Hamilton's principle/Mathematics/The Hamilton-Jacobi equation

Plain-language version: Waves and rays

The Hamilton-Jacobi equation is a single first-order partial differential equation whose complete solution integrates the entire mechanical problem. It completes the sequence $n$ second-order equations, then $2n$ first-order equations, then one partial differential equation — and it is the point at which mechanics acquires a wave description, which is why this page is the ancestor of the Schrödinger equation.

The idea

Wavefronts and the rays orthogonal to them.

Hamilton's equations are easy in one case: if $H\equiv 0$, then every coordinate and momentum is constant and the motion is trivial. So look for a change of variables, preserving the canonical form (7) of the Hamiltonian and Hamilton's equations, that transforms the system into that case. All the difficulty moves into finding the transformation.

Such a change is a canonical transformation, and one way to generate one is by a function $S(t,q,P)$ of the old coordinates and the new momenta. The generating relations are

$p_i=\frac{\partial S}{\partial q_i},Q_i=\frac{\partial S}{\partial P_i},K=H+\frac{\partial S}{\partial t}\mathrm{(}\mathrm{1}\mathrm{)}$

with $K$ the new Hamiltonian. Demanding $K=0$ and substituting the first relation into the third gives the equation.

Statement

From n second-order equations to one partial differential equation.

$\frac{\partial S}{\partial t}+H(t,q_1,\dots ,q_n,\frac{\partial S}{\partial q_1},\dots ,\frac{\partial S}{\partial q_n})=0\mathrm{(}\mathrm{2}\mathrm{)}$

This is the Hamilton-Jacobi equation for the function $S(t,q_1,\dots ,q_n)$. It is first order in all $n+1$ variables and generally nonlinear, since $H$ is typically quadratic in the momenta.

A complete integral is a solution containing $n$ non-additive constants ${\alpha }_1,\dots ,{\alpha }_n$. Given one, the motion follows by differentiation alone: setting the new momenta to the ${\alpha }_i$, the second relation of (1) gives

$\frac{\partial S}{\partial {\alpha }_i}={\beta }_i=\text{const}\mathrm{(}\mathrm{3}\mathrm{)}$

which are $n$ algebraic relations determining the $q_i$ as functions of $t$ and the $2n$ constants $\alpha ,\beta$. No integration is required at this stage. That is Jacobi's theorem, and it is the payoff: the entire integration of the mechanical problem is traded for one partial differential equation.

Separation of variables

The trade is worth making only when (2) can be solved, and the practical route is separation. If $H$ carries no explicit $t$, put

$S(t,q)=W(q)-Et\mathrm{(}\mathrm{4}\mathrm{)}$

and (2) reduces to the time-independent form

$H(q,\frac{\partial W}{\partial q})=E\mathrm{(}\mathrm{5}\mathrm{)}$

$W$ is Hamilton's characteristic function, and it is the abbreviated action treated on Abbreviated action: Maupertuis versus Hamilton. Each cyclic coordinate separates further, contributing a term ${\alpha }_k{q}_k$ to $W$. A system that separates completely is integrable, and the Kepler problem on Central force and the Kepler problem is the classical instance.

Whether a given system separates depends on the coordinates chosen, and no general procedure finds them. Most systems separate in no coordinates at all.

The optical analogy

Equation (5) with $H=p^2/2m+V$ reads

$|\mathrm{\nabla }W{|}^2=2m(E-V(q))\mathrm{(}\mathrm{6}\mathrm{)}$

Compare the eikonal equation of geometrical optics, $|\mathrm{\nabla }\Psi {|}^2=n^2(q)$, with $\Psi$ the optical path and $n$ the refractive index. The two are the same equation with

$n(q)⟷\sqrt{2m(E-V(q))}\mathrm{(}\mathrm{7}\mathrm{)}$

So surfaces of constant $W$ are wavefronts and trajectories are the rays orthogonal to them. Mechanics is geometrical optics for a medium of refractive index (7).

Hamilton saw this before he wrote the mechanics. He notes that the dynamical principle is only another form of the idea already applied to optics in his theory of systems of rays. The optical papers came first and the mechanics is their transcription. This is developed on the optical-mechanical analogy.

The consequence that matters. Geometrical optics is the short-wavelength limit of wave optics. If mechanics is geometrical optics, then classical mechanics should be the short-wavelength limit of some wave mechanics, and equation (2) should be the limit of a wave equation. Substituting $\psi =A\exp (iS/\hslash )$ into the Schrödinger equation and keeping the leading order in $\hslash$ returns exactly (2). Schrödinger constructed the wave equation by running that argument backwards.

The gap between Hamilton in 1834 and Schrödinger in 1926 was ninety-two years, and the mathematics was in place the whole time. What was missing was any reason to believe the wavelength was not zero.

Conjugate points return

$S$ is not globally single-valued. Where extremals through a common starting point refocus, several trajectories reach the same $(t,q)$, $S$ becomes multivalued, and the derivatives in (1) fail. Those are precisely the conjugate points of the Jacobi condition and conjugate points, seen from the wave side: they are the caustics of the ray family, where the wavefront folds.

The same singularities appear in every setting the analogy touches: a focus in optics, a caustic in the ray picture, a conjugate point in the variational theory, and the location of the phase correction in the path integral on the Feynman path integral. One phenomenon, four names.

Hamilton's pair

Hamilton derived two partial differential equations rather than one, a final and an initial equation, because his characteristic function depends on both endpoints. In his notation they are

$\frac{1}{2}\Sigma \frac{1}{m}\{{\left(\frac{\delta V}{\delta x}\right)}^2+{\left(\frac{\delta V}{\delta y}\right)}^2+{\left(\frac{\delta V}{\delta z}\right)}^2\}=U+H\mathrm{(}\mathrm{8}\mathrm{)}$

and the same expression in the initial coordinates equal to $U_0+H$.

Equation (8) is (5) with $V$ for $W$ and $U+H$ for $E$, allowing for his sign convention on the force-function. The pairing is the subject of the next page.

Next

Action as a function of its endpoints takes Hamilton's own route: define the action on the actual motion, as a function of where it starts and ends, and read the mechanics off its derivatives.