Conservative dynamics and the calculus of variations

Starting with elementary calculus of variations and Legendre trans-form, it is shown how the mathematical structures of conservative dynam-ics (Poincaré-Cartan integral invariant, symplectic structure, Hamiltonian form of the equations) arise from the simple computation of the variations of an action integral. The study of simple examples of integrable geodesic flows on the 2-torus then leads to the notion of Lagrangian submanifolds and to the Hamilton-Jacobi equation, whose relation to the Hamiltonian vector-field is the first step of the duality between particles and waves. The two last lectures are a brief introduction to KAM and weak KAM theories which describe what remains of complete integrability for more general hamiltonians, in particular for perturbations of integrable convex hamiltonians.1 Due to the format of the course, only the easy results are proved while the harder (or longer) ones are admitted without remorse. This is, I hope, justified by the fact that it allows to give in a very short time some idea of the general architecture of the theory. 1The first three lectures are directly inspired by the course [C1] on weak KAM theory given by the author in Barcelona in July 2004. The last one owes much to the paper [Fe] and to discussions with Jacques Féjoz. 1 1 From the calculus of variations to the struc-tures of conservative dynamics 1.