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.
Highlights
Symplectic Sturm theory has a lot of predecessor, like Morse, Lidskii, Bott, Edwards, Givental who proved the Lagrangian nonoscillation of the Picard-Fuchs equation for hyperelliptic integrals
We stress on the fact that classical comparison theorems for conjugate and focal points in Riemannian manifolds and more generally on Lorenzian manifolds but for timelike geodesics, requires curvature assumptions or Morse index arguments
Remark 4.8 The basic idea behind the proof of Proposition 4.7 is to perturb the path s → Ls in order to get regular crossing. Once this has been done, for concluding, it is enough to prove that the local contribution at each crossing instant to the spectral flow is the opposite of the local contribution to the Maslov index
Summary
Symplectic Sturm theory has a lot of predecessor, like Morse, Lidskii, Bott, Edwards, Givental who proved the Lagrangian nonoscillation of the Picard-Fuchs equation for hyperelliptic integrals. They, describe the rotation of a straight line through the origin of the phase plane of the
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