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

Abstract

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 γ=γ(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma =\\gamma (t)$$\\end{document} 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.