Sturm theory with applications in geometry and classical mechanics

TL;DR

This paper generalizes classical Sturm theorems to symplectic geometry, linking them to the Maslov index, and applies these results to geometry and celestial mechanics.

Contribution

It extends symplectic Sturm theory from optical Hamiltonians to general Hamiltonians, with applications in geometry and mechanics.

Findings

Generalized Sturm theorems to broader Hamiltonian systems.

Connected Sturm theory with Maslov index properties.

Applied results to geometry of conjugate points and celestial mechanics.

Abstract

Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index. Starting from the celebrated paper of Arnol'd on symplectic Sturm theory for optical Hamiltonians, we provide a generalization of his results to general Hamiltonians. We finally apply these results for detecting some geometrical information about the distribution of conjugate and focal points on semi-Riemannian manifolds and for studying the geometrical properties of the solutions space of singular Lagrangian systems arising in Celestial Mechanics.