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CONTRACTIBLE PERIODIC ORBITS OF LAGRANGIAN SYSTEMS

Part of: Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Hamiltonian and Lagrangian mechanics

Published online by Cambridge University Press:  30 January 2019

MIGUEL PATERNAIN Open the ORCID record for MIGUEL PATERNAIN [Opens in a new window]
MIGUEL PATERNAIN*
Affiliation:
Universidad de la República, Centro de Matemática, Facultad de Ciencias, Iguá 4225, 11400 Montevideo, Uruguay email miguel@cmat.edu.uy
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Abstract

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We consider a convex Lagrangian $L:\mathit{TM}\rightarrow \mathbb{R}$ quadratic at infinity with $L(x,0)=0$ for every $x\in M$ and such that the 1-form $\unicode[STIX]{x1D703}$ defined by $\unicode[STIX]{x1D703}_{x}(v)=L_{v}(x,0)v$ is not closed. We show that for every number $a<0$, there is a contractible (nonconstant) periodic orbit with action $a$. We also obtain estimates of the period and energy of such periodic orbits.

Keywords

convex Lagrangianaction functionalperiodic orbit

MSC classification

Primary: 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
Secondary: 70H12: Periodic and almost periodic solutions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 3 , June 2019 , pp. 445 - 453
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

The author was supported by an Anii grant.

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