Hamilton–Jacobi Theory

Hamilton–Jacobi theory rewrites Hamiltonian mechanics as a first-order nonlinear partial differential equation for an action function. The central object is Hamilton’s principal function $S$, whose gradients give momenta and whose equation is

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

This is classical mechanics in action-phase form. It is the mathematical bridge from Hamiltonian trajectories to WKB phases, semiclassical propagators, stationary phase, and path-integral saddle points.

Helpful background. Hamiltonian Mechanics Review introduces $H$, and Canonical Transformations explains generating functions.

Action functions and wavefronts

Instead of solving for one trajectory at a time, Hamilton–Jacobi theory describes a family of trajectories through an action function. Its derivatives give momenta. In WKB theory the same function sets the wavefronts, while a separate transport equation determines their amplitude.

Hamilton’s Principal Function

Fix an initial endpoint $(q_a,t_a)$ and a final endpoint $(q_b,t_b)$. Suppose a classical trajectory $\gamma$ connects them. Hamilton’s principal function is the classical action evaluated on that trajectory:

$$S(q_b,t_b;q_a,t_a) = \int_{t_a}^{t_b} L(q_\gamma,\dot q_\gamma,t)\,dt.$$

When there is more than one classical trajectory connecting the endpoints, $S$ is branch-dependent. Semiclassical formulas then sum over branches.

The endpoint derivatives generate the endpoint momenta:

$$p_{b,i} = \frac{\partial S}{\partial q_b^i}, \qquad p_{a,i} = - \frac{\partial S}{\partial q_a^i}.$$

The final-time derivative gives the final Hamiltonian:

$$\frac{\partial S}{\partial t_b} = - H(q_b,p_b,t_b).$$

These identities are why the same action function appears in canonical transformations, propagator phases, and stability determinants.

The Hamilton–Jacobi Equation

Suppress the fixed initial endpoint and write the principal function as $S(q,t)$. Since

$$p_i = \frac{\partial S}{\partial q^i},$$

and

$$\frac{\partial S}{\partial t} = - H(q,p,t),$$

one obtains the Hamilton–Jacobi equation:

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

For a particle of mass $m$ in a potential $V(q,t)$,

$$H(q,p,t) = \frac{p^2}{2m}+V(q,t),$$

so the equation becomes

$$\frac{\partial S}{\partial t} + \frac{1}{2m} \left( \nabla S \right)^2 + V(q,t) = 0.$$

This equation is first order in time but nonlinear in $S$. Its characteristics are the classical Hamiltonian trajectories.

Time-Independent Systems

If the Hamiltonian has no explicit time dependence, one often separates

$$S(q,t) = W(q;E)-Et.$$

The function $W$ is Hamilton’s characteristic function. Substituting into the Hamilton–Jacobi equation gives

$$H\left( q, \frac{\partial W}{\partial q} \right) = E.$$

For one-dimensional motion with

$$H(x,p) = \frac{p^2}{2m}+V(x),$$

the characteristic equation is

$$\frac{1}{2m} \left( \frac{dW}{dx} \right)^2 + V(x) = E.$$

Thus

$$\frac{dW}{dx} = \pm \sqrt{2m(E-V(x))} = \pm p(x),$$

and

$$W_\pm(x;E) = \pm \int^x p(x';E)\,dx'.$$

This is exactly the action integral that appears in leading one-dimensional WKB wavefunctions.

Free Particle Example

For a free particle in one dimension, let

$$T=t-t_0, \qquad S(x,t;x_0,t_0) = \frac{m(x-x_0)^2}{2T}.$$

The final momentum is

$$\frac{\partial S}{\partial x} = \frac{m(x-x_0)}{T} = p.$$

The time derivative is

$$\frac{\partial S}{\partial t} = - \frac{m(x-x_0)^2}{2T^2} = - \frac{p^2}{2m}.$$

Therefore

$$\frac{\partial S}{\partial t} + \frac{1}{2m} \left( \frac{\partial S}{\partial x} \right)^2 = 0,$$

which is the free-particle Hamilton–Jacobi equation.

This same $S$ is the phase in the exact free-particle propagator:

$$K_{\mathrm{free}}(x,t;x_0,t_0) \propto \exp\left[ \frac{i}{\hbar} \frac{m(x-x_0)^2}{2(t-t_0)} \right].$$

Complete Integrals and Canonical Transformations

A solution $S(q,\alpha,t)$ depending on $n$ constants $\alpha_i$ is a complete integral when $\det(\partial^2S/\partial q^i\partial\alpha_j)\ne0$ locally. This condition lets us solve the momentum equations for the parameters. It can be used as a generating function for a canonical transformation:

$$p_i = \frac{\partial S}{\partial q^i}, \qquad \beta_i = \frac{\partial S}{\partial \alpha_i}.$$

Here $\beta_i$ are the new coordinates and $\alpha_i$ are their conjugate momenta. The Hamilton–Jacobi equation makes the transformed Hamiltonian $K=H+\partial_tS$ vanish, so both sets are constants of motion. This is the classical reason Hamilton–Jacobi theory is sometimes described as a method for integrating Hamilton’s equations.

For quantum mechanics, the more important lesson is structural: action functions can generate canonical maps, and their derivatives encode classical momenta and stability. Semiclassical formulas use exactly this derivative information.

WKB Leading Phase

In stationary WKB, one writes

$$\psi(q) = A(q) \exp\left( \frac{i}{\hbar}W(q) \right).$$

For the Schrödinger equation

$$\left[ - \frac{\hbar^2}{2m}\nabla^2 + V(q) \right]\psi = E\psi,$$

the leading order in $\hbar$ gives

$$\frac{1}{2m} \left( \nabla W \right)^2 + V(q) = E.$$

This is the time-independent Hamilton–Jacobi equation. The next order gives a transport equation for $A(q)$, which supplies the WKB amplitude.

Thus the WKB slogan “phase equals action divided by $\hbar$” has a precise meaning: the leading phase function solves the classical Hamilton–Jacobi equation.

Semiclassical Propagator Bridge

The semiclassical propagator has the schematic form

$$K_{\mathrm{sc}}(q_b,t_b;q_a,t_a) \sim \sum_\gamma A_\gamma \exp\left( \frac{i}{\hbar} S_\gamma(q_b,t_b;q_a,t_a) \right),$$

where $S_\gamma$ is Hamilton’s principal function for the classical trajectory branch $\gamma$.

The second derivatives of $S_\gamma$ determine how nearby trajectories focus or spread. In the Van Vleck determinant, one encounters

$$D_\gamma = \det\left( - \frac{\partial^2S_\gamma} {\partial q_b\,\partial q_a} \right).$$

The phase comes from Hamilton–Jacobi theory; the prefactor comes from the quadratic fluctuation and stability information around the classical path.

Eikonal Viewpoint

The Hamilton–Jacobi equation is also an eikonal equation: it determines surfaces of constant action. The momentum

$$p_i = \frac{\partial S}{\partial q^i}$$

annihilates every tangent displacement within an action surface: $p_i\,\delta q^i=\delta S=0$. In Euclidean coordinates a metric identifies this covector with the normal vector $\nabla S$, just as a wave vector is normal to a wavefront.

This analogy is useful but limited. Quantum WKB wavefronts also carry amplitudes, phases, boundary conditions, caustic repairs, and connection formulas. The classical eikonal equation supplies only the leading phase. Multidimensional WKB Preview derives the associated ray-tube transport law and shows where the configuration-space projection becomes singular.

Common Mistakes

• Treating $S(q,t)$ as an arbitrary phase rather than a solution of a nonlinear PDE.
• Forgetting that $S$ can be multivalued when several classical trajectories reach the same endpoint.
• Confusing Hamilton’s principal function $S(q,t)$ with Hamilton’s characteristic function $W(q;E)$.
• Dropping the endpoint signs $p_b=\partial S/\partial q_b$ and $p_a=-\partial S/\partial q_a$.
• Assuming the leading Hamilton–Jacobi phase includes the WKB amplitude.
• Using the Hamilton–Jacobi equation at caustics without changing representation or using a uniform approximation.
• Interpreting a semiclassical sum over branches as a classical probability mixture.

Exercises

1. Verify the Hamilton–Jacobi equation for the free-particle principal function

$$S(x,t;x_0,t_0) = \frac{m(x-x_0)^2}{2(t-t_0)}.$$

Solution

Let $T=t-t_0$. Then

$$\frac{\partial S}{\partial x} = \frac{m(x-x_0)}{T},$$

and

$$\frac{\partial S}{\partial t} = - \frac{m(x-x_0)^2}{2T^2}.$$

Therefore

$$\frac{\partial S}{\partial t} + \frac{1}{2m} \left( \frac{\partial S}{\partial x} \right)^2 = - \frac{m(x-x_0)^2}{2T^2} + \frac{m(x-x_0)^2}{2T^2} = 0.$$

2. For $H=p^2/(2m)+V(x)$, derive the time-independent Hamilton–Jacobi equation and solve for $dW/dx$.

Solution

Use $S(x,t)=W(x;E)-Et$. The Hamilton–Jacobi equation gives

$$-E + \frac{1}{2m} \left( \frac{dW}{dx} \right)^2 + V(x) = 0.$$

Thus

$$\left( \frac{dW}{dx} \right)^2 = 2m(E-V(x)),$$

so

$$\frac{dW}{dx} = \pm\sqrt{2m(E-V(x))}.$$

3. Explain why the WKB leading phase satisfies a Hamilton–Jacobi equation.

Solution

Insert

$$\psi(q) = A(q) \exp\left( \frac{i}{\hbar}W(q) \right)$$

into the stationary Schrödinger equation. The leading order in $\hbar$ comes from differentiating the exponential phase twice. It gives

$$\frac{1}{2m} \left( \nabla W \right)^2 + V(q) = E,$$

which is the time-independent Hamilton–Jacobi equation.

4. Why can Hamilton’s principal function be multivalued for fixed endpoints?

Solution

The fixed-endpoint boundary-value problem can have more than one classical trajectory. Each branch has its own classical action and endpoint momenta. Therefore the principal function is branch-dependent, and semiclassical propagators sum over the branches as amplitudes.

References

• V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315-397, 1972.
• M. Brack and R. K. Bhaduri, Semiclassical Physics, Westview Press, 2003.