Non-integrability of anisotropic quasi homogeneous Hamiltonian ...

PRE-PUBLICACIONES del
seminario matematico 2002
Non-integrability of anisotropic
quasi homogeneous
Hamiltonian systems
M. Arribas
A. Elipe
A. Riaguas
“garcia de galdeano”
n. 22
seminario
matemático
garcía de galdeano
Universidad de Zaragoza

Non-integrability of anisotropic quasi
homogeneous Hamiltonian systems
M. Arribas a,b , A. Elipe a , A. Riaguas a,b
a Grupo de Mecánica Espacial. Universidad de Zaragoza. 50009 Zaragoza. Spain
b Dpto. Matemática Aplicada. Universidad de Zaragoza. 50009 Zaragoza. Spain
Abstract
We consider the anisotropic Hamiltonian systems which potential is made of a finite
sum of homogeneous parts of arbitrary degree. For this problem, we prove for two
and three degrees of freedom, that there are no more meromorphic integrals than
the Hamiltonian itself, except for the classical integrable cases.
Key words: Non-integrability; Anisotropic problem; Homogeneous potential
1 Introduction
Central force fields are among the first forces one meets in Mechanics. Indeed,
they represent integrable systems, for they are systems of only one degree of
freedom, and besides, fundamental motions like the Keplerian one, harmonic
oscillators and diffusors belong to this class of forces. More complex problems
are represented by adding perturbations to these more simple problems above
mentioned.
One of the first central perturbation added to the Kepler problem was proposed
by Newton in Liber II, Propositio XLIV of his Philosophiae Naturalis
Principia Mathematica [15]. Indeed, a short sentence at the end of Corollarium
2 of Propositio XLV reveals that Newton considered at one time explaining
departures in the orbit of the moon from a mere Keplerian orbit as the effect
of an additional central repulsion. Thus, the potential is of the type
U(r) = − A r − B
2r 2 , (1)
where A and B are parameters, independent of the radial distance r. This potential
represents the class of what Deprit [7] dubbed quasi-Keplerian systems.
Preprint submitted to Elsevier Science 3 December 2002

Deprit obtained analytical solutions by means of a canonical transformation
— the torsion— that converts the quasi-Keplerian Hamiltonian into a pure
Keplerian one.
We meet several physical problems with this potential, for instance, it represents
the intermediary problem in the artificial satellite theory [7,2] after
some simplifications based on Lie transformations are done. Another problem
of this type is the so called Maneff’s potential [16]
U(r) = − Gm 1m 2
r
(
1 + 3G(m 1 + m 2 )
2c 2 r
)
,
where G is the Gaussian constant, m 1 and m 2 the masses of two particles, and
c is the light speed. This simple post-Newtonian non-relativistic potential may
be used for describing the secular motions of the perihelia of the inner planets.
Some authors [2,17,8] addressed the attention to this problem. In particular,
a solution valid even for collisions is presented in [17]; however, the solution is
split into three cases, elliptic, parabolic and hyperbolic (which actually means
energy < 0, = 0 or > 0, respectively). Recently, a general solution that is valid
for whatever value of the energy has been given [4] by means of a universal
formulation with the Stumpff functions.
There are problems in which the potential (1) represents only the two first
terms in potentials that are given by polynomials in the inverse of the radial
distance, that is, potentials of the type
U = − ∑
1≤i≤n
a i
r i , (2)
that receive the name of quasihomogeneous potentials. We face this type of
potentials, for instance, in the motion of a particle attracted by a spheroid and
such that the particle moves on the equatorial plane [11,6,10,14]. In this case,
the influence of the spheroid’s oblateness leads to a potential in the form of Eq.
(2). Although the problem represented by this potential is integrable, several
works have been devoted to the finding of its solution in a simple way. Let us
mention for instance, the torsion–transformation [7], or the linearization and
regularization techniques developed in [3,5].
The anisotropic Kepler problem (AKP) was profusely studied by Gutzwiller
(see [13] and references therein) to find connections between classical and
quantum mechanics. Essentially, the anisotropic Kepler may be formulated as
the motion of a particle under the potential of the form
U = −
A
(µq 2 1 + q 2 2) 1/2 , 2

that is to say, the motion takes place on a space in which the forces act
differently in every direction.
Later on, this analysis has been extended to the so called anisotropic Maneff’s
problem (AMP), which potential is given by
1
U = −
(µq1 2 + q2) − b
2 1/2 (µq1 2 + q2) . (3)
2
At difference of the isotropic potential, the angular momentum is not an integral
of the Hamiltonian corresponding to the anisotropic problem, that is of
two degrees of freedom, and then the integrability is an open question. Diacu
[9] claims that “there are strong arguments favouring the nonintegrability of
the system.” From our part, we prove here that this is the case, and even
more, that Hamiltonian systems with anisotropic potential of the type
U = − ∑
1≤i≤n
A i (µ i q 2 1 + q 2 2) −i/2 , (4)
with µ i parameters (that without loss of generality we can assume 0 ≤ µ i ≤ 1)
are non-integrable, in the sense that they do not admit more meromorphic
integrals than the Hamiltonian itself. Note that the anisotropic Kepler and
Maneff cases are but particular cases of the potential (4), and note also that
the limit cases µ i = 1 or µ i = 0 are integrable. The same result is proved for
the 3-D problem.
2 The 2-D problem
Let us consider the planar problem, that is, the motion on the plane (q 1 , q 2 )
of a particle of unit mass that is moving under the gravity field (4). Then, the
Hamiltonian function is
H = 1 2 (p2 1 + p 2 2) − ∑
1≤i≤n
A i (µ i q 2 1 + q 2 2) −i/2 , (5)
with A i , µ i parameters, and n ∈ N finite.
This problem is a generalization of both the AKP and AMP above mentioned.
These problems seem to be nonintegrable, and there are indications of it, for
instance, that heteroclinic orbits exist in the zero-energy case (see [9]). Computing
Poincaré surfaces of section gives also strong evidences of it. Figure
1 shows a suite of Poincaré sections for the AMP in the q 2 − p 2 plane, with
3

q 1 = 0, for b = 1, energy E = 0.5 and several values of µ. For high values
of µ, the motion is quite regular and it is strongly dominated by the central
stable periodic orbit. Nevertheless, as µ decreases the cross section is different;
indeed, for µ = 0.86 the central elliptic points has bifurcated through a pitchfork
bifurcation, and now it is converted into hyperbolic and two new elliptic
equilibria appear on the q 2 -axis. At the same time a chaotic region appears,
and it is wider as the parameter µ decreases in such a way that for µ = 0.52
the whole region is chaotic.
There is an important theorem of Yoshida [21] concerning the nonintegrability
of problems in two degrees of freedom for homogeneous potentials, and that
gives sufficient conditions for the non-integrability for homogeneous potentials
(of degree k) with two degrees of freedom. This theorem says:
Theorem 1 (Yoshida[21]) If IC λ is in the region S k , then there cannot
exist an additional analytic integral. The non-integrable region S k can be computed
as follows: (Note that when k = 0, ±2, such regions are not defined.)
i) k ≥ 3: S k = {λ < 0, 1 < λ < k − 1, k + 2 < λ < 3k − 2, . . . ,
j(j − 1)k/2 + j < λ < j(j + 1)k/2 − j, . . .},
ii) S 1 = R − {0, 1, 3, 6, 10, . . . , j(j + 1)/2, . . .},
iii) S −1 = R − {−1, 0, −2, −5 . . . , −j(j + 1)/2 + 1, . . .},
iv) k ≤ −3: S k = {λ > 1, 0 > λ > −|k| + 2, −|k| − 1 > λ > −3|k| + 3, . . .
−j(j − 1)|k|/2 − (j − 1) > λ > −j(j + 1)|k|/2 + (j + 1), . . .}.
We cannot apply this theorem directly, since our potential (5) is not homogeneous,
but is made of the sum of homogeneous terms V (x, y) = ∑ k V k (x, y).
Precisely for this type of potentials Yoshida [22] gave another interesting result.
From the lowest part V kmin and highest part V kmax we can compute the
λ kmin , the lowest IC, and λ kmax , the highest IC, them
Theorem 2 (Yoshida[22]) If either λ kmin ∈ S kmin or λ kmax ∈ S kmax
holds, then there cannot exist an analytical integral.
In other words, the theorem states that if the lowest part or the highest part
is shown to be non-integrable, then the total system is also non-integrable.
Let us consider the maximum order, kmax, of homogeneity in the sum (5),
that is, kmax= −1,
4

Following the above result, we can take as a similarity-invariant Hamiltonian
the one given by the potential:
V 1 (q) = −
A 1
√
µ 1 q 2 1 + q 2 2
, (6)
that is homogeneous of order k = −1. According to Yoshida’s result [21],
the non-integrability depends on finding a solution c = (c 1 , c 2 ) of the system
c = ∇V 1 (c).
√
Since the gradient is ∇V 1 = (A 1 µ 1 x/D, A 1 y/D), with D = µ 1 q1 2 + q2, 2 a
solution of the system c = ∇V 1 (c) is simply c = (c 1 , c 2 ) = (0, A 1/3
1 ).
The components of the Hessian of V 1 evaluated at this particular solution
are Hess 1,1 = µ 1 , Hess 1,2 = −2, Hess 2,1 = 0, and Hess 2,2 = −2, hence, its
eigenvalues are λ 1 = −2 and λ 2 = µ 1 .
Note that, since the function V 1 is homogeneous of degree k = −1, the first
eigenvalue is λ 1 = k −1 = −2, and the second is λ 2 = Tr[HessV 1 (c)] −(k −1),
that is precisely the integrability coefficient (IC) defined by Yoshida [21].
In our case, the IC= µ 1 ∈ S −1 , since we considered 0 < µ 1 < 1, thus, we
conclude that the Hamiltonian (5) does not have an additional analytical
integral.
Note that although we considered in the potential of (5) terms ranging from
1 ≤ i ≤ n, we could obtain similar result for −n ≤ i ≤ −1, that is, for
terms with positive degrees of homogeneity. Note also that for the limit cases
µ 1 = {0, 1}, the IC= µ 1 ∉ S −1 , and the theorem does not guarantee the
non-integrability, but it is well known that these two cases are integrable.
3 The 3-D problem
Let us now consider the integrability of the anisotropic problem in three degrees
of freedom. The potential is assumed to be a finite sum of homogeneous
terms in the form
U = − ∑
1≤i≤n
A i (µ i q 2 1 + ν i q 2 2 + q 2 3) −i/2 , (7)
where now at each term of the sum we introduce two positive anisotropy
parameters µ i and ν i , that without loss of generality, we can assume that at
5

least one of them, say µ i , is 0 < µ i < 1. In this way, we generalize the classical
AKP and AMP by introducing two anisotropic directions.
Since Yoshida’s theorem [22] was proved for only two degrees of freedom, we
cannot apply it. However, Arribas and Elipe [1] proved the following result,
valid for n degrees of freedom:
Proposition 1 Let us consider the Hamiltonian
H = 1 2 p2 + V m (q) + V k (q) (8)
where V m (q) and V k (q) are homogeneous functions of degree m and k respectively,
then there exists a symplectic transformation that converts this Hamiltonian
into an auxiliary one in the form
H h = 1 2 P2 + V k (Q). (9)
Such a symplectic transformation, in the extended space phase, is of the type
p 0 = α a P 0 , q 0 = α b Q 0 , p = α c P, q = α d Q, (10)
with α = constant, and the exponents have to be conveniently chosen, depending
on the degrees of homogeneity. Thus, if we prove that the auxiliary
Hamiltonian (9) has no meromorphic integral besides the Hamiltonian function
itself, then the original Hamiltonian (8) neither is integrable. For details,
the reader is addressed to [1].
Based on this proposition, we proceed to study the non-integrability of the
Hamiltonian
H = 1 2 (p2 1 + p 2 2 + p 2 3) − ∑
1≤i≤n
A i (µ i q 2 1 + ν i q 2 2 + q 2 3) −i/2 , (11)
by means of an auxiliary Hamiltonian, for instance the one in which the potential
is homogeneous of degree k = −1,
V 1 (q 1 , q 2 , q 3 ) = −A i (µ 1 q 2 1 + ν 1 q 2 2 + q 2 3) −1/2 ,
obtained through the symplectic transformation (10), resulting into a Hamiltonian
H = 1 2 (p2 1 + p 2 2 + p 2 3) + V 1 (q 1 , q 2 , q 3 ). (12)
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Again, it is essential the finding of a straight-line solution, with constants
c = (c 1 , c 2 , c 3 ), that are solution of the algebraic equation c = ∇V 1 (c). It is
easy to see that c = (0, 0, A 1/3
1 ) is a solution of it.
The Hessian of the potential evaluated at these constants is
⎛ ⎞
µ 1 0 0
HessV 1 (c) =
⎜ 0 ν 1 0
,
⎟
⎝ ⎠
0 0 −2
and its eigenvalues are λ 1 = µ 1 , λ 2 = ν 1 and λ 3 = −2. Note that again, and
because the degree of homogeneity of the potential (k = −1), there results
that one of the eigenvalues is λ 3 = k − 1 = −2.
At difference of the 2-D problem, we cannot apply Yoshida’s theorem, that was
formulated for only two degrees of freedom. Fortunately, Morales and Ramis
[19,20] gave another result that is a generalization of the previous one to n
degrees of freedom. (This result generalize the one of Yoshida even for n = 2).
For our degree of homogeneity (k = −1), this result may be formulated as
Theorem 3 (Morales and Ramis [19,20]) If the Hamiltonian system (12)
is completely integrable (with holomorphic or meromorphic) first integrals,
then each pair (k, λ i ) belongs to one of the following lists:
1) (k, p + p(p − 1)k/2),
.
18) (k, 1 2 ( k−1
k
+ p(p + 1)k)),
where p is an arbitrary integer.
N.B. We put only the cases 1) and 18), since they are the only ones of interest
for our degree of homogeneity. Details about this theorem may be found in
the book [20].
Thus, according with this theorem, if our problem is completely integrable,
the eigenvalues µ 1 , and ν 1 must be of the form
p − p(p − 1)/2,
or
1
(2 − p(p + 1)), with p ∈ Z.
2
These values depending on p are always either a negative integer, or 0 or 1.
But we assumed that the anisotropic parameters µ i and ν i were positive, thus,
7

the problem (12) has no meromorphic integrals besides the Hamiltonian itself,
except for the cases in which the parameters µ i , ν i are zero or 1, and it is well
known that are integrable cases.
Now, starting for the non integrability of the auxiliary problem, we can prove
the non existence of rational integrals for the original Hamiltonian. Following
the steps in [12], let us suppose that there exits a rational integral of (11):
I(p,q, t) = constant. If we apply the canonical transformation given by (10),
the expression in the new variables will be:
I ∗ (P,Q, t) = αm pol m (P,Q, t)
α n pol n (P,Q, t) = constant,
where pol m , pol n represent polynomials of degree m and n in (P,Q, t).
Then, the function Î(P,Q, t) = lim α→∞ α s I ∗ (P,Q, t), where s is an adequate
value such that guarantees that the limit exists, is a rational integral of the
auxiliary Hamiltonian H 1 (12). But we just proved that this Hamiltonian have
not meromorphic integrals, in particular, rational integrals, thus neither does
H.
Mondéjar [18] proved that for a family of Hamiltonian systems H ɛ depending
continuously of a real parameter ɛ, under ciertain conditions, if for a given
value ɛ 0 of the parameter the corresponding Hamiltonian system H ɛ0 is not
completely integrable, then for values in a neighborhood of ɛ 0 the Hamiltonian
system will not be completely integrable.
In our case, we can assume that our potential is of the form V = ∑ 1≤i≤n ɛ n V n ,
with ɛ a small parameter, and we just proved the nonintegrability of the Hammiltonian
containing only the first term, but {ɛ 2 , . . . ɛ n } ∈ B, with B an open
neighborhood of ɛ. Hence the whole Hamiltonian for the potential (7) is not
completely integrable.
4 Conclusions
We prove that general anisotropic Hamiltonian systems which potential is
made of a finite sum of homogeneous parts of arbitrary degree do not have
more meromorphic integrals than the Hamiltonian itself, except for the classical
integrable cases. In particular, the anisotropic Kepler (AKP) and Maneff
(AMP) problems are no integrable.
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Acknowledgments
Partially supportted by the Ministerio de Ciencia y Tecnología (#ESP2002-
023929).
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Fig. 1. Poincaré sections (q 1 = 0, q 2 , p 2 ) of the Maneff’s problem for several values
of the anisotropic parameter µ. We can see the transition from the regular motion
(µ = 1, integrable) to chaotic motion, passing though a pitchfork bifurcation.
10