The differential geometric structure of general mechanical systems from the Lagrangian point of view

Abstract

Several differential-geometric points of view on analytical mechanics of systems with a finite number of degrees of freedom are developed in generality, emphasizing Cartan’s calculus of differential forms and Ehresmann’s theory of jet spaces. The classical theory of Lagrange’s equations with external forces and constraints (‘‘holonomic’’ or ‘‘nonholonomic’’) is put into an invariant and coordinate-free form. The relation between this ‘‘Lagrangian’’ and the ‘‘Hamiltonian-symplectic’’ approach, which is that most extensively used in the contemporary mathematical physics literature, is also developed.