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arXiv · 2307.07883

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

Erasmo Caponio·Dario Corona·Roberto Giambò·Paolo Piccione
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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 $K$. By utilizing the invariance of the Lagrangian 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'', 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.

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Erasmo Caponio, Dario Corona, Roberto Giambò, Paolo Piccione. 2023-07-15. Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge. https://doi.org/10.1007/s10231-024-01424-4

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