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This article is cited in 6 scientific papers (total in 6 papers)
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.
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
Linking options:
https://www.mathnet.ru/eng/sm9291https://doi.org/10.1070/SM9291 https://www.mathnet.ru/eng/sm/v211/i1/p32
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Abstract page: | 762 | Russian version PDF: | 150 | English version PDF: | 55 | References: | 80 | First page: | 72 |
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