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Nonexistence of invariant graphs in all supercritical energy levels of mechanical Lagrangians in T 2

Authors:	Rafael O Ruggiero

Institution:	1. Pontifícia Universidade Católica do Rio de Janeiro, PUC-Rio, Dep. de Matemática, Rua Marqués de S?o Vicente, 225, Gávea, RJ BRAZIL

Abstract:	Let (T², g) be a smooth Riemannian structure in the torus T². We show that given ε > 0 and any C^∞ function U : T² → ℝ there exists a C¹ function U_ε with Lipschitz derivatives that is ε-C⁰ close to U for which there are no continuous invariant graphs in any supercritical energy level of the mechanical Lagrangian L_ε : TT² → ℝ given by $L{\left( {p,\upsilon } \right)} = \frac{1} {2}g{\left( {\upsilon ,\upsilon } \right)} - U_{\varepsilon } {\left( p \right)}$ . We also show that given n ∈ ℕ, the set of C^∞ potentials U : T² → ℝ for which there are no continuous invariant graphs in any supercritical energy level E ≤ n of $L{\left( {p,\upsilon } \right)} = \frac{1} {2}g{\left( {\upsilon ,\upsilon } \right)} - U{\left( p \right)}$ is C⁰ dense in the set of C^∞ functions. Partially supported by CNPq, FAPERJ-Cientistas do nosso estado.

Keywords:	invariant graph mechanical Lagrangian critical level
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