Editing Legendre transformation (section)

==Legendre transformation on manifolds==

Let <math display="inline">M</math> be a [[smooth manifold]], let <math>E</math> and <math display="inline">\pi : E\to M</math> be a [[vector bundle]] on <math>M</math> and its associated [[bundle projection]], respectively. Let <math display="inline">L : E\to \R</math> be a smooth function. We think of <math display="inline">L</math> as a [[Lagrangian mechanics|Lagrangian]] by analogy with the classical case where <math display="inline">M = \R</math>, <math display="inline">E = TM = \Reals \times \Reals </math> and <math display="inline">L(x,v) = \frac 1 2 m v^2 - V(x)</math> for some positive number <math display="inline">m\in \Reals</math> and function <math display="inline">V : M \to \Reals</math>.

As usual, the [[dual bundle|dual]] of <math display="inline">E</math> is denote by <math display="inline">E^*</math>. The fiber of <math display="inline">\pi</math> over <math display="inline">x\in M</math> is denoted <math display="inline">E_x</math>, and the restriction of <math display="inline">L</math> to <math display="inline">E_x</math> is denoted by <math display="inline">L|_{E_x} : E_x\to \R</math>. The ''Legendre transformation'' of <math display="inline">L</math> is the smooth morphism<math display="block">\mathbf F L : E \to E^*</math> defined by <math display="inline">\mathbf FL(v) = d(L|_{E_x})_v \in E_x^*</math>, where <math display="inline">x = \pi(v)</math>. Here we use the fact that since <math display="inline">E_x</math> is a vector space, <math display="inline">T_v(E_x)</math> can be identified with <math display="inline">E_x</math>.
In other words, <math display="inline">\mathbf FL(v)\in E_x^*</math> is the covector that sends <math display="inline">w\in E_x</math> to the directional derivative <math display="inline">\left.\frac d {dt}\right|_{t=0} L(v + tw)\in \R</math>.

To describe the Legendre transformation locally, let <math display="inline">U\subseteq M</math> be a coordinate chart over which <math display="inline">E</math> is trivial. Picking a trivialization of <math display="inline">E</math> over <math display="inline">U</math>, we obtain charts <math display="inline">E_U \cong U \times \R^r</math> and <math display="inline">E_U^* \cong U \times \R^r</math>. In terms of these charts, we have <math display="inline">\mathbf FL(x; v_1, \dotsc, v_r) = (x; p_1,\dotsc, p_r)</math>, where <math display="block">p_i = \frac {\partial L}{\partial v_i}(x; v_1, \dotsc, v_r)</math> for all <math display="inline">i = 1, \dots, r</math>. If, as in the classical case, the restriction of <math display="inline">L : E\to \mathbb R</math> to each fiber <math display="inline">E_x</math> is strictly convex and bounded below by a positive definite quadratic form minus a constant, then the Legendre transform <math display="inline">\mathbf FL : E\to E^*</math> is a diffeomorphism.<ref name="CdS2008">Ana Cannas da Silva. ''Lectures on Symplectic Geometry'', Corrected 2nd printing. Springer-Verlag, 2008. pp. 147-148. {{ISBN|978-3-540-42195-5}}.</ref> Suppose that <math display="inline">\mathbf FL</math> is a diffeomorphism and let <math display="inline">H : E^* \to \R</math> be the "[[Hamiltonian mechanics|Hamiltonian]]" function defined by <math display="block">H(p) = p \cdot v - L(v),</math> where <math display="inline">v = (\mathbf FL)^{-1}(p)</math>. Using the natural isomorphism <math display="inline">E\cong E^{**}</math>, we may view the Legendre transformation of <math display="inline">H</math> as a map <math display="inline">\mathbf FH : E^* \to E</math>. Then we have<ref name="CdS2008"/> <math display="block">(\mathbf FL)^{-1} = \mathbf FH.</math>