0 k Ž Ž m y k . j 1k j 2myky1z 2
ks0
qk j 1ky 1j 2my kz 1 . .
Ž 32 .
Let r ,1rr be the eigenvalues of the monodromy matrix D0 ŽT . of the unperturbed NVE Ž14.. This matrix is called non-resonant if the eigenvalues are not roots of unit. It is known w2;3x that, when the integral Ž31. is expressed in the variables that diagonalize the monodromy matrix, either there is a resonance condition in the eigenvalues, i.e. an integer n exists, such that r n s 1, or the integral, at t s t 0 , is of the form m m
t 0 j 12
F 0 Ž t0 . s f 0 Ž . 2
m
Ž t 0 . j 22 Ž t 0 .
Ž 33 .
for some even integer m. This is proved by replacing j Ž t 0 q T . s D0) ŽT . j Ž t 0 . in the equation F 0 Ž t 0 q T . s F 0 Ž t 0 . and using the fact that f 0 k are T-periodic functions of time. We are going to assume that the monodromy matrix of the unperturbed NVE D0 ŽT . is non-resonant, i.e. we are going to exclude the case where the eigenvalues e " v T are roots of unit. Since these roots are dense on the unit circle, this assumption cannot be checked for imaginary v , and for this reason we assumed that v is real, i.e. that the unperturbed SLS is unstable with respect to normal variations. Thus, the integral F 0 Ž t 0 . of the unperturbed NVE is in our case of the form Ž33.. Now we equate
F 1Ž t0 q T . s F 1Ž t0 .
m 2
which are the coefficients of all the monomials except j 1 q1j 2 y1, j 1 y1j 2 q1 , j 1 j 2 . The coefficient f 1 Ž t 0 . may be different from zero, but is eliminated from Ž34.. The remaining non-zero coefficients of the above monomials satisfy the equations m f m Ž d r q a ry1 . s 0 , 2 02 m f 1Ž m q1. r 2 q f 0 m c r y f 1Ž m q1. s 0 , 2 2 2 2 m f 1Ž m y1. ry2 q f 0 m bry1 y f 1Ž m y1. s 0 Ž 35 . 2 2 2 2 m 2
m 2
m 2
m 2
m 2
m 2
m 2
f 1 k j 1k j 2myk
Ý
and replace j Ž t 0 q T ., z Ž t 0 q T . by the solutions Ž21. and Ž25. for t s t 0 q T. Due to the fact that only f 0 is different from zero, the arrangement of the terms of the polynomials in Ž32., after substituting j and z by their solutions Ž21. and Ž25. respectively, leads to m m m f1 k Ž t0 . s 0 ; k / y 1 , , q 1 , 2 2 2
Ž 34 .
where the values of f 0 , f 1Ž q1. , f 1Ž y1. are considered for t s t 0 , since they are T-periodic functions. Notice also that a,b,c,d that enter in the equations are given by Ž28. with upper limit t s t 0 q T, since F 1 , and therefore j and z , are evaluated at this t. Since the integral of the NVE of the perturbed SLS is non-trivial, for suitable t 0 it holds that f 0 Ž t 0 . / 0. Therefore Eq. Ž35a. gives m 2
m 2
m 2
m 2
d r q a ry1 s 0 . It is easily seen from Ž28. that this equation is satisfied identically. In Eqs. Ž35b. and Ž35c. f 0 / 0, f 1Ž q1. , f 1Ž y1. are univalued functions of t 0 and, by solving with respect to c and b respectively, we obtain m 2
m 2
t 0qT
Ht
m 2
b Ž s y t 0 . e 2 v Ž syt 0 . ds s const.s c1 ,
0
t 0qT
Ht
b Ž s y t 0 . ey2 v Ž syt 0 . ds s const.s c 2
Ž 36 .
0
where the constants c1 , c 2 depend on f 0 Ž t 0 ., f 1Ž " 1.Ž t 0 . and also on v ,T. However, the integrals on the left-hand sides can be taken from t 0 to t 0 q T along any path in the complex time plane. If the composition of two such paths encircles a pole either m 2
m 2
E. Meletlidour Physics Letters A 270 (2000) 47–54
of the function b Ž t .e 2 v t or the function b Ž t .ey2 v t with nonzero residue, then either of these two integrals along the different paths attains different values. This contradicts the fact that, according to Ž36. these integrals must equal to constants, which is a necessary condition for the existence of an additional independent integral of motion of the perturbed Hamiltonian system. If therefore either of the functions b Ž t .e " 2 v t has a pole with non-zero residue, the perturbed Hamiltonian system does not possess a second integral of motion, analytic with respect to ´ and holomorphic with respect to x, y, p x , p y at least in an open neighbourhood of the SLS and independent of H. Note that, since functions e " v t are analytic, the poles of the integrands coincide with the poles of b . Thus we have proved the following
53
non-integrability by applying Mel’nikov’s w25x method. Moreover, the parts of lower and higher degree in the potential are by themselves integrable, so neither Yoshida’s w27x theorem is applicable. The straight line solution of the unperturbed and perturbed Hamiltonians on the x-axis is Žsee e.g. w31x, p. 207. x Ž t y t 0 . s Ccn l Ž t y t 0 . ,k where l,t 0 are arbitrary constants, while k 2 s Ž l2 y 1.rŽ2 l2 . and C 2 s l2 y 1. This is a periodic solution with period T s 4 K Ž k .rl, where K is the complete elliptic integral of the first kind. We see that V02 y y s y1 so the SLS is vertically unstable and we evaluate
b Ž t . s yV1 y y Ž x Ž t y t 0 . ,0,0 . s y2Ccn l Ž t y t 0 . ,k .
Theorem 1. Consider the Hamiltonian H s 12 Ž p x2 q p y2 . q V01 Ž x . q V02 Ž y . q ´ V1 Ž x , y, ´ . where V0 1 ,V0 2 ,V1 are holomorphic functions of (x,y,p x ,p y ) g C 4 and analytic with respect to ´ g R , for which the following conditions are satisfied: (i) V0 2 y (0) s 0, (ii) = V1(x,0,´ ) s 0, (iii) V0 2 y y (0) is strictly negatiÕe. Then, if either of the functions V1 y y e 2 v t or V1 y y e y 2 v t where v s (yV0 2 y y )1r 2 , eÕaluated along the unperturbed straight-line solution, haÕe at least one pole in the complex time domain with non-zero residue, the Hamiltonian H is non-integrable, i.e. it does not haÕe an integral of motion, independent of H, holomorphic in an open neighbourhood of the straight line solution and analytic with respect to ´ g R , in an open interÕal of ´ around zero.
5. Application and conclusions We will apply the previous results to the system described by the Hamiltonian H s 12 Ž p x2 q p y2 . q 12 x 2 q 14 x 4 y 12 y 2 q ´ xy 2 .
Ž 37 . Note that, since the integrable part of H does not possess a homoclinic loop, one cannot prove its
Since this function has simple poles on the complex t-plane, then expŽ"v t . b Ž t . have also simple poles and the Hamiltonian H is non-integrable for ´ / 0 in an open neighbourhood around zero. In order to attain the above results, we have used the period-T monodromy matrix of the NVE of the unperturbed SLS. The form of the Hamiltonian has simplified the calculations considerably. The criterion falls, as has been expected, on the branching of the solutions of the perturbed NVE. On the other hand, we do not want the SLS to branch at all, since we assumed that the unperturbed SLS is preserved in the same form under the perturbation. We have not used the monodromy group of the NVE of the unperturbed SLS, since it is trivial, i.e. it is the unit matrix and first order perturbations do not guarantee that the perturbed monodromy group possesses a non-resonant element. In the case where the NVE of the unperturbed SLS have non-trivial monodromy group, the calculations would be, although straightforward, much more sophisticated. It is certainly a more general case that can be applied to a general class of perturbed integrable Hamiltonians but on the other hand its applicability is restrained by the need of knowing the solution of the unperturbed NVE which has no more constant coefficients but is a Hill equation. However, the present case cannot be treated through this general method, due to the triviality of the monodromy group.
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Acknowledgements The author wishes to thank Drs. S. Ichtiaroglou and S. Tanabe for very fruitful discussions. This work was partially supported by a post-doctoral fellowship of the Greek Scholarship Foundation ŽI.K.Y..
References w1x H. Poincare, Nouvelles de la Mechanique ´ 1892, Les Methodes ´ ´ Celeste, vol. I, Gauthier-Villars, Paris. English translation: ´ Goroff D.L. ŽEd.., New Methods in Celestial Mechanics, American Institute of Physics, 1993. w2x S.L. Ziglin, Func. Anal. Appl. 16 Ž1983. 181. w3x S.L. Ziglin, Func. Anal. Appl. 17 Ž1983. 6. w4x H. Yoshida, Physica D 29 Ž1987. 128. w5x J.J. Moralez-Ruiz, C. Simo, ´ J. Diff. Eq. 107 Ž1994. 140. w6x J.J. Morales-Ruiz, C. Ramis, 1998a, Galoisian obstructions to integrability of Hamiltonian systems Žpreprint.. w7x J.J. Morales-Ruiz, C. Ramis, 1998b, Galoisian obstructions to integrability of Hamiltonian systems II Žpreprint.. w8x H. Ito, Kodai Math. J. 8 Ž1995. 120. w9x H. Ito, ZAMP 38 Ž1987. 459. w10x H. Ito, Bol. Soc. Bras. Mat. 21 Ž1990. 95. w11x S. Ichtiaroglou, J. Phys. A: Math. Gen. 22 Ž1989. 3461. w12x M.A. Almeida, I.C. Moreira, H. Yoshida, J. Phys. A: Math. Gen. 25 Ž1992. L227. w13x M. Irigoyen, C. Simo, ´ Celest. Mech. Dyn. Astr. 55 Ž1993. 281.
w14x J.-P. Franc¸oise, M. Irigoyen, Geom. Phys. 10 Ž1993. 231. w15x J.M. Ferrandiz, M.E. Sansaturio, Phys. Lett. A 207 Ž1995. ´ 180. w16x J.M. Ferrandiz, M.E. Sansaturio, I. Vigo, Phys. Lett. A 221 ´ Ž1996. 153. w17x M.E. Sansaturio, I. Vigo-Aguiar, J.M. Ferrandiz, J. Phys. A: ´ Math. Gen. 30 Ž1997. 5869. w18x M.E. Sansaturio, I. Vigo-Aguiar, J.M. Ferrandiz, in: B.A. ´ Steves, A.E. Roy ŽEds.., The Dynamics of Small Bodies in the Solar System, Kluwer, the Netherlands, 1999, pp. 295– 302. w19x M.I. Vigo-Aguiar, 1999, No integrabilidad del problema del satelite, Doctor’s Thesis, Universidad de Alicante. ´ w20x R.C. Churchill, D.L. Rod, J. Diff. Eq. 76 Ž1988. 91. w21x R.C. Churchill, D.L. Rod, SIAM J. Math. Annal. 2 Ž1991. 1790. w22x E. Meletlidou, S. Ichtiaroglou, Physica D 71 Ž1994. 261. w23x E. Meletlidou, S. Ichtiaroglou, J. Phys. A 27 Ž1994. 3919. w24x E. Meletlidou, S. Ichtiaroglou, Phys. Lett. A 188 Ž1994. 157. w25x V.K. Mel’nikov, Trans. Moscow Math. Soc. 12 Ž1963. 3. w26x V.V. Kozlov, J. Appl. Math. Mech. 42 Ž1979. 420. w27x H. Yoshida, Commun. Math. Phys. 116 Ž1988. 529. w28x G.D. Birkhoff, 1927, Dynamical Systems, Colloquium Publications, vol. IX, Am. Math. Soc., Providence, Rhode Island, Acta Math. 50. w29x V.V. Kozlov, 1996, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, Berlin. w30x K.R. Meyer, G.R. Hall, 1992, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer, New York. w31x H.T. Davis, 1962, Introduction to Nonlinear Differential and Integral Equations, Dover, New York.