Geometry of lagrangian and hamiltonian formalisms in the dynamics of strings

Abstract

The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration space is TM, is based on the existence of canonical symplectic isomorphisms of double vector bundles T∗TM, T∗T∗M, and TT∗M, where the symplectic structure on TT∗M is the tangent lift of the canonical symplectic structure T∗M. We show that there exists an analogous picture in the dynamics of objects for which the configuration space is nTM, if we make use of certain structures of graded bundles of degree n, i.e. objects generalizing vector bundles (for which n = 1). For instance, the role of TT∗M is played in our approach by the manifold ^nTM^nT∗M, which is canonically a graded bundle of degree n over ^nTM. Dynamics of strings and the Plateau problem in statics are particular cases of this framework