On the density of mechanical Lagrangians in $T^{2}$ without continuous invariant graphs in all supercritical energy levels

Rafael O. Ruggiero

doi:10.3934/dcdsb.2008.10.661

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2008, Volume 10, Issue 2&3: 661-679. Doi: 10.3934/dcdsb.2008.10.661

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On the density of mechanical Lagrangians in $T^{2}$ without continuous invariant graphs in all supercritical energy levels

Rafael O. Ruggiero^1,

1.
Departamento de Matemática, Pontificia Universidade Católica do Rio de Janeiro, Rua Marqués de São Vicente 225, Gávea, Rio de Janeiro

Received: August 31, 2006

Revised: May 31, 2007

Published: August 2008

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Abstract

We show that the set of $C^{\infty}$ mechanical Lagrangians $L(p,v)$ in $T^{2}$ without continuous invariant graphs in all supercritical energy levels is dense in the $C^{1}$ topology. A mechanical Lagrangian $L: T$$T^{2} \rightarrow \mathbb R$ is a function in the tangent space of the torus $T$$T^{2}$ given by $L(p,v)=\frac{1}{2}g(v,v)-U(p)$, where $g$ is a Riemannian metric and $U$ is a smooth potential.

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Mathematics Subject Classification: 37J40, 37J50, 53D25.

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