Fiber derivative

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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 = \frac{d}{ds}|_{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.

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