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.
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).
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
\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://mathscinet.ams.org/mathscinet-getitem?mr=3357278}
\zmath{https://zbmath.org/?q=an:06488659}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2015RCD....20..309M}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84934963061}
This publication is cited in the following 7 articles:
Athanasios C. Tzemos, George Contopoulos, “Integrals of Motion in Time-periodic Hamiltonian Systems:
The Case of the Mathieu Equation”, Regul. Chaotic Dyn., 26:1 (2021), 89–104
Tzemos A.C. Contopoulos G., “Order and Chaos in Time Periodic Hamiltonian Systems”, Physica D, 419 (2021), 132847
Boris S. Bardin, Víctor Lanchares, “Stability of a One-degree-of-freedom Canonical System in the Case of Zero Quadratic and Cubic Part of a Hamiltonian”, Regul. Chaotic Dyn., 25:3 (2020), 237–249
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
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
A. P. Markeev, “Ob ustoichivosti nepodvizhnykh tochek otobrazhenii, sokhranyayuschikh ploschad”, Nelineinaya dinam., 11:3 (2015), 503–545
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
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