The Legendre transform

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.