Sturm theory with applications in geometry and classical 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.

Renormalized oscillation theory for regular linear non-Hamiltonian systems

“…(See, e.g,. [1,4,5,6,7,8,9,15,16,21]). On the other hand, in both the Hamiltonian and non-Hamiltonian settings, if the target space is not Dirichlet then such monotonicity is not assured.…”