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

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

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