Minimax periodic orbits of convex Lagrangian systems on complete Riemannian manifolds

DOI10.1007/S12220-022-01004-ZzbMATH Open1494.58006arXiv2009.11004OpenAlexW4289711879WikidataQ114220960 ScholiaQ114220960MaRDI QIDQ2162146

Wenmin Gong

Publication date: 5 August 2022

Published in: The Journal of Geometric Analysis (Search for Journal in Brave)

Abstract: In this paper we study the existence of periodic orbits with prescribed energy levels of convex Lagrangian systems on complete Riemannian manifolds. We extend the existence results of Contreras by developing a modified minimax principal to a class of Lagrangian systems on noncompact Riemannian manifolds, namely the so called $lsh$ Lagrangian systems. In particular, we prove that for almost every $kin(0,c_u(L))$ the exact magnetic flow associated to a $lsh$ Lagrangian has a contractible periodic orbit with energy $k$. We also discuss the existence and non-existence of closed geodesics on the product Riemannian manifold $RimesM$.

Full work available at URL: https://arxiv.org/abs/2009.11004

zbMATH Keywords

periodic orbitsRiemannian manifoldsmagnetic geodesicsconvex Lagrangiansgeneralized minimax principle

Mathematics Subject Classification ID

Variational principles in infinite-dimensional spaces (58E30) Symplectic aspects of Floer homology and cohomology (53D40) Variational problems in applications to the theory of geodesics (problems in one independent variable) (58E10) Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems (37J46)

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Linking and closed orbits
• On periodic orbits in cotangent bundles of non-compact manifolds
• Closed magnetic geodesics on \(S^2\)
• Closed orbits of a charge in a weakly exact magnetic field
• Periodic magnetic geodesics on almost every energy level via variational methods
• On the Rabinowitz Floer homology of twisted cotangent bundles
• Periodic orbits of twisted geodesic flows and the Weinstein-Moser theorem
• Action minimizing invariant measures for positive definite Lagrangian systems
• Hamiltonian dynamics on convex symplectic manifolds
• Closed characteristics on non-compact hypersurfaces in \(\mathbb {R}^{2n}\)
• Closed geodesics on complete surfaces
• Closed nonself-intersecting extremals of multivalued functionals
• On the existence of closed geodesics on noncompact Riemannian manifolds
• Closed geodesics on non-compact Riemannian manifolds
• Lagrangian graphs, minimizing measures and Mañé's critical values
• The Palais-Smale condition and Mañé's critical values
• Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles
• Floer homology in disk bundles and symplectically twisted geodesic flows
• Weak KAM theorem on non compact manifolds
• Applications of Hofer's geometry to Hamiltonian dynamics
• The Palais-Smale condition on contact type energy levels for convex Lagrangian systems
• Homotopy theory of infinite dimensional manifolds
• Closed magnetic geodesics on closed hyperbolic Riemann surfaces
• The Hamiltonian formalism and a many-valued analogue of Morse theory
• A.D. Alexandrov spaces with curvature bounded below
• Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles
• Floer homology in the cotangent bundle of a closed Finsler manifold and noncontractible periodic orbits
• CONTRACTIBLE PERIODIC ORBITS OF LAGRANGIAN SYSTEMS
• Existence of periodic solutions of Hamiltonian systems on almost every energy surface
• Action potential and weak KAM solutions

Related Items (3)

Infinitely many noncontractible closed magnetic geodesics on non-compact manifolds ⋮ Periodic orbits on Riemannian manifolds under the action of an at most quadratic potential ⋮ Ultimate periodicity of orbits for min-max systems

This page was built for publication: Minimax periodic orbits of convex Lagrangian systems on complete Riemannian manifolds