Legendre Transformation

Idea#

The Legendre transformation of a (convex) real-valued function \(f\) on one of its variable, say, \(x\), is a way to express the family of tangent lines of the graph of \(f\) by the conjugate variable of \(x\).

A simplest example should demonstrate the idea. On \(x_0\), the tangent to \(f(x_0)\) is given by the line \(y=f'(x_0)x + b\) where \(f(x_0) = f'(x_0) x_0 + b\) with \(b = f(0)\). Now assume that \(f'\) is invertible (which is the case when \(f\) is convex), then \(x_0 = f'^{-1}(p)\), with \(p = f'(x_0)\). Hence we have that \(b=f(f'^{-1}(p)) - p f'^{-1}(p) \equiv - f^\star(p)\). We call \(f^\star\) the Legendre transformation of \(f\).

The family of tangent lines of the graph of \(f\) is parametrized by the slope \(p\), given by \(y = px - f^\ast(p)\). It is the solution to the equation \(F(x,y,p) = y + f^\star (p) - px = 0\).

Generalization to Manifolds#

The Legendre transformation of a smooth function \(L:TM\to\mathbb{R}\) from the tangent bundle \(\pi: TM\to M\) of a manifold \(M\) is the smooth morphism \(\mathbf{F}L: TM\to T^\ast M\) from the tangent bundle to the cotangent bundle given by \(\mathbf{F}L(v) = d(L|_{TM_x})_v\) where \(x = \pi(v)\).

Locally this is just saying that \(\mathbf{F}L(x;v) =(x;\partial L / \partial v) = (x;p)\).

Involutiveness#

The Legendre transformation is involutive. The graph of the original function is the envelop of this family of tangent lines. Demanding \(\partial F/\partial p = 0\), eliminating \(p\) from the equation \(F(x,y,p) =0\) and \(\partial F/\partial p =0\), we have

\[y = x\cdot {{f^\star}'}^{-1}(x)-f^\star({{f^\star}'}^{-1}(x)).\]

Identifying \(y\) with \(f(x)\) and observe that the RHS is the Legendre transformation of \(f^\star\). \(f = f^{\star\star}\).