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.

References

• Marsden, Jerrod E.; Ratiu, Tudor (1998). Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems