Matematicheskii Sbornik
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb., 2020, Volume 211, Number 1, Pages 32–59 (Mi msb9291)

First integrals and asymptotic trajectories

V. V. Kozlov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: We discuss the relationship between the singular points of an autonomous system of differential equations and the critical points of its first integrals. Applying the well-known Splitting Lemma, we introduce local coordinates in which the first integral takes a “canonical” form. These coordinates make it possible to introduce a quasihomogeneous structure in some neighbourhood of any singular point and so to prove general theorems on the existence of asymptotic trajectories which go into or out of that singular point. We consider quasihomogeneous truncations of the original system of differential equations and show that if the singular point is isolated, the quasihomogeneous system is Hamiltonian. For a general mechanical system with two degrees of freedom, we prove a theorem on the instability of an equilibrium when it is neither a local minimum nor a local maximum of the potential energy.
Bibliography: 21 titles.

Keywords: splitting lemma, quasihomogeneous system, asymptotic trajectory, Hamiltonian system, gyroscopic stabilization.

 Funding Agency Grant Number Russian Science Foundation 19-71-30012 This research was funded by a grant from the Russian Science Foundation (project no. 19-71-30012).

DOI: https://doi.org/10.4213/sm9291

Full text: PDF file (664 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2020, 211:1, 29–54

Bibliographic databases:

UDC: 517.925.51+517.93
MSC: Primary 34D05, 58K05; Secondary 58K05

Citation: V. V. Kozlov, “First integrals and asymptotic trajectories”, Mat. Sb., 211:1 (2020), 32–59; Sb. Math., 211:1 (2020), 29–54

Citation in format AMSBIB
\Bibitem{Koz20} \by V.~V.~Kozlov \paper First integrals and asymptotic trajectories \jour Mat. Sb. \yr 2020 \vol 211 \issue 1 \pages 32--59 \mathnet{http://mi.mathnet.ru/msb9291} \crossref{https://doi.org/10.4213/sm9291} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4045697} \elib{https://elibrary.ru/item.asp?id=45498441} \transl \jour Sb. Math. \yr 2020 \vol 211 \issue 1 \pages 29--54 \crossref{https://doi.org/10.1070/SM9291} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000522111300001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85087456204}