Periodic Orbits of Twisted Geodesic Flows and The Weinstein-Moser Theorem

2007/05/13 by Ginzburg, Viktor L., Gurel, Basak Z. · 2 citations
#37J45 #53D40 #70H12 #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.0705.1818

Abstract

In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more general theorem concerning periodic orbits of autonomous Hamiltonian flows near Morse-Bott non-degenerate, symplectic extrema. Namely, we show that all energy levels near such extrema carry periodic orbits, provided that the ambient manifold meets certain topological requirements. This result is a partial generalization of the Weinstein-Moser theorem. The proof of the generalized Weinstein-Moser theorem is a combination of a Sturm-theoretic argument and a Floer homology calculation.

Cited by

• Electromagnetic curvature via Jacobi-Maupertuis and beyond
• The growth rate of closed prime magnetic geodesics on closed contact manifolds

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