Fiber derivative

In the context of Lagrangian mechanics, the fiber derivative is used to convert between the Lagrangian and Hamiltonian forms. In particular, if formula_1 is the configuration manifold then the Lagrangian formula_2 is defined on the tangent bundle formula_3 and the Hamiltonian is defined on the cotangent bundle formula_4—the fiber derivative is a map formula_5 such that
where formula_7 and formula_8 are vectors from the same tangent space. When restricted to a particular point, the fiber derivative is a Legendre transformation.