Normally stable hamiltonian systems

Kenneth R. Meyer; Jesús F. Palacián; Patricia Yanguas

doi:10.3934/dcds.2013.33.1201

Article Contents

2013, Volume 33, Issue 3: 1201-1214. Doi: 10.3934/dcds.2013.33.1201

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Normally stable hamiltonian systems

1.
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221
2.
Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona

Received: April 30, 2011

Revised: October 31, 2011

Published: February 2013

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Abstract

We study the stability of an equilibrium point of a Hamiltonian system with $n$ degrees of freedom. A new concept of stability called normal stability is given which applies to a system in normal form and relies on the existence of a formal integral whose quadratic part is positive definite. We give a necessary and sufficient condition for normal stability. This condition depends only on the quadratic terms of the Hamiltonian. We relate normal stability with formal stability and Liapunov stability. An application to the stability of the $L_4$ and $L_5$ equilibrium points of the spatial circular restricted three body problem is given.

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Mathematics Subject Classification: Primary: 34C20, 34C25, 37J40; Secondary: 70F10, 70K65.

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