The Hamilton–Jacobi Equation and Lagrangian Submanifolds

In Hamiltonian mechanics, a trajectory is a curve in phase space. In Hamilton–Jacobi theory, we seek to describe an entire family of trajectories at once. The key is to find a function $S(q, t)$ satisfying

$$\frac{\partial S}{\partial t} + H\!\left(q, \frac{\partial S}{\partial q}, t\right) = 0 .$$

This equation turns the dynamical problem into a first-order nonlinear partial differential equation, but its deeper meaning comes from symplectic geometry.

1. The Action as a Generating Function

Let $S(q, t)$ be the action evaluated along a classical trajectory. When the endpoint varies, the differential of the action is

$$\mathrm{d} S = p_i \, \mathrm{d} q^i - H \, \mathrm{d} t .$$

Thus $p_i = \partial S / \partial q^i$, while $\partial S / \partial t = -H$. The Hamilton–Jacobi equation is precisely the compatibility condition for these two relations.

2. Lagrangian Submanifolds

The graph of the differential of a function $S(q)$,

$$L_S = \left\{\left(q, p\right) \in T^*Q \;\middle|\; p_i = \frac{\partial S}{\partial q^i}\right\},$$

is a Lagrangian submanifold of $T^*Q$. Solving the Hamilton–Jacobi equation can therefore be understood as finding Lagrangian submanifolds with the appropriate evolution under the Hamiltonian flow.

3. From Trajectories to Families of Trajectories

The usual Hamilton equations determine a single trajectory from initial data, whereas a complete integral $S(q, \alpha, t)$ encodes a family of trajectories labeled by parameters $\alpha$. Geometric language reinterprets “integrating the equations of motion” as “constructing an appropriate Lagrangian foliation.”

This perspective also explains why Hamilton–Jacobi theory connects naturally to geometrical optics, the WKB approximation, and semiclassical quantum mechanics: all of them study the Lagrangian geometry defined by a phase function.