Hamilton-Jacobi Theory
Hamilton-Jacobi Theory
The Hamilton-Jacobi equation represents the ultimate expression of the canonical transformation programme: find a CT that makes all new coordinates cyclic, so that the new Hamiltonian $K=0$ (or a constant). If the generating function $S(q,\alpha,t)$ (called Hamilton's principal function) satisfies $$H\!\left(q_1,\ldots,q_n,\frac{\partial S}{\partial q_1},\ldots,\frac{\partial S}{\partial q_n},t\right)+\frac{\partial S}{\partial t}=0,$$ then the canonical transformation generated by $S$ reduces the motion to quadrature. The $n$ new momenta $\alpha_i$ are constants of integration, and the new coordinates $\beta_i=\partial S/\partial\alpha_i$ are also constant. The motion is then recovered by inverting these algebraic relations.
For time-independent Hamiltonians, $S=W(q,\alpha)-Et$ where $W$ (Hamilton's characteristic function) satisfies the time-independent Hamilton-Jacobi equation: $H(q,\partial W/\partial q)=E$. This is a first-order PDE for $W$; its complete integral (containing $n$ constants $\alpha_i$ with $\alpha_1=E$) encodes the complete solution of the mechanical problem. The HJ equation is the classical limit of the quantum-mechanical Schrödinger equation: writing $\psi=e^{iS/\hbar}$ and taking $\hbar\to 0$ recovers the HJ equation from the Schrödinger equation,
revealing that classical trajectories are the rays of a quantum wave.When the HJ equation is separable — when $S$ factors as $S=\sum_i S_i(q_i,\alpha)$ — each $S_i$ satisfies an ODE. Separability is the hallmark of an integrable system. The Kepler problem, the isotropic harmonic oscillator, and the motion on an ellipsoid are all separable in appropriate coordinates. The separation constants become the action variables of the system.
Definition — Hamilton's Principal Function
Hamilton's principal function $S(q,\alpha,t)$ is a type-$F_2$ generating function (with $\alpha_i$ as the new constant momenta) that satisfies the HJ equation. The solution of the mechanical problem follows from $p_i=\partial S/\partial q_i$ and $\beta_i=\partial S/\partial\alpha_i=\text{const}$.
Theorem — Hamilton-Jacobi Equation
The HJ equation is the PDE $\partial S/\partial t + H(q,\partial S/\partial q,t)=0$. A complete integral $S(q,\alpha_1,\ldots,\alpha_n,t)$ containing $n$ non-trivial constants $\alpha_i$ provides the complete solution of Hamilton's equations via $p_i=\partial S/\partial q_i$ and $\beta_i=\partial S/\partial\alpha_i=\text{const}$.
Example 1: Free Particle
$H=p^2/(2m)$. The HJ equation is $\frac{1}{2m}(\partial S/\partial x)^2+\partial S/\partial t=0$. Try $S=\alpha x - \alpha^2 t/(2m)$. Then $p=\partial S/\partial x=\alpha=\text{const}$ (momentum conserved), and $\beta=\partial S/\partial\alpha=x-\alpha t/m$, giving $x=\beta+pt/m$. This is Newton's first law: $x(t)=x_0+v_0 t$, recovered elegantly from the HJ equation.
Example 2: Harmonic Oscillator via HJ
$H=p^2/(2m)+m\omega^2 q^2/2=E$. The time-independent HJ equation: $\frac{1}{2m}(W')^2+\frac{m\omega^2 q^2}{2}=E$. Solving: $W=\int\sqrt{2mE-m^2\omega^2 q^2}\,dq$. The action variable $J=\oint p\,dq=\oint W'\,dq=\pi\sqrt{2mE}\cdot\sqrt{2E/(m\omega^2)}/1=2\pi E/\omega$, giving $E=\omega J/2\pi$. The angle variable advances as $\theta=\omega t$, recovering simple harmonic motion.
Example 3: Kepler Problem Separation
In polar coordinates, $H=\frac{1}{2m}(p_r^2+p_\theta^2/r^2)-k/r=E$. Separation: $S=S_r(r)+S_\theta(\theta)-Et$. $\theta$ is cyclic: $S_\theta=l\theta$ (angular momentum). Then $S_r=\int\sqrt{2mE+2mk/r-l^2/r^2}\,dr$. The orbit follows from $\partial S/\partial l=\text{const}$, yielding the conic sections of Kepler's laws. The radial period $T=2\pi mk/\sqrt{-2mE^3}\cdot\frac{1}{l}\cdot l=2\pi m k(-2mE)^{-3/2}$ gives Kepler's third law $T^2\propto a^3$.
Example 4: WKB Approximation & Quantum Connection
Writing the quantum wave function $\psi=A e^{iS/\hbar}$ and substituting into the Schrödinger equation $-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=i\hbar\dot{\psi}$: to leading order in $\hbar$, the imaginary part gives the HJ equation for $S$, while the real part gives the continuity equation $\partial|A|^2/\partial t+\nabla\cdot(|A|^2\nabla S/m)=0$. The WKB approximation uses this to compute quantum tunnelling rates, energy levels ($\oint p\,dq=(n+1/2)h$), and semiclassical propagators — all rooted in the classical HJ theory.
Practice Problems
- Solve the HJ equation for a particle falling under gravity: $H=p^2/(2m)+mgq$. Find $S(q,\alpha,t)$ and recover $q(t)$ from the relation $\beta=\partial S/\partial\alpha$.
- Show that for a separable Hamiltonian $H=\sum_i H_i(q_i,p_i)$, the HJ equation separates into independent ODEs for each $S_i(q_i)$, and the separation constants are the action variables.
- Use the time-independent HJ equation to find the orbit of a particle in the potential $V(r)=-k/r+l^2/(2mr^2)$ (effective 1D problem). Show the orbit closes only for $V=-k/r$.
- The action variable is $J=\frac{1}{2\pi}\oint p\,dq$. Compute $J$ for a particle of energy $E$ in the box $V=0$ for $0
- Explain in physical terms why the HJ equation is the classical limit of the Schrödinger equation. What plays the role of wavelength in classical mechanics?
Show Answer Key
1. $H = p^2/(2m)+mgq$. HJ: $\frac{\partial S}{\partial t}+\frac{1}{2m}(\frac{\partial S}{\partial q})^2+mgq=0$. Try $S = W(q)-\alpha t$: $\frac{1}{2m}(W')^2+mgq=\alpha$. $W'=\sqrt{2m(\alpha-mgq)}$, $W=\int\sqrt{2m(\alpha-mgq)}\,dq = -\frac{(2m(\alpha-mgq))^{3/2}}{3m^2 g}$. $\beta = \partial S/\partial\alpha = t-\frac{\sqrt{2m(\alpha-mgq)}}{mg}$. Solving: $q(t) = \alpha/(mg)-\frac{g}{2}(t-\beta)^2$. ✓
2. If $H = \sum H_i(q_i,p_i)$, try $S = \sum S_i(q_i) - Et$. Then $\sum H_i(q_i,S_i') = E$. Since each term depends on a different $q_i$, each must equal a constant $\alpha_i$ with $\sum\alpha_i = E$. The action variables are $J_i = \frac{1}{2\pi}\oint \frac{\partial S_i}{\partial q_i}dq_i$, and $\alpha_i = \alpha_i(J_i)$.
3. Effective 1D: $H_{\text{eff}} = \frac{p_r^2}{2m}+\frac{l^2}{2mr^2}-\frac{k}{r}$. HJ gives $p_r = \sqrt{2m(E+k/r)-l^2/r^2}$. The orbit integral $\phi = \int\frac{l/r^2}{\sqrt{2m(E+k/r)-l^2/r^2}}dr$ gives $r(\phi) = \frac{l^2/(mk)}{1+e\cos\phi}$ (conic section). The orbit closes because the $r$ and $\phi$ periods are commensurate (Bertrand's theorem: only $1/r$ and $r^2$ potentials give closed orbits).
4. In the box: $p = \sqrt{2mE}$ (constant magnitude). $J = \frac{1}{2\pi}\oint p\,dq = \frac{1}{2\pi}\cdot 2a\cdot\sqrt{2mE} = \frac{a\sqrt{2mE}}{\pi}$. Solving: $E = \frac{J^2\pi^2}{2ma^2}$. Quantum: $E_n = \frac{n^2\pi^2\hbar^2}{2ma^2}$. Correspondence: $J = n\hbar$ (Bohr-Sommerfeld quantization). ✓
5. In the limit $\hbar\to 0$, the Schrödinger equation $i\hbar\partial\psi/\partial t = H\psi$ with $\psi = A e^{iS/\hbar}$ reduces to the HJ equation $\partial S/\partial t + H(q,\nabla S) = 0$ (the eikonal/WKB limit). The phase $S/\hbar$ plays the role of action. The wavelength $\lambda = 2\pi\hbar/p$ (de Broglie) → 0 in the classical limit, analogous to geometric optics as the short-wavelength limit of wave optics.