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