Abstract
AbstractWe study the integrability of 2D Hamiltonian systems $$H_\mu =\frac{1}{2}(p_1^2+p_2^2) + \omega (p_1q_2-p_2q_1) -\tfrac{\mu }{r}+ V(q_1,q_2)$$
H
μ
=
1
2
(
p
1
2
+
p
2
2
)
+
ω
(
p
1
q
2
-
p
2
q
1
)
-
μ
r
+
V
(
q
1
,
q
2
)
, where $$r^2=q_1^2+q_2^2$$
r
2
=
q
1
2
+
q
2
2
, and potential $$V(q_1,q_2)$$
V
(
q
1
,
q
2
)
is a homogeneous rational function of integer degree k. The main result states that under very general assumptions: $$\mu \omega \ne 0$$
μ
ω
≠
0
, $$|k|>2$$
|
k
|
>
2
and $$V(1,\mathrm {i})\ne 0$$
V
(
1
,
i
)
≠
0
or $$V(-1,\mathrm {i})\ne 0$$
V
(
-
1
,
i
)
≠
0
, the system is not integrable. It was obtained by combining the Levi-Civita regularization, differential Galois methods, and the so-called coupling constant metamorphosis transformation. The proof that the regularized version of the problem is not integrable contains the most important theoretical results, which can be applied to study integrability of other problems.
Publisher
Springer Science and Business Media LLC
Subject
Electrical and Electronic Engineering,Applied Mathematics,Mechanical Engineering,Ocean Engineering,Aerospace Engineering,Control and Systems Engineering
Reference32 articles.
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