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 Nelin. Dinam., 2014, Volume 10, Number 4, Pages 447–464 (Mi nd456)

This article is cited in 5 scientific papers (total in 5 papers)

Simplifying the structure of the third and fourth degree forms in the expansion of the Hamiltonian with a linear transformation

Anatoly P. Markeev

A. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101-1, Moscow, 119526, Russia

Abstract: We consider the canonical differential equations describing the motion of a system with one degree of freedom. The origin of the phase space is assumed to be an equilibrium position of the system. It is supposed that in a sufficiently small neighborhood of the equilibrium Hamiltonian function can be represented by a convergent series. This series does not include terms of the second degree, and the terms of the third and fourth degrees are independent of time. Linear real canonical transformations leading the terms of the third and fourth degrees to the simplest forms are found. Classification of the systems in question being obtained on the basis of these forms is used in the discussion of the stability of the equilibrium position.

Keywords: Hamiltonian system, canonical transformation, stability.

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UDC: 531.36
MSC: 70H05,70H15,70E50
Received: 04.11.2014
Revised: 20.11.2014

Citation: Anatoly P. Markeev, “Simplifying the structure of the third and fourth degree forms in the expansion of the Hamiltonian with a linear transformation”, Nelin. Dinam., 10:4 (2014), 447–464

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\Bibitem{Mar14} \by Anatoly~P.~Markeev \paper Simplifying the structure of the third and fourth degree forms in the expansion of the Hamiltonian with a linear transformation \jour Nelin. Dinam. \yr 2014 \vol 10 \issue 4 \pages 447--464 \mathnet{http://mi.mathnet.ru/nd456} 

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This publication is cited in the following articles:
1. A. P. Markeev, “Ob ustoichivosti nepodvizhnykh tochek otobrazhenii, sokhranyayuschikh ploschad”, Nelineinaya dinam., 11:3 (2015), 503–545
2. Anatoly P. Markeev, “On the Birkhoff Transformation in the Case of Complete Degeneracy of the Quadratic Part of the Hamiltonian”, Regul. Chaotic Dyn., 20:3 (2015), 309–316
3. Boris S. Bardin, Victor Lanchares, “On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy”, Regul. Chaotic Dyn., 20:6 (2015), 627–648
4. A. P. Markeev, “On the stability of periodic trajectories of a planar Birkhoff billiard”, Proc. Steklov Inst. Math., 295 (2016), 190–201
5. Rodrigo Gutierrez, Claudio Vidal, “Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case”, Regul. Chaotic Dyn., 22:7 (2017), 880–892
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