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 Regul. Chaotic Dyn., 2016, Volume 21, Issue 5, Pages 510–521 (Mi rcd200)

Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field

Alexey V. Ivanov

Saint-Petersburg State University, Universitetskaya nab. 7/9, Saint-Petersburg, 199034 Russia

Abstract: We study connecting orbits of a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential $U(q, t) = f(t)V (q)$. It is assumed that the factor $f(t)$ tends to $\infty$ as $t\to\pm\infty$ and vanishes at a unique point $t_{0} \in \mathbb{R}$. Let $X_{+}$, $X_{-}$ denote the sets of isolated critical points of $V(x)$ at which $U(x, t)$ as a function of $x$ distinguishes its maximum for any fixed $t > t_{0}$ and $t < t_{0}$, respectively. Under nondegeneracy conditions on points of $X_\pm$ we prove the existence of infinitely many doubly asymptotic trajectories connecting $X_{-}$ and $X_{+}$.

Keywords: connecting orbits, homoclinic and heteroclinic orbits, nonautonomous Lagrangian system, variational method

DOI: https://doi.org/10.1134/S1560354716050026

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Bibliographic databases:

MSC: 37J45, 34C37, 70H03
Accepted:09.08.2016
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Citation: Alexey V. Ivanov, “Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field”, Regul. Chaotic Dyn., 21:5 (2016), 510–521

Citation in format AMSBIB
\Bibitem{Iva16} \by Alexey V. Ivanov \paper Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field \jour Regul. Chaotic Dyn. \yr 2016 \vol 21 \issue 5 \pages 510--521 \mathnet{http://mi.mathnet.ru/rcd200} \crossref{https://doi.org/10.1134/S1560354716050026} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000385167300002} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84990943363} 

• http://mi.mathnet.ru/eng/rcd200
• http://mi.mathnet.ru/eng/rcd/v21/i5/p510

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Alexey V. Ivanov, “Connecting Orbits near the Adiabatic Limit of Lagrangian Systems with Turning Points”, Regul. Chaotic Dyn., 22:5 (2017), 479–501
2. V A. Ivanov, “Transversal connecting orbits of Lagrangian systems with turning points: Newton-Kantorovich method”, 2018 Days on Diffraction (DD), eds. O. Motygin, A. Kiselev, L. Goray, A. Kazakov, A. Kirpichnikova, M. Perel, IEEE, 2018, 149–154
3. Alexey V. Ivanov, “On Transversal Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field: the Newton – Kantorovich Approach”, Regul. Chaotic Dyn., 24:4 (2019), 392–417