Integrability of the generalised Hill problem

Combot, Thierry; Maciejewski, Andrzej J.; Przybylska, Maria

doi:10.1007/s11071-021-07040-8

Cited by 2 publications

(2 citation statements)

References 21 publications

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“…This case is excluded in our theorem. To study its integrability we developed other method, see [8].…”

Section: Then For An Arbitrary H It Does Not Admit An Additional Firs...mentioning

confidence: 99%

“…Let us assume the system given by Hamiltonian (6) satisfying assumptions of this theorem is integrable with a first integral I (q, p, μ). Then, the system given by Hamiltonian K μ (u, v), see (8), is also integrable with corresponding first integral J μ (u, v). Note that we can set K (u, v, α) = K μ (u, v) with α = 4μ, and then we can write…”

Section: Proofsmentioning

confidence: 99%

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Integrability of Hamiltonian systems with gyroscopic term

Przybylska

Maciejewski

2022

Nonlinear Dyn

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We 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.

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“…This case is excluded in our theorem. To study its integrability we developed other method, see [8].…”

Section: Then For An Arbitrary H It Does Not Admit An Additional Firs...mentioning

confidence: 99%

Section: Proofsmentioning

confidence: 99%

Integrability of Hamiltonian systems with gyroscopic term

Przybylska

Maciejewski

2022

Nonlinear Dyn

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Chaos and integrability of relativistic homogeneous potentials in curved space

Szumiński,

Przybylska,

Maciejewski

2024

Nonlinear Dyn

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Relativistic Hamiltonian systems of n degrees of freedom in static curved spaces are considered. The source of space-time curvature is a scalar potential $$V(\varvec{q})$$ V ( q ) . In the limit of weak potential $$2V(\varvec{q})/mc^2\ll 1$$ 2 V ( q ) / m c 2 ≪ 1 , and small momentum $$|\varvec{p} |/ mc\ll 1$$ | p | / m c ≪ 1 , these systems transform into the corresponding non-relativistic flat Hamiltonian’s with the standard sum of kinetic energy plus potential $$V(\varvec{q})$$ V ( q ) . We compare the dynamics of the classical and the corresponding relativistic curved counterparts on examples of important physical models: the Hénon–Heiles system, the Armbruster–Guckenheimer–Kim galactic system and swinging Atwood’s machine. Our main results are formulated for relativistic Hamiltonian systems with homogeneous potentials of non-zero integer degree k of homogeneity. First, we show that the integrability of a non-relativistic flat Hamiltonian with a homogeneous potential is a necessary condition for the integrability of its relativistic counterpart in curved space-time with the same homogeneous potential or with a non-homogeneous potential that the lowest homogenous part coincides with this homogeneous potential. Moreover, we formulate necessary integrability conditions for relativistic Hamiltonian systems with homogeneous potentials in curved space-time. These conditions were obtained from analysis of the differential Galois group of variational equations along a particular straight-line solution defined by a non-zero vector $$\varvec{d}$$ d satisfying $$V'(\varvec{d})=\gamma \varvec{d}$$ V ′ ( d ) = γ d for a certain $$\gamma \ne 0$$ γ ≠ 0 . They are very strong: if the relativistic system is integrable in the Liouville sense, then either $$k=\pm 2$$ k = ± 2 , or all non-trivial eigenvalues of the re-scaled Hessian $$\gamma ^{-1}V''(\varvec{d})$$ γ - 1 V ′ ′ ( d ) are either 0, or 1. Certain integrable relativistic systems are presented.

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Integrability of the generalised Hill problem

Cited by 2 publications

References 21 publications

Integrability of Hamiltonian systems with gyroscopic term

Integrability of Hamiltonian systems with gyroscopic term

Chaos and integrability of relativistic homogeneous potentials in curved space

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