Fiber derivative

In the context of Lagrangian Mechanics the fiber derivative is used to convert between the Lagrangian and Hamiltonian forms. In particular, if Q is the configuration manifold then the Lagrangian L is defined on the tangent bundle T Q and the Hamiltonian is defined on the cotangent bundle T ∗ Q —the fiber derivative is a map F L : T Q → T ∗ Q such that

where v and w are vectors from the same tangent space. When restricted to a particular point, the fiber derivative is a Legendre transformation.