Web of families of periodic orbits of the generalized hill problem

Batkhin, A. B.

doi:10.1134/s1064562414060064

Cited by 8 publications

(11 citation statements)

References 5 publications

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“…Usually in celestial mechanics, bodies with non-negative masses are considered. However, Batkhin [13] proposed to consider problems where some masses are negative. In the Hill problem with body mass equal to −1 (called the anti-Hill problem), families of periodic solutions are extensions of families of periodic solutions to the classical Hill problem.…”

Section: Generalized Problemsmentioning

confidence: 99%

“…For recent results in this direction for the restricted three-body problem, see [9,15,52,53]. For the Hill problem, see [54] ( § 10.4), [55] ( § 19), [11][12][13]36,37,45,[56][57][58]. However, families of invariant tori were not studied yet.…”

Section: Skeletonsmentioning

confidence: 99%

“…The method of generalized problems admitting in Celestial Mechanics bodies with negative masses [13]. Such problems have unified complete families of periodic solutions, which facilitates their calculation.…”

mentioning

confidence: 99%

“…Such problems have unified complete families of periodic solutions, which facilitates their calculation. Example: generalized Hill's problem [13].…”

mentioning

confidence: 99%

“…Poincaré [14] wrote about the benefits of such "skeletons". Examples: the Hill problem [13] and partly the restricted three-body problem [9,15]. 7.…”

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confidence: 99%

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Survey of Eight Modern Methods of Hamiltonian Mechanics

Bruno

Batkhin

2021

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Here we describe eight new methods, arisen in the last 60 years, to study solutions of a Hamiltonian system with n degrees of freedom. The first six of them are intended for systems with small parameters or without them. The methods allow to find families of periodic solutions and families of invariant n-dimensional tori by means of analytic computation near a stationary solution, near a periodic solution and near an invariant torus, using the corresponding normal form of a Hamiltonian. Then we can continue the founded families by means of numerical computation. In a Hamiltonian system without parameters, only periodic solutions and invariant n-dimensional tori form one-parameter families. The last two methods are intended for systems with not small parameters, which do not depend on time. They allow computing sets of parameters, which guarantee the stability of some solutions for linear (method seven) and nonlinear (method eight) systems. We do not consider chaotic behaviors, but only regular ones.

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Section: Generalized Problemsmentioning

confidence: 99%

Section: Skeletonsmentioning

confidence: 99%

mentioning

confidence: 99%

“…Such problems have unified complete families of periodic solutions, which facilitates their calculation. Example: generalized Hill's problem [13].…”

mentioning

confidence: 99%

“…Poincaré [14] wrote about the benefits of such "skeletons". Examples: the Hill problem [13] and partly the restricted three-body problem [9,15]. 7.…”

mentioning

confidence: 99%

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Survey of Eight Modern Methods of Hamiltonian Mechanics

Bruno

Batkhin

2021

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Integrability of the generalised Hill problem

2021

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We consider a certain two-parameter generalisation of the planar Hill lunar problem. We prove that for nonzero values of these parameters the system is not integrable in the Liouville sense. For special choices of parameters the system coincides with the classical Hill system, the integrable synodical Kepler problem or the integrable parametric Hénon system. We prove that the synodical Kepler problem is not super-integrable, and that the parametric Hénon problem is super-integrable for infinitely many values of the parameter.

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Bifurcations of Periodic Solutions of a Hamiltonian System with a Discrete Symmetry Group

Batkhin

2020

Program Comput Soft

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Web of families of periodic orbits of the generalized hill problem

Cited by 8 publications

References 5 publications

Survey of Eight Modern Methods of Hamiltonian Mechanics

Survey of Eight Modern Methods of Hamiltonian Mechanics

Integrability of the generalised Hill problem

Bifurcations of Periodic Solutions of a Hamiltonian System with a Discrete Symmetry Group

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