Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge (Q6594156)

This paper deals with the problem of finding solutions with fixed energy for an autonomous Lagrangian system with a finite number of degrees of freedom, subject to two-point or periodic boundary conditions. The authors consider an indefinite Lagrangian \(L\) on a manifold \(M\) (say, \(L : TM \longrightarrow \mathcal{R}\) that is invariant under a one-dimensional group of local diffeomorphisms generated by a complete vector field \(K\). The Noether charge associated to \(K\) can be simply defined as \(K^V(L)\), where \(K^V\) denotes the vertical lift of \(K\) to \(TM\). Indeed, the authors could simplify their notations just using, for instance, those in the book [the reviewer and \textit{P. R. Rodrigues}, Methods of differential geometry in analytical mechanics. Amsterdam etc.: North-Holland (1989; Zbl 0687.53001)]. The additional assumption is that the Noether charge should be linear in each tangent space \(T_xM\). \N\NThe aim of the paper is to investigate the solutions to the Euler-Lagrange equations of \(L\) that connect a point \(p\) to a flow line \(\gamma\) of \(K\) and have fixed energy \(\kappa\). Indeed, Theorem 5.1 gives a necessary and sufficient condition to fulfill this condition. Let us say that, when \(L\) is positively homogeneous of degree 2, then the authors obtain a generalization of Fermat's principle in a stationary spacetime. At the end of the paper, they discuss the case when the Noether charge is affine. The results are interesting in general relativity.