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On geodesic flows with symmetries and closed magnetic geodesics on orbifolds

Part of: Symplectic geometry, contact geometry Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Variational problems in infinite-dimensional spaces Global differential geometry

Published online by Cambridge University Press:  20 November 2018

LUCA ASSELLE  and
FELIX SCHMÄSCHKE
LUCA ASSELLE
Affiliation:
Justus Liebig Universität Giessen, Mathematisches Institut, Arndtstrasse 2, Raum 102, D-35392 Giessen, Germany email luca.asselle@ruhr-uni-bochum.de
FELIX SCHMÄSCHKE
Affiliation:
Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany email felix.schmaeschke@math.hu-berlin.de
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Abstract

Let $Q$ be a closed manifold admitting a locally free action of a compact Lie group $G$. In this paper, we study the properties of geodesic flows on $Q$ given by suitable G-invariant Riemannian metrics. In particular, we will be interested in the existence of geodesics that are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.

Keywords

magnetic flowsperiodic orbitsRabinowitz action functionalsymplectic reduction

MSC classification

Primary: 37J15: Symmetries, invariants, invariant manifolds, momentum maps, reduction 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 53D20: Momentum maps; symplectic reduction
Secondary: 53C35: Symmetric spaces 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 6 , June 2020 , pp. 1480 - 1509
Copyright
© Cambridge University Press, 2018

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