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September  2008, 10(2&3, September): 661-679. doi: 10.3934/dcdsb.2008.10.661

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, Brazil

Received  September 2006 Revised  June 2007 Published  June 2008

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.
Keywords: Mather set., critical level, mechanical Lagrangian, Invariant graph.
Mathematics Subject Classification: 37J40, 37J50, 53D2.
Citation: Rafael O. Ruggiero. On the density of mechanical Lagrangians in $T^{2}$ without continuous invariant graphs in all supercritical energy levels. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 661-679. doi: 10.3934/dcdsb.2008.10.661
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