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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

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

Bibliographic databases:

UDC: 517.925.51+517.93
MSC: Primary 34D05, 58K05; Secondary 58K05
Received: 10.06.2019

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
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