To describe the Legendre transformation locally, let U ⊆ ⊆ M be a coordinate chart over which E is trivial. Picking a trivialization of E over U , we obtain charts E U ≅ ≅ U × × R r and E U ∗ ∗ ≅ ≅ U × × R r . In terms of these charts, we have F L ( x ; v 1 , … … , v r ) = ( x ; p 1 , … … , p r ) , where p i = ∂ ∂ L ∂ ∂ v i ( x ; v 1 , … … , v r ) for all i = 1 , … … , r . If, as in the classical case, the restriction of L : E → → R to each fiber E x is strictly convex and bounded below by a positive definite quadratic form minus a constant, then the Legendre transform F L : E → → E ∗ ∗ is a diffeomorphism. Suppose that F L is a diffeomorphism and let H : E ∗ ∗ → → R be the " Hamiltonian " function defined by H ( p ) = p ⋅ ⋅ v − − L ( v ) , where v = ( F L ) − − 1 ( p ) . Using the natural isomorphism E ≅ ≅ E ∗ ∗ ∗ ∗ , we may view the Legendre transformation of H as a map F H : E ∗ ∗ → → E . Then we have ( F L ) − − 1 = F H .