Preface

This monograph is devoted to presenting in detail a few selected applications of critical point theory, and in particular Morse theory, to Lagrangian dynamics. A Lagrangian system is defined by a configuration space M , which has the structure of a smooth manifold, and a Lagrangian function L defined on the tangent bundle of M and, in the non-conservative case, depending upon time as well. In JosephLouis Lagrange’s reformulation of classical mechanics, the Lagrangian function is given by the difference of kinetic and potential energy. The motion of the associated mechanical system is described by the Euler-Lagrange equations, a system of second-order ordinary differential equations that involves the Lagrangian. The principle of least action, which in different settings is even anterior to Lagrange’s work, states that the curves that are solutions of the Euler-Lagrange equations admit a variational characterization: they are extremal points of a functional, the action, associated to the Lagrangian. The development of critical point theory in the nineteenth and twentieth century provided a powerful machinery to investigate dynamical questions in Lagrangian systems, such as existence, multiplicity or uniqueness of solutions of the Euler-Lagrange equations with prescribed boundary conditions. In this monograph, we will consider closed configuration spaces M and we will focus on the class of so-called Tonelli Lagrangians: these are smooth Lagrangian functions L : R×TM → R that, when restricted to the fibers of TM , have positive definite Hessian and superlinear growth. We will normally restrict ourselves to those Tonelli Lagrangians L whose solutions of the Euler-Lagrange equations are defined for all times, a condition that is always fulfilled when the time-derivative of the Lagrangian is suitably bounded. The importance of the Tonelli class is twofold. From the variational point of view, the Tonelli assumptions imply existence and regularity of action minimizing curves joining given points in the configuration space. From a symplectic point of view, Tonelli Lagrangians constitute the broadest family of fiberwise convex Lagrangian functions for which the Lagrangian-Hamiltonian duality holds. These generalities on Tonelli Lagrangians, together with a brief introduction to the Lagrangian and Hamiltonian formalism, will be the subject of Chapter 1.