On the existence of Euler-Lagrange orbits satisfying the conormal boundary conditions

Luca Asselle

On the existence of Euler-Lagrange orbits satisfying the conormal boundary conditions

Dynamical Systems 2016-11-28 v2 Symplectic Geometry

Authors: Luca Asselle

Abstract

Let $(M,g)$ be a closed Riemannian manifold, $L: TM\rightarrow \mathbb R$ be a Tonelli Lagrangian. Given two closed submanifolds $Q_0$ and $Q_1$ of $M$ and a real number $k$ , we study the existence of Euler-Lagrange orbits with energy $k$ connecting $Q_0$ to $Q_1$ and satisfying the conormal boundary conditions. We introduce the Ma\~n\'e critical value which is relevant for this problem and discuss existence results for supercritical and subcritical energies. We also provide counterexamples showing that all the results are sharp.

Keywords

Euler-Lagrange orbits conormal boundary conditions Tonelli Lagrangian Mañé critical value Riemannian manifold

Cite

@article{arxiv.1507.05883,
  title  = {On the existence of Euler-Lagrange orbits satisfying the conormal boundary conditions},
  author = {Luca Asselle},
  journal= {arXiv preprint arXiv:1507.05883},
  year   = {2016}
}

Comments

In this revised version some sharper examples have been added in Section 6 and some gaps have been eliminated. To appear in Journal of Functional Analysis

On the existence of Euler-Lagrange orbits satisfying the conormal boundary conditions

Abstract

Keywords

Cite

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