| Full text | |
| Author(s): |
Caponio, Erasmo
;
Corona, Dario
;
Giambo, Roberto
;
Piccione, Paolo
Total Authors: 4
|
| Document type: | Journal article |
| Source: | Annali di Matematica Pura ed Applicata; v. 203, n. 4, p. 32-pg., 2024-04-23. |
| 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 gamma = gamma ( 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. (AU) | |
| FAPESP's process: | 22/13010-3 - Orthogonal geodesics in manifolds with singular boundary. Applications to the theory of minimal surfaces. |
| Grantee: | Paolo Piccione |
| Support Opportunities: | Research Grants - Visiting Researcher Grant - International |
| FAPESP's process: | 22/16097-2 - Modern methods in differential geometry and geometric analysis |
| Grantee: | Paolo Piccione |
| Support Opportunities: | Research Projects - Thematic Grants |