Editing Legendre transformation (section)

===Analytical mechanics===
A Legendre transform is used in [[classical mechanics]] to derive the [[Hamiltonian mechanics|Hamiltonian formulation]] from the [[Lagrangian mechanics|Lagrangian formulation]], and conversely. A typical Lagrangian has the form

<math display="block">L(v,q)=\tfrac{1}2\langle v,Mv\rangle-V(q),</math>
where <math>(v,q)</math> are coordinates on {{math|'''R'''<sup>''n''</sup> × '''R'''<sup>''n''</sup>}}, {{mvar|M}} is a positive definite real matrix, and
<math display="block">\langle x,y\rangle = \sum_j x_j y_j.</math>

For every {{mvar|q}} fixed, <math>L(v, q)</math> is a convex function of <math>v</math>, while <math>V(q)</math> plays the role of a constant.

Hence the Legendre transform of <math>L(v, q)</math> as a function of <math>v</math> is the Hamiltonian function,
<math display="block">H(p,q)=\tfrac {1}{2} \langle p,M^{-1}p\rangle+V(q).</math>

In a more general setting, <math>(v, q)</math> are local coordinates on the [[tangent bundle]] <math>T\mathcal M</math> of a manifold <math>\mathcal M</math>. For each {{mvar|q}}, <math>L(v, q)</math> is a convex function of the tangent space {{math|''V<sub>q</sub>''}}. The Legendre transform gives the Hamiltonian <math>H(p, q)</math> as a function of the coordinates {{math|(''p'', ''q'')}} of the [[cotangent bundle]] <math>T^*\mathcal M</math>; the inner product used to define the Legendre transform is inherited from the pertinent canonical [[symplectic vector space|symplectic structure]]. In this abstract setting, the Legendre transformation corresponds to the [[tautological one-form]].{{Explain|date=April 2023}}