Intuitive

Concrete

For a very nice explanation of the origin of the Hamilton-Jacobi equation and its meaning see chapter 2 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan

The Hamilton-Jacobi equation is essentially a dispersion relation for a complex wave. This is easy to see in the context of non-relativistic quantum mechanics. If a quantum amplitude is expressed in the form ψ = Rexp(iS/h¯), then the Hamilton-Jacobi equation relates p = ∂S/∂q to E = −∂S/∂t by the condition p2(q) = 2m(E −V). This is a relation between the wave vector k = p/h¯ and the frequency ω = E/h¯ of the “matter wave” associated with the particle.chapter 2 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan

Take note that while the Hamilton equation for time-evolution is closely related to the Heisenberg equation in quantum mechanics, the Hamilton-Jacobi equation is closely connected to the Schrödinger equation. See, for example, Section 4.8.1 in Tong's lecture notes.

Abstract

We consider a manifold $Q$ and a Lagrangian $L \colon TQ\rightarrow\mathbb{R}$. Now, we define Hamilton's principal function \[ W \colon Q\times\mathbb{R}\times Q\times\mathbb{R}\longrightarrow\mathbb{R} \] by \[ W(q_0,t_0; q_1, t_1) = \inf_{q\in\Upsilon} S(q) \] where \[ \Upsilon=\bigl\{ q \colon [t_0,t_1]\rightarrow Q,\, q(t_0)=q_0,\text{ \& }q(t_1)=q_1 \bigr\} \] and \[ S(q) = \int_{t_0}^{t_1} L\Bigl(q(t),\dot{q}(t)\Bigr)\,dt . \] Here $W$ is the least action for some path from $(q_0,t_0)$ to $(q_1,t_1)$. We have \[ \frac{\partial}{\partial q_1^i}W(q_0,q_1) = (p_1)_i, \]

where $p_1$ is the momentum of the particle going from $q_0$ to $q_1$, at time $t_1$, and $$ \frac{\partial W}{\partial q_0^i}= -(p_0)_i ,\qquad\text{(-momentum at time $t_0$)} $$ $$ \frac{\partial W}{\partial t_1}= -H_1,\qquad\text{(-energy at time $t_1$)} $$ $$ \frac{\partial W}{\partial t_0}= H_0,\qquad\text{(+energy at time $t_0$)} $$

Take note that $H_1=H_0$ since energy is conserved. These last four equations are the Hamilton–Jacobi equations.

Why is it interesting?