paper

Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge

arXiv:2307.07883 · doi:10.1007/s10231-024-01424-4

Abstract

We consider an autonomous, indefinite Lagrangian admitting an infinitesimal symmetry whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point to a flow line of the infinitesimal generator . By utilizing the invariance of the Lagrangian under the flow of , we simplify the problem into a two-point boundary problem. Consequently, we derive an equation that involves the differential of the ``arrival time'', seen as a functional on the infinite dimensional manifold of connecting paths satisfying the semi-holonomic constraint defined by the Noether charge. When the Lagrangian is positively homogeneous of degree two in the velocities, the resulting equation establishes a variational principle that extends the Fermat's principle in a stationary spacetime. Furthermore, we also analyze the scenario where the Noether charge is affine.

36 pages; fully revised version: we show that the Lagrangians considered include a class of Lorentz-Finsler metrics which do not reduce to Lorentzian metrics; we added a new appendix where we deal with pseudocoercivity in connection with global hyperbolicity. To appear on Ann. Mat. Pura App

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