arXiv subjects

Daniel Offin

Publications and source records attributed to Daniel Offin.

5 recordsLinked to original sources

arXiv2026

Morse Theory and Stability of Periodic Billiard Trajectories

In a billiard table with $C^2$ boundary, trajectories are critical points of the negative length functional on the space of admissible paths in the billiard table. The goal of this paper is twofold: first, to prove theorems which aid in computing the Morse index of trajectories; second, to use the Morse index to determine the stability of periodic trajectories. To the first end, we prove discrete analogs to the Morse index theorem for arbitrary trajectories, which are critical points with respect to fixed-endpoint variations, and for periodic trajectories, which are critical points with respect to periodic variations. To the second end, we prove a criterion for linear stability of periodic trajectories depending on the Morse index of the second iterate of the trajectory, and prove a sufficient condition for a billiard table to carry a linearly stable 2-periodic trajectory.

math.DS↗

arXiv2024

Minimization and Hyperbolicity

In this paper we study the relationship between the strict locally minimizing orbits for time dependent lagrangian systems and hyperbolicity properties of the corresponding lagrangian flow.

math.DS↗

arXiv2020

Sturm theory with applications in geometry and classical mechanics

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.

math.DG↗

arXiv2019

Keplerian orbits through the Conley-Zehnder index

It was discovered by Gordon in 1977 that Keplerian ellipses in the plane are minimizers of the Lagrangian action and spectrally stable as periodic points of the associated Hamiltonian flow. The aim of this paper is to give a homotopy theoretical proof of these results through a self-contained, explicit and simple computation of the Conley-Zehnder index. The techniques developed in this paper can be used to investigate the higher dimensional case of Keplerian ellipses, where the classical variational proof no longer applies.

math.DS↗

arXiv2018

Stability of Periodic Orbits by Conley-Zehnder index theory

We give a necessary and sufficient condition for strong stability of low dimensional Hamiltonian systems, in terms of the iterates of a closed orbit and the Conley-Zehnder index. Applications to Mathieu equation and stable harmonic oscillations for forced pendulum type equations are considered as applications of the main result.

math.DS↗