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 $$TQ$$, and the Hamiltonian is defined on the cotangent bundle $$T^* Q$$—the fiber derivative is a map $$\mathbb{F}L:TQ \rightarrow T^* Q$$ such that


 * $$\mathbb{F}L(v) \cdot w = \left. \frac{d}{ds} \right|_{s=0} L(v+sw)$$,

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