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

Erasmo Caponio, Dario Corona, Roberto Giambò, Paolo Piccione

Published 2023-07-15Version 1

We consider an autonomous, indefinite Lagrangian $L$ admitting an infinitesimal symmetry $K$ 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 $p$ to a flow line $\gamma=\gamma(t)$ of $K$ that does not cross $p$. By utilizing the invariance of $L$ under the flow of $K$, we simplify the problem into a two-point boundary problem. Consequently, we derive an equation that involves the differential of the ``arrival time'' $t$, seen as a functional on the infinite dimensional manifold of connecting paths satisfying the semi-holonomic constraint defined by the Noether charge. When $L$ is positively homogeneous of degree $2$ 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.