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 Regul. Chaotic Dyn., 2015, Volume 20, Issue 3, Pages 309–316 (Mi rcd45)

On the Birkhoff Transformation in the Case of Complete Degeneracy of the Quadratic Part of the Hamiltonian

Anatoly P. Markeev

RAS Institute for Problems in Mechanics ul. Vernadskogo 101, str. 1, Moscow, 119526 Russia

Abstract: A time-periodic one-degree-of-freedom system is investigated. The system is assumed to have an equilibrium point in the neighborhood of which the Hamiltonian is represented as a convergent series. This series does not contain any second-degree terms, while the terms up to some finite degree $l$ do not depend explicitly on time. An algorithm for constructing a canonical transformation is proposed that simplifies the structure of the Hamiltonian to terms of degree $l$ inclusive.
As an application, a special case is considered when the expansion of the Hamiltonian begins with third-degree terms. For this case, sufficient conditions for instability of the equilibrium are obtained depending on the forms of the fourth and fifth degrees.

Keywords: Hamiltonian system, canonical transformations, stability

 Funding Agency Grant Number Russian Science Foundation 14-21-00068 The study was financed by the grant from the Russian Science Foundation (Project No.14-21-00068) and conducted at the Moscow Aviation Institute (National Research University).

DOI: https://doi.org/10.1134/S1560354715030077

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MSC: 70H05, 70H15, 70E50
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Citation: 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

Citation in format AMSBIB
\Bibitem{Mar15} \by Anatoly P. Markeev \paper On the Birkhoff Transformation in the Case of Complete Degeneracy of the Quadratic Part of the Hamiltonian \jour Regul. Chaotic Dyn. \yr 2015 \vol 20 \issue 3 \pages 309--316 \mathnet{http://mi.mathnet.ru/rcd45} \crossref{https://doi.org/10.1134/S1560354715030077} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3357278} \zmath{https://zbmath.org/?q=an:06488659} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2015RCD....20..309M} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84934963061} 

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Citing articles on Google Scholar: Russian citations, English citations
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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. 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
3. 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
4. V. V. Basov, A. S. Chermnykh, “Two-dimensional homogeneous cubic systems: classification and normal forms-III”, Vestnik St. Petersburg Univ. Math., 50:2 (2017), 97–110