Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-25T22:45:56.063Z Has data issue: false hasContentIssue false
  • English
  • Français

On geodesic flows with symmetries and closed magnetic geodesics on orbifolds

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

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbondandolo, A.. Lectures on the free period Lagrangian action functional. J. Fixed Point Theory Appl. 13(2) (2013), 397–430.Google Scholar
Abbondandolo, A., Asselle, L., Benedetti, G., Mazzucchelli, M. and Taimanov, I. A.. The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere. Adv. Nonlinear Stud. 1 (2017), 1–17.Google Scholar
Abbondandolo, A., Macarini, L., Mazzucchelli, M. and Paternain, G. P.. Infinitely many periodic orbits of exact magnetic flows on surfaces for almost every subcritical energy level. J. Eur. Math. Soc. (JEMS) 19(2) (2017), 551–579.Google Scholar
Abbondandolo, A., Macarini, L. and Paternain, G. P.. On the existence of three closed magnetic geodesics for subcritical energies. Comment. Math. Helv. 90(1) (2015), 155–193.Google Scholar
Adem, A., Leida, J. and Ruan, Y.. Orbifolds and Stringy Topology. Cambridge University Press, New York, 2007.Google Scholar
Asselle, L.. On the existence of Euler–Lagrange orbits satisfying the conormal boundary conditions. J. Funct. Anal. 271 (2016), 3513–3553.Google Scholar
Asselle, L. and Benedetti, G.. Infinitely many periodic orbits in non-exact oscillating magnetic fields on surfaces with genus at least two for almost every low energy level. Calc. Var. Partial Differential Equations 54(2) (2015), 1525–1545.Google Scholar
Asselle, L. and Benedetti, G.. The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles. J. Topol. Anal. 8(3) (2016), 545–570.Google Scholar
Asselle, L. and Benedetti, G.. On the periodic motions of a charged particle in an oscillating magnetic flows on the two-torus. Math. Z. 286(3–4) (2016), 843–859.Google Scholar
Asselle, L. and Mazzucchelli, M.. On Tonelli periodic orbits with low energy on surfaces. Trans. Amer. Math. Soc. (2017), https://doi.org/10.1090/tran/7185.Google Scholar
Asselle, L. and Schmäschke, F.. On the existence of closed magnetic geodesics via symplectic reduction. J. Fixed Point Theory Appl. 20 (2018), article 49.Google Scholar
Benedetti, G.. The contact property for symplectic magnetic fields on s 2. Ergod. Th. & Dynam. Sys. 36(3) (2016), 682–713.Google Scholar
Benedetti, G. and Zehmisch, K.. On the existence of periodic orbits for magnetic systems on the two-sphere. J. Mod. Dyn. 9 (2015), 141–146.Google Scholar
Bloch, A. M., Marsden, J. E., Krishnaprasad, P. S. and Murray, Richard M.. Nonholonomic mechanical systems with symmetry. Arch. Ration. Mech. Anal. 136 (1996), 21–99.Google Scholar
Bredon, G. E.. Introduction to Compact Transformation Groups (Pure and Applied Mathematics, 46). Academic Press, New York, 1972.Google Scholar
Bredon, G. E.. Sheaf Theory, second edn. Springer, New York, 1997.Google Scholar
Contreras, G.. The Palais–Smale condition on contact type energy levels for convex Lagrangian systems. Calc. Var. Partial Differential Equations 27(3) (2006), 321–395.Google Scholar
Davison, C. M. and Dullin, H. R.. Geodesic flow on three-dimensional ellipsoids with equal semi-axes. Regul. Chaotic Dyn. 12(2) (2007), 172–197.Google Scholar
Dragomir, G. C.. Closed geodesics on orbifolds of nonpositive or nonnegative curvature. Geom. Dedicata 172(1) (2014), 199–211.Google Scholar
Fedorov, Y. N. and Jovanović, B.. Integrable nonholonomic geodesic flows on compact Lie groups. Topological Methods in the Theory of Integrable Systems. Camb. Sci. Publ., Cambridge, 2006, pp. 115–152.Google Scholar
Félix, Y., Halperin, S. and Thomas, J.-C.. Rational Homotopy Theory. Springer, New York, 2001.Google Scholar
Frauenfelder, U.. The Arnold-Givental conjecture and moment Floer homology. Int. Math. Res. Not. IMRN 42 (2008), 2179–2269.Google Scholar
Ginzburg, V. L.. On Closed Trajectories of a Charge in a Magnetic Field (An Application of Symplectic Geometry, 8). Cambridge University Press, Cambridge, 1996, pp. 131–148.Google Scholar
Ginzburg, V. L. and Gürel, B. Z.. Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem. Comment. Math. Helv. 84 (2009), 865–907.Google Scholar
Griffiths, P. and Harris, J.. Principles of Algebraic Geometry. Wiley, New York, 1978.Google Scholar
Guruprasad, K. and Haefliger, A.. Closed geodesics on orbifolds. Topology 45(3) (2006), 611–641.Google Scholar
Klingenberg, W.. Lectures on Closed Geodesics. Springer, Berlin, 1978.Google Scholar
Klingenberg, W.. Riemannian Geometry. De Gruyter, Berlin, New York, 1995.Google Scholar
Kobayashi, S. and Nomizu, K.. Foundations of Differential Geometry. Wiley Classics Library, New York, 1996.Google Scholar
Lange, C.. On the existence of closed geodesics on two-orbifolds. Pacific J. Math. 294(2) (2018), 453–472.Google Scholar
Mardsen, J. and Ratiu, T.. Introduction to Mechanics and Symmetry. Springer, New York, 1998.Google Scholar
Ol’shanskii, A. Yu. and Sapir, M. V.. Non-amenable finitely presented torsion-by-cyclic groups. Publ. Math. Inst. Hautes Études Sci. 96 (2002), 43–169.Google Scholar
Schneider, M.. Closed magnetic geodesics on S 2. J. Differential Geom. 87(2) (2011), 343–388.Google Scholar
Schneider, M.. Alexandrov-embedded closed magnetic geodesics on S 2. Ergod. Th. & Dynam. Sys. 32(4) (2012), 1471–1480.Google Scholar
Schneider, M.. Closed magnetic geodesics on closed hyperbolic Riemann surfaces. Proc. Lond. Math. Soc. (3) 105(2) (2012), 424–446.Google Scholar
Struwe, M.. Existence of periodic solutions of Hamiltonian systems on almost every energy surface. Bol. Soc. Brasil. Mat. (N.S.) 20(2) (1990), 49–58.Google Scholar
Taimanov, I. A.. Non-self-intersecting closed extremals of multivalued or not everywhere positive functionals. Izv. Akad. Nauk SSSR Ser. Mat. 55(2) (1991), 367–383.Google Scholar
Taimanov, I. A.. Closed extremals on two-dimensional manifolds. Uspekhi Mat. Nauk 47(2(284)) (1992), 143–185 223.Google Scholar