Physics:Fiber derivative

In the context of Lagrangian mechanics, the fiber derivative is used to convert between the Lagrangian and Hamiltonian forms. In particular, if [math]\displaystyle{ Q }[/math] is the configuration manifold then the Lagrangian [math]\displaystyle{ L }[/math] is defined on the tangent bundle [math]\displaystyle{ TQ }[/math] , and the Hamiltonian is defined on the cotangent bundle [math]\displaystyle{ T^* Q }[/math]—the fiber derivative is a map [math]\displaystyle{ \mathbb{F}L:TQ \rightarrow T^* Q }[/math] such that

[math]\displaystyle{ \mathbb{F}L(v) \cdot w = \left. \frac{d}{ds} \right|_{s=0} L(v+sw) }[/math],

where [math]\displaystyle{ v }[/math] and [math]\displaystyle{ w }[/math] are vectors from the same tangent space. When restricted to a particular point, the fiber derivative is a Legendre transformation.

References

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