# American Institute of Mathematical Sciences

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

 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.
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
 [1] Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751 [2] Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260 [3] Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of discrete mechanical systems by stages. Journal of Geometric Mechanics, 2016, 8 (1) : 35-70. doi: 10.3934/jgm.2016.8.35 [4] Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029 [5] Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems. Journal of Geometric Mechanics, 2010, 2 (1) : 69-111. doi: 10.3934/jgm.2010.2.69 [6] Rodolfo Ríos-Zertuche. Characterization of minimizable Lagrangian action functionals and a dual Mather theorem. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2615-2639. doi: 10.3934/dcds.2020143 [7] Boris Hasselblatt. Critical regularity of invariant foliations. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 931-937. doi: 10.3934/dcds.2002.8.931 [8] Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1811-1827. doi: 10.3934/dcds.2014.34.1811 [9] Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185 [10] Bin Dong, Aichi Chien, Yu Mao, Jian Ye, Fernando Vinuela, Stanley Osher. Level set based brain aneurysm capturing in 3D. Inverse Problems and Imaging, 2010, 4 (2) : 241-255. doi: 10.3934/ipi.2010.4.241 [11] Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems and Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029 [12] Bertuel Tangue Ndawa. Infinite lifting of an action of symplectomorphism group on the set of bi-Lagrangian structures. Journal of Geometric Mechanics, 2022, 14 (3) : 409-426. doi: 10.3934/jgm.2022006 [13] Kaizhi Wang. Action minimizing stochastic invariant measures for a class of Lagrangian systems. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1211-1223. doi: 10.3934/cpaa.2008.7.1211 [14] Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483 [15] Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807 [16] Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 633-646. doi: 10.3934/dcds.2002.8.633 [17] Nils Ackermann, Thomas Bartsch, Petr Kaplický. An invariant set generated by the domain topology for parabolic semiflows with small diffusion. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 613-626. doi: 10.3934/dcds.2007.18.613 [18] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems and Imaging, 2021, 15 (2) : 315-338. doi: 10.3934/ipi.2020070 [19] Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390 [20] Zhenlin Guo, Ping Lin, Guangrong Ji, Yangfan Wang. Retinal vessel segmentation using a finite element based binary level set method. Inverse Problems and Imaging, 2014, 8 (2) : 459-473. doi: 10.3934/ipi.2014.8.459

2021 Impact Factor: 1.497