Periodic solutions of Lagrangian systems on a compact manifold. (English) Zbl 0605.58034

Benci, Vieri

J. Differ. Equations 63, 135-161 (1986).

The author studies periodic solutions of prescribed period when the Lagrangian system is constrained to a compact manifold. This fact allows him to use many techniques developed in the theory of closed geodesics on Riemannian compact manifolds [cf. the reviewer, Foundations of Global Nonlinear Analysis, Teubner-Texte zur Mathematik, Vol. 86, Leipzig (1986)]. He considers M to be a smooth n-dimensional manifold with TM its tangent bundle. He then considers a time periodic Lagrangian of periodic T, \({\mathcal L}_ t: TM\to R\), and he searches for T-periodic solutions of the Lagrange equations, which in local coordinates are \[ (1)\quad \frac{d}{dt}\frac{\partial {\mathcal L}}{\partial \dot q}(t,q,\dot q)- \frac{\partial {\mathcal L}}{\partial q}(t,q,\dot q)=0,\quad i=1,2,...,n. \] The author proves that if the fundamental group of M is finite, then (1) has infinitely many T-periodic solutions, if \({\mathcal L}_ t\) satisfies certain physically reasonable assumptions. The author makes extensive use of the Palais-Smale condition (C).

Reviewer: T.Rassias

MSC:

37G99	Local and nonlocal bifurcation theory for dynamical systems
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Keywords:

Sobolev space; Lie group; Abelian group; periodic solutions; Lagrangian system; compact manifold; Palais-Smale condition

References:

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