A dynamical condition for differentiability of Mather's average action

Alexandre Rocha; Mário Jorge Dias Carneiro

doi:10.3934/jgm.2014.6.549

Article Contents

2014, Volume 6, Issue 4: 549-566. Doi: 10.3934/jgm.2014.6.549

This issue Previous Article Nonlinear constraints in nonholonomic mechanics Next Article

A dynamical condition for differentiability of Mather's average action

Alexandre Rocha^1, and
Mário Jorge Dias Carneiro^2,

1.
IEF, Campus UFV-Florestal, Universidade Federal de Viçosa, Florestal, MG 35690-000
2.
ICEX, Universidade Federal de Minas Gerais, Belo Horizonte, MG 30161-970

Received: September 30, 2013

Revised: July 31, 2014

Published: November 2014

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Abstract

We prove the differentiability of Mather's average action on all rotation vectors of measures whose supports are contained in a Lipschitz Lagrangian asymptotically isolated graph, invariant by Tonelli Hamiltonians. We also show the relationship between differentiability of $\beta $ and local integrability of the Hamiltonian flow.

Keywords:

Mather's theory,
differentiability of Mather's average action,
integrability of the Hamiltonian flow.

Mathematics Subject Classification: Primary: 37J50, 37J15; Secondary: 37J35.

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