Английская Википедия: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 <math>Q</math> is the configuration manifold then the Lagrangian <math>L</math> is defined on the tangent bundle <math>TQ</math> , and the Hamiltonian is defined on the cotangent bundle <math>T^* Q</math>—the fiber derivative is a map <math>\mathbb{F}L:TQ \rightarrow T^* Q</math> such that

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

where <math>v</math> and <math>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

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