Let M be a smooth manifold, let E and π π : E → → M be a vector bundle on M and its associated bundle projection, respectively. Let L : E → → R be a smooth function. We think of L as a Lagrangian by analogy with the classical case where M = R , E = T M = R × × R and L ( x , v ) = 1 2 m v 2 − − V ( x ) for some positive number m ∈ ∈ R and function V : M → → R .