Legendre transform

This post has been migrated from my old blog, the math-physics learning seminar.

Yesterday, I gave an introductory talk on Hamiltonian mechanics and symplectic geometry. The starting point is the Legendre transform. First, begin with a configuration space (Q). The Lagrangian (\mathcal{L}) is a smooth function on (TQ). In local coordinates (q^i) on (Q), we have coordinates ((q_i, v_i)) on (TQ), where the (v^i) are the components of the tangent vector

(v = v_i \partial_i \in T_q Q). Typically, the Lagrangian will be of the form

[ \mathcal{L}(q,v) = \frac{1}{2} g(v,v) - V(q), ]

where (g) is some metric on (Q). Now we introduce new coordinates (p_i) defined by

[ p_i = \frac{\partial \mathcal{L}}{\partial v^i}. ]

If (\mathcal{L}) is (strictly?) convex in (v) then we can solve for (v^i) as a function of ((q^i, p_j)). It is easy to check that the (p_i) transform as covectors, and so this gives a diffeomorphism (TQ \to T^\ast Q)(which depends on (\mathcal{L})). For example, in the above Lagrangian,

[ \frac{\partial \mathcal{L}}{\partial v} = g(v, -), ]

which is just the dual of (v) with respect to the metric (g). So for Lagrangians of this form, the map (TQ \to T^\ast Q) is just the one given by the metric.

Now comes the interesting part. There is a natural way to turn (\mathcal{L}), which is a function on (TQ), into a function (H) on (T^\ast Q), in such a way that if we repeat this process, we will get back the original function (\mathcal{L}) on (TQ). This is the Legendre transform:

[ \mathcal{H} = pv - L. ]

Now suppose we have a curve (q(t), \dot{q}(t) \in TQ) that satisfies the Euler-Lagrange equations. Then by the identification (TQ = T^\ast Q), this gives a curve ((q(t), p(t)) \in T^\ast Q). What equation does it satisfy? We have

[ \frac{d}{dt} p = \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial v} = \frac{\partial \mathcal{L}}{\partial q} = -\frac{\partial H}{\partial q}, ]

and

[ \frac{d}{dt}q = v = \frac{\partial H}{\partial p}. ]

These are Hamilton’s equations, and they say that the curve (\gamma = (q(t), p(t)) \in T^\ast Q) is just an integral curve of the symplectic gradient of (H)! So classical mechanics is really just about flows of Hamiltonian vector fields on symplectic manifolds.