The Hamilton-Jacobi Equation as the Eikonal Limit: WKB and Geometric Optics

Intuition Beginner

Imagine standing at one side of a landscape and wanting to walk to the other side by the fastest route. The travel time depends on your path -- uphill is slower, downhill is faster. You could solve Newton's equations for your body and trace out the full trajectory. But there is a shortcut: compute the "shortest-time function" $T(\mathbf{x})$ that records the minimum travel time from your starting point to any position $\mathbf{x}$. This function encodes all the optimal routes implicitly -- the slope of $T$ at any point tells you which direction to walk.

The Hamilton-Jacobi equation applies this idea to all of mechanics. Instead of tracking a particle's position and momentum along a trajectory, you track a single function $S(q,t)$ called Hamilton's principal function (the action evaluated along the optimal path from a starting point to the configuration $q$ at time $t$). This function satisfies one equation -- the Hamilton-Jacobi equation -- and once you have a complete solution $S$, you can read off every possible trajectory without solving any further differential equations. Every initial condition corresponds to a different choice of integration constants in $S$.

The connection to optics is the deep part. Think of the level sets of $S$ -- the surfaces where $S$ has a fixed value -- as wavefronts. Particle trajectories are always perpendicular to these wavefronts, pointing in the direction where $S$ increases most steeply. This is exactly how light rays behave: light rays are perpendicular to surfaces of constant phase (wavefronts), and they bend when the medium changes. In a region of high refractive index, light slows down and the wavefronts crowd together; in a region of steep potential, the particle slows down and the $S$-wavefronts crowd together. The mathematics is the same.

In the early 19th century, Hamilton studied optics first and then realised that the same framework -- rays following the gradient of a "wavefront" function -- applies to mechanical trajectories following the gradient of the action. The eikonal equation of geometric optics (which governs how light rays bend through a medium with varying refractive index) is the same equation as the Hamilton-Jacobi equation, with the action playing the role of the optical path length. This optico-mechanical analogy is not a metaphor or a post-hoc observation. It is the historical route by which Hamilton discovered his equations, and it is the bridge that Schrodinger crossed a century later to discover the wave equation of quantum mechanics.

Light rays as particle trajectories

In geometric optics, light travels along rays. The rays are determined by the refractive index $n(\mathbf{x})$ of the medium. Fermat's principle states that light chooses the path that makes the optical path length stationary:

$$\delta \left(\text{optical path length}\right)=0.$$

The optical path length is the ordinary distance weighted by the refractive index -- light takes longer to traverse a given distance in a denser medium. The eikonal function $\varphi (\mathbf{x})$ records the optical path length from a source to the point $\mathbf{x}$. Its gradient gives the direction of the light ray:

$$\text{ray direction}=\text{gradient of }\varphi .$$

The eikonal equation itself is the constraint that the squared magnitude of this gradient equals $n^2$:

$${\left|\text{gradient of }\varphi \right|}^2=n(\mathbf{x}{)}^2.$$

Now replace "light" with "particle" and "optical path length" with "action." The particle's trajectory follows the gradient of the action function $W(q)$, and the constraint on $W$ is:

$${\left|\text{gradient of }W\right|}^2=2m(E-V(q)).$$

The right side is $2m(E-V)$, which plays the role of $n^2$. The correspondence is exact: $n^2\leftrightarrow 2m(E-V)$, optical path length $\leftrightarrow$ abbreviated action, ray equation $\leftrightarrow$ particle trajectory. A region where $V$ is large corresponds to a region of low refractive index -- the particle speeds up, just as light speeds up in a less dense medium. A turning point where $E=V$ corresponds to the edge of a shadow region where light cannot reach.

Visual Beginner

Figure: Wavefronts (level sets of the action $S$) and particle trajectories for a free particle. The wavefronts are parallel planes of constant $q-pt/m$, and the trajectories are straight lines perpendicular to them.

When the potential is zero, the wavefronts are flat and the trajectories are straight lines. When the potential varies, the wavefronts curve and the trajectories bend -- exactly like light bending through a graded-index lens. At a turning point (where $E=V$), the wavefronts pile up and the spacing between them goes to zero, just as light waves bunch up at the boundary of a shadow.

Figure: The optico-mechanical analogy as a two-column comparison. Left column: optics -- source, wavefronts, rays, refractive index n(x), eikonal phi, Fermat's principle. Right column: mechanics -- initial condition, level sets of S, trajectories, potential V(q), action W, Maupertuis' principle. Arrows connect corresponding entries.

Worked example Beginner

For a free particle in one dimension with Hamiltonian $H=p^2/(2m)$, the Hamilton-Jacobi equation says: the rate of change of $S$ with respect to time equals minus the Hamiltonian, with the momentum $p$ replaced by the slope of $S$ in $q$. That is:

$$\frac{1}{2m}{\left(\frac{dS}{dq}\right)}^2+\frac{dS}{dt}=0.$$

Try a solution of the form $S=W(q)-Et$ where $E$ is a constant (the energy). The time rate of change of $S$ is $-E$ and the slope in $q$ is $dW/dq$. Substituting:

$$\frac{1}{2m}{\left(\frac{dW}{dq}\right)}^2=E,\text{so}\frac{dW}{dq}=\pm \sqrt{2mE}.$$

The slope of $W$ is constant, so $W=\pm \sqrt{2mE}q+\text{const}$, giving $S=\pm \sqrt{2mE}q-Et+\alpha$ where $\alpha$ is an integration constant. The momentum is $p=dS/dq=\pm \sqrt{2mE}$, and the position follows from the relation $\beta =-dS/dE=t\mp q/\sqrt{2E/m}$, giving $q(t)=\pm \sqrt{2E/m}t+q_0$. The free-particle trajectory -- uniform motion at constant velocity -- is recovered without ever solving a differential equation. Differentiation of the action function did all the work.

The wavefronts of $S$ are the lines of constant $\pm \sqrt{2mE}q-Et$, which move to the right (or left) at constant speed $\sqrt{2E/m}$. The particle trajectory is perpendicular to these wavefronts -- a straight line at constant velocity, just as a light ray in a uniform medium is a straight line perpendicular to flat wavefronts.

Check your understanding Beginner

Formal definition Intermediate+

Let $H(q,p,t)$ be a Hamiltonian on $T^{\text{astQ}}\times \mathbb{R}$. Hamilton's principal function $S(q,\alpha ,t)$ is the value of the action integral

$$S(q,\alpha ,t)={\int }_{t_0}^{t}L(q(t^{\prime }),\dot{q}(t^{\prime }),t^{\prime })d{t}^{\prime }$$

evaluated along the unique classical trajectory from the initial condition parametrised by $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$ at time $t_0$ to the final configuration $q$ at time $t$.

The Hamilton-Jacobi equation from the generating function $F_2$

The Hamilton-Jacobi equation arises as the condition that a type-2 generating function 09.05.02 transforms the Hamiltonian to zero. A type-2 generating function $F_2(q,P,t)$ produces the canonical transformation

$$p_i=\frac{\partial F_2}{\partial q^i},Q^i=\frac{\partial F_2}{\partial P_i},K(Q,P,t)=H(q,p,t)+\frac{\partial F_2}{\partial t}.$$

Choose the new momenta to be constants $P_i={\alpha }_i$ and set $F_2=S(q,\alpha ,t)$. The goal is to make the new Hamiltonian vanish: $K=0$. The condition $K=0$ reads:

$$\boxed{H\left(q^1,\dots ,q^n,\frac{\partial S}{\partial q^1},\dots ,\frac{\partial S}{\partial q^n},t\right)+\frac{\partial S}{\partial t}=0.}$$

This is the Hamilton-Jacobi equation. When $K=0$, the new Hamilton's equations are ${\dot{\alpha }}_i=0$ and ${\dot{\beta }}^i=0$: all new variables are constants, and the mechanical problem is solved. The old variables are recovered via $p_i=\partial S/\partial q^i$ and ${\beta }^i=\partial S/\partial {\alpha }_i$.

A complete solution (or complete integral) of the Hamilton-Jacobi equation is a solution $S(q^1,\dots ,q^n,{\alpha }_1,\dots ,{\alpha }_n,t)$ depending on $n$ independent non-additive constants ${\alpha }_1,\dots ,{\alpha }_n$ such that

$$\det \left(\frac{{\partial }^{2}S}{\partial q^i\partial {\alpha }_j}\right)\ne 0.$$

The non-degeneracy condition ensures that the transformation $(q,p)\leftrightarrow (\alpha ,\beta )$ is invertible via the implicit function theorem.

Hamilton-Jacobi theorem. If $S(q,\alpha ,t)$ is a complete solution of the Hamilton-Jacobi equation, then the $2n$ equations

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

where $\beta =({\beta }^1,\dots ,{\beta }^n)$ are $n$ new constants, implicitly define the general solution $(q(t),p(t))$ of Hamilton's equations parametrised by the $2n$ constants $(\alpha ,\beta )$.

The eikonal equation from the wave equation

For a time-independent Hamiltonian $H(q,p)$, separate $S=W(q)-Et$ where $W$ is Hamilton's characteristic function and $E$ is the energy. The time-independent Hamilton-Jacobi equation is

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

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

$$\frac{1}{2m}{\left|\frac{\partial W}{\partial q}\right|}^2+V(q)=E,\text{i.e.,}{\left|\nabla W\right|}^2=2m(E-V(q)).$$

This is the eikonal equation. To see its origin from wave theory, consider the scalar wave equation (or Helmholtz equation) at frequency $\omega$:

$${\nabla }^2\psi +k^2{n}^2(\mathbf{x})\psi =0,$$

where $k=\omega /c$ is the wave number and $n(\mathbf{x})$ is the refractive index. Substitute the ansatz $\psi =A(\mathbf{x})e^{ik\varphi (\mathbf{x})}$ where $\varphi$ is the eikonal (optical path length). Computing ${\nabla }^2\psi$ and collecting powers of $k$:

$$k^2\left(|\nabla \varphi {|}^2-n^2\right)\psi +ik\left(2\nabla \varphi \cdot \nabla A+A{\nabla }^2\varphi \right)+{\nabla }^{2}A=0.$$

In the short-wavelength limit $k\to \infty$ (equivalently $\lambda \to 0$), the leading-order $O(k^2)$ term dominates and yields:

$$|\nabla \varphi {|}^2=n^2(\mathbf{x}).$$

This is the eikonal equation of geometric optics. The $O(k)$ terms give the transport equation for the amplitude $A$, expressing energy conservation along the rays. The $O(1)$ terms contain diffraction corrections that geometric optics ignores. The identical structure appears in the WKB limit of the Schrodinger equation with $k\sim 1/\hslash$.

The WKB method

The Wentzel-Kramers-Brillouin method connects the Hamilton-Jacobi equation to quantum mechanics. Write the wave function as

$$\psi (q,t)=A(q,t)\exp \left(\frac{i}{\hslash }S(q,t)\right).$$

Substitute into the time-dependent Schrodinger equation $i\hslash {\partial }_t\psi =\hat{H}\psi$. For the Hamiltonian $\hat{H}=-{\hslash }^2{\nabla }^2/(2m)+V(q)$, collecting powers of $\hslash$:

• Order ${\hslash }^0$: $\frac{1}{2m}|\nabla S{|}^2+V+\partial S/\partial t=0$. This is the Hamilton-Jacobi equation.
• Order ${\hslash }^1$: $2\nabla S\cdot \nabla A+A{\nabla }^{2}S+\partial A/\partial t=0$. This is the transport equation for the amplitude $A$, expressing probability conservation along the classical trajectories.

The leading-order equation is the Hamilton-Jacobi equation. This is the precise sense in which classical mechanics is the $\hslash \to 0$ limit of quantum mechanics: the phase of the quantum wave function satisfies the classical Hamilton-Jacobi equation at leading order, and the corrections are organised in powers of $\hslash$.

Hamilton's optico-mechanical analogy

Hamilton's analogy between optics and mechanics is summarised in the following correspondence:

Optics	Mechanics
Eikonal $\varphi (\mathbf{x})$	Characteristic function $W(q)$
Refractive index $n(\mathbf{x})$	$\sqrt{2m(E-V(q))}$
Light rays	Particle trajectories
Wavefronts (level sets of $\varphi$)	Level sets of $W$
Fermat's principle	Maupertuis' principle
Wave equation	Schrodinger equation

Fermat's principle states that light travels along paths of stationary optical path length $\int nds$. Maupertuis' principle states that a mechanical trajectory at fixed energy $E$ makes the abbreviated action $\int pdq$ stationary. These are the same equation:

$$\int nds=\int \sqrt{2m(E-V)}|dq|=\int pdq.$$

The eikonal equation $|\nabla \varphi {|}^2=n^2$ and the time-independent HJ equation $|\nabla W{|}^2=2m(E-V)$ are the same equation with different names. Hamilton recognised this in 1833 and it is the structural reason that quantum mechanics was discovered through an analogy with optics.

Separation of variables. If the Hamiltonian permits a complete solution of the form $S={\sum }_i{S}_i(q^i,\alpha ,t)$ where each $S_i$ depends on only one coordinate, the Hamilton-Jacobi equation separates into $n$ independent ODEs. This occurs when the Hamiltonian has $n$ functionally independent commuting first integrals -- the condition for Liouville integrability 09.06.02.

Key derivation Intermediate+

Derivation: the Hamilton-Jacobi equation from the action

Let $(q(t),p(t))$ be a solution of Hamilton's equations with initial conditions $(q_0,p_0)$ at time $t_0$. Define the action as a function of the endpoint:

$$S(q,t)={\int }_{t_0}^{t}L(q,\dot{q},t^{\prime })d{t}^{\prime }.$$

Vary the endpoint $(q,t)$ while keeping the initial point fixed. The variation gives (using the fact that the trajectory satisfies the Euler-Lagrange equations, so the bulk variation vanishes):

$$\delta S=p_i\delta q^i-H\delta t.$$

The boundary term from varying the endpoint position gives $p_i\delta q^i$, and varying the endpoint time gives $-H\delta t$ (the explicit $L\delta t$ contribution cancels the $-H\delta t$ from the canonical momentum relation). Therefore

$$\frac{\partial S}{\partial q^i}=p_i,\frac{\partial S}{\partial t}=-H(q,p,t)=-H\left(q,\frac{\partial S}{\partial q},t\right).$$

Rearranging:

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

This is the Hamilton-Jacobi equation. The momentum is recovered as $p={\nabla }_{q}S$.

Derivation: characteristics of the HJ equation are Hamilton's equations

The method of characteristics for the first-order PDE $F(q,{\nabla }_{q}S,S,t)=0$ gives the characteristic ODEs

$${\dot{q}}^i=\frac{\partial F}{\partial p_i},{\dot{p}}_i=-\frac{\partial F}{\partial q^i},\dot{S}=p_i{\dot{q}}^i-F.$$

For the Hamilton-Jacobi equation, $F=H(q,p,t)+\partial S/\partial t$, and the characteristic equations are:

$${\dot{q}}^i=\frac{\partial H}{\partial p_i},{\dot{p}}_i=-\frac{\partial H}{\partial q^i},\dot{S}=p_i{\dot{q}}^i-H=L.$$

The first two are exactly Hamilton's equations. The third states that $S$ changes at rate $L$ along the trajectory -- consistent with $S$ being the action integral. The characteristic curves of the HJ equation are the classical trajectories. This is why the Hamilton-Jacobi equation encodes the full dynamics: its characteristic curves are the solutions of Hamilton's equations.

Derivation: the WKB approximation in detail

Write the one-dimensional wave function as $\psi (x,t)=A(x,t)e^{iS(x,t)/\hslash }$. Compute the derivatives:

$$\frac{\partial \psi }{\partial t}=\left(\frac{1}{A}\frac{\partial A}{\partial t}+\frac{i}{\hslash }\frac{\partial S}{\partial t}\right)\psi ,$$ $$\frac{{\partial }^2\psi }{\partial x^2}=\left(\frac{1}{A}\frac{{\partial }^{2}A}{\partial x^2}-\frac{1}{{\hslash }^2}{\left(\frac{\partial S}{\partial x}\right)}^2+\frac{2i}{\hslash }\frac{1}{A}\frac{\partial A}{\partial x}\frac{\partial S}{\partial x}+\frac{i}{\hslash }\frac{{\partial }^{2}S}{\partial x^2}\right)\psi .$$

Substituting into $i\hslash {\partial }_t\psi =-\frac{{\hslash }^2}{2m}{\partial }_x^2\psi +V\psi$ and separating real and imaginary parts:

Real part ($O({\hslash }^0)$ and $O({\hslash }^{-2})$): The dominant terms (order ${\hslash }^{-2}$ and ${\hslash }^0$) give

$$\frac{1}{2m}{\left(\frac{\partial S}{\partial x}\right)}^2+V+\frac{\partial S}{\partial t}=\frac{{\hslash }^2}{2m}\frac{1}{A}\frac{{\partial }^{2}A}{\partial x^2}.$$

At leading order ($\hslash \to 0$), the right side vanishes and this is the Hamilton-Jacobi equation. The next-order correction involves the curvature of the amplitude envelope.

Imaginary part ($O({\hslash }^{-1})$):

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

This is the transport equation: a continuity equation expressing conservation of probability density $\rho =|A{|}^2$ along the flow with velocity $v=(1/m)\partial S/\partial x$. The classical trajectories (characteristics of the HJ equation) are the streamlines of this flow.

Bridge. The Hamilton-Jacobi equation builds toward the Schrodinger equation 12.02.01 via the WKB correspondence -- the phase of the quantum wave function satisfies the classical Hamilton-Jacobi equation at leading order, and the corrections are organised in powers of $\hslash$. The foundational reason this works is that the canonical transformation generated by $S$ is the classical shadow of the unitary time-evolution operator $U(t)=e^{-i\hat{H}t/\hslash }$: the classical action is the logarithm of the quantum propagator. This appears again in the eikonal equation of geometric optics 10.08.04, where the short-wavelength limit of the wave equation produces the same Hamilton-Jacobi structure, and the central insight is that Hamilton's mechanics was originally modelled on geometric optics -- the analogy is not a post-hoc observation but the historical route by which Hamilton discovered his equations.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib does not contain the Hamilton-Jacobi equation, the theory of generating functions for canonical transformations, the eikonal equation, or the WKB method in a formalised form. The closest available infrastructure includes symplectic linear algebra, smooth manifold theory, cotangent bundles, and basic ODE existence theory. The theory of first-order PDEs and their characteristics is absent. Formalising even the statement of the Hamilton-Jacobi theorem -- that a complete solution of the HJ equation generates the general solution of Hamilton's equations -- would require:

1. The method of characteristics for first-order PDEs on manifolds.
2. The theory of generating functions (type-2 canonical transformations and the relation between $F_2$ and $S$).
3. The inverse function theorem in the setting of canonical coordinates, applied to the non-degeneracy condition $\det ({\partial }^{2}S/\partial q\partial \alpha )\ne 0$.
4. The eikonal equation and its derivation from the wave equation via the short-wavelength limit, which requires asymptotic analysis and PDE infrastructure.
5. The WKB expansion and its connection to the HJ equation, which requires formal power series in $\hslash$ and singular perturbation theory.
6. The Bohr-Sommerfeld quantisation conditions and the Maslov index, which require the theory of Lagrangian submanifolds and their quantisation.

This is a substantial undertaking at the frontier of what Mathlib currently supports. The symplectic geometry prerequisites (the symplectic form, Hamiltonian vector fields, canonical transformations) are themselves under development.

Advanced results Master

Semiclassical quantisation: Bohr-Sommerfeld and the Maslov index

The Bohr-Sommerfeld quantisation condition arises from requiring the WKB wave function to be single-valued over a closed classical orbit. For a one-dimensional bound state with turning points $q_1$ and $q_2$ (where $E=V(q_i)$), the WKB wave function in the classically allowed region is

$$\psi (q)\sim \frac{C}{\sqrt{p(q)}}\exp \left(\frac{i}{\hslash }{\int }_{q_1}^{q}p(q^{\prime })d{q}^{\prime }\right)+\frac{D}{\sqrt{p(q)}}\exp \left(-\frac{i}{\hslash }{\int }_{q_1}^{q}p(q^{\prime })d{q}^{\prime }\right),$$

where $p(q)=\sqrt{2m(E-V(q))}$. The matching conditions at the turning points (where $p\to 0$ and the WKB approximation breaks down) were analysed by Langer (1937) using Airy-function asymptotics. The result is that the wave function in the allowed region must be a cosine with a phase shift of $\pi /4$ at each turning point. Single-valuedness of the wave function around the full libration then requires:

$$\oint pdq=\left(n+\frac{1}{2}\right)h,n=0,1,2,\dots$$

The $1/2$ is the Maslov correction: each turning point contributes $\pi /4$ of extra phase, and two turning points give $\pi /2=2\pi \cdot (1/4)$, hence the extra $1/2$ quantum number. Without the Maslov correction, the naive Bohr-Sommerfeld condition $\oint pdq=nh$ gives incorrect ground-state energies (it predicts $E_0=0$ for the harmonic oscillator, while the correct answer is $E_0=\hslash \omega /2$).

The Maslov index $\mu$ generalises this to arbitrary Lagrangian submanifolds in phase space. For a closed orbit on the energy surface, the corrected quantisation condition is:

$$\frac{1}{2\pi \hslash }\oint pdq=n+\frac{\mu }{4},$$

where $\mu$ counts the number of caustics (points where the projection from the Lagrangian submanifold to configuration space is singular) encountered along the orbit. For a generic libration with two smooth turning points, $\mu =2$, giving $n+1/2$. For a rotational orbit on a torus (no turning points), $\mu =0$.

The Maslov index was introduced by Victor Maslov in 1965 and given a rigorous cohomological interpretation by Arnold (1967) and Hormander (1971): $\mu$ is the Maslov class of the Lagrangian submanifold, an element of $H^1(\Lambda ,\mathbb{Z})$ where $\Lambda$ is the Lagrangian submanifold.

Geometric optics limit of Maxwell's equations

The full Maxwell equations in a medium with spatially varying refractive index $n(\mathbf{x})$ are:

$$\nabla \times \mathbf{E}=-\frac{1}{c}\frac{\partial \mathbf{B}}{\partial t},\nabla \times \mathbf{H}=\frac{1}{c}\frac{\partial \mathbf{D}}{\partial t},\nabla \cdot \mathbf{D}=0,\nabla \cdot \mathbf{B}=0,$$

with the constitutive relations $\mathbf{D}=n^2\mathbf{E}$ and $\mathbf{B}=\mathbf{H}$ (in Gaussian units for a non-magnetic dielectric). For monochromatic fields at frequency $\omega$, substitute the geometric optics ansatz:

$$\mathbf{E}(\mathbf{x},t)=\mathbf{e}(\mathbf{x})e^{i{k}_0\varphi (\mathbf{x})-i\omega t},\mathbf{H}(\mathbf{x},t)=\mathbf{h}(\mathbf{x})e^{i{k}_0\varphi (\mathbf{x})-i\omega t},$$

where $k_0=\omega /c$ and $\varphi$ is the eikonal. The amplitudes $\mathbf{e}$ and $\mathbf{h}$ are expanded in powers of $1/k_0$. At leading order $O(k_0^2)$, Maxwell's equations reduce to:

$${\left(\nabla \varphi \right)}^2=n^2(\mathbf{x}),\mathbf{e}\cdot \nabla \varphi =0,\mathbf{h}\cdot \nabla \varphi =0.$$

The first is the eikonal equation; the second and third state that $\mathbf{E}$ and $\mathbf{H}$ are transverse to the ray direction $\nabla \varphi$. At next order $O(k_0)$, the transport equations for the field amplitudes give the intensity variation along the rays, expressing energy conservation in the ray picture. The $O(1)$ corrections contain diffraction, polarisation effects, and the first departures from geometric optics.

The ray equations derived from the eikonal equation are:

$$\frac{d\mathbf{x}}{ds}=\frac{\nabla \varphi }{n},\frac{d}{ds}\left(n\frac{d\mathbf{x}}{ds}\right)=\nabla n,$$

where $s$ is the arc length along the ray. These are Hamilton's equations in disguise: with the "Hamiltonian" $H(\mathbf{x},\mathbf{p})=|\mathbf{p}{|}^2/(2n^2)$ and the "momentum" $\mathbf{p}=\nabla \varphi$, the ray equations are $\dot{\mathbf{x}}=\partial H/\partial \mathbf{p}$ and $\dot{\mathbf{p}}=-\partial H/\partial \mathbf{x}$.

Catastrophe optics

Caustics are the singularities of geometric optics -- the surfaces where neighbouring rays focus and the intensity predicted by geometric optics diverges. The classification of caustic singularities is the subject of catastrophe optics (Berry and Upstill 1980, Kravtsov and Orlov 1990), which applies Thom's catastrophe theory to the eikonal function.

Near a caustic, the eikonal function $W(q)$ has a degenerate critical point: the Jacobian ${\partial }^{2}W/\partial q^i\partial {\alpha }_j$ vanishes, and the map from the Lagrangian submanifold to configuration space is singular. The Maslov canonical operator repairs the geometric optics prediction by replacing the divergent amplitude with a uniform asymptotic expansion involving Airy functions (for fold caustics) or Pearcey functions (for cusp caustics), depending on the codimension of the singularity.

The standard classification (Arnold 1972) identifies the generic caustic singularities in $n$-dimensional configuration space with the $A_k$ ($k\ge 2$), $D_k$ ($k\ge 4$), and $E_6,E_7,E_8$ catastrophe germs. For $n\le 4$, only $A_2$ (fold), $A_3$ (cusp), $A_4$ (swallowtail), $D_4$ (hyperbolic and elliptic umbilic), and $A_5$ are stable. The diffraction patterns near each type of caustic are universal -- they depend only on the catastrophe type and not on the details of the medium.

In mechanics, the same classification governs the focal structures of particle trajectories. A system of trajectories in configuration space develops caustics wherever the HJ generating function has a degenerate Hessian, and the semiclassical wave function near these caustics is described by the same Airy and Pearcey integrals.

Fermat's principle as the Hamilton-Jacobi equation

Fermat's principle states that the optical path length ${\int }_a^{b}n(\mathbf{x})ds$ is stationary along light rays. In the language of Hamilton-Jacobi theory, the optical path length is the eikonal $\varphi (\mathbf{x})$, and Fermat's principle is the variational principle associated with the eikonal equation:

$${\left|\nabla \varphi \right|}^2=n^2(\mathbf{x}).$$

The ray equations are the characteristics of this PDE, and the rays are orthogonal to the wavefronts (level sets of $\varphi$). This is the optical version of Maupertuis' principle: the abbreviated action $\int pdq$ is stationary along mechanical trajectories, and the trajectories are orthogonal to the level sets of $W$.

The two principles are the same:

$${\int }_a^{b}nds={\int }_a^b\sqrt{2m(E-V)}|dq|={\int }_a^{b}pdq,$$

with the identification $n^2\leftrightarrow 2m(E-V)$. Fermat's principle is Maupertuis' principle, seen through the lens of the optico-mechanical analogy. The variational principle, the first-order PDE for the generating function, and the characteristic equations form a trinity that is the same in both optics and mechanics.

The eikonal equation and Hamiltonian mechanics on configuration space

The time-independent Hamilton-Jacobi equation $H(q,\nabla W)=E$ defines a Hamilton-Jacobi surface in phase space: the graph of $p=\nabla W(q)$ is an $n$-dimensional Lagrangian submanifold of the $2n$-dimensional phase space. The trajectories are the characteristics of the HJ equation restricted to this submanifold.

The projection of this Lagrangian submanifold to configuration space is the configuration-space shadow of the dynamics. Where the projection is regular (the Jacobian ${\partial }^{2}W/\partial q^i\partial {\alpha }_j$ is non-singular), each point in configuration space lies on exactly one trajectory and the geometric optics picture is valid. Where the projection is singular (the Jacobian vanishes), multiple trajectories focus at the same point, the amplitude diverges, and diffraction effects become important. This is the catastrophe structure described above.

The Lagrangian submanifold defined by $p=\nabla W(q)$ is the phase-space object that unifies the Hamilton-Jacobi equation, the eikonal equation, and the WKB approximation: it is the classical phase-space structure whose quantisation produces the quantum wave function.

Synthesis. The Hamilton-Jacobi equation is the bridge between classical mechanics and wave mechanics. The central insight is that particle trajectories are the characteristics (gradient lines) of a scalar function $S$ satisfying a first-order PDE, and the Schrodinger equation is the second-order PDE whose short-wavelength limit recovers this first-order PDE. The analogy with optics is not decorative but structural: Hamilton's original formulation of mechanics was explicitly modelled on geometric optics, and the discovery that the Schrodinger equation stands to the Hamilton-Jacobi equation as wave optics stands to geometric optics was the founding insight of quantum mechanics. The Maslov index, the WKB expansion, catastrophe optics, and the Bohr-Sommerfeld quantisation conditions all build toward this correspondence from the classical side, while the path integral 12.10.01 builds toward it from the quantum side. The Hamilton-Jacobi equation is the load-bearing structure of the classical-quantum correspondence, and its separation of variables is the bridge from classical integrability 09.06.02 to the exact solvability of quantum systems with the same symmetry group.

Full proof set Master

Proposition 1 (The Hamilton-Jacobi theorem). Let $S(q,\alpha ,t)$ be a complete solution of the Hamilton-Jacobi equation $H(q,{\nabla }_{q}S,t)+{\partial }_{t}S=0$, with $\det ({\partial }^{2}S/\partial q^i\partial {\alpha }_j)\ne 0$. Define $p_i=\partial S/\partial q^i$ and ${\beta }^i=\partial S/\partial {\alpha }_i$. Then $(q(t),p(t))$ defined implicitly by ${\beta }^i=\partial S/\partial {\alpha }_i$ and $p_i=\partial S/\partial q^i$ satisfies Hamilton's equations for $H$.

Proof. The implicit function theorem guarantees that ${\beta }^i=\partial S/\partial {\alpha }_i(q,\alpha ,t)$ can be solved for $\alpha =\alpha (q,\beta ,t)$ (hence also for $q=q(\alpha ,\beta ,t)$) in a neighbourhood of any point where $\det ({\partial }^{2}S/\partial q\partial \alpha )\ne 0$.

Since $\beta$ and $\alpha$ are constants by construction, differentiate ${\beta }^i=\partial S/\partial {\alpha }_i$ with respect to $t$ along the trajectory:

$$0=\frac{d}{dt}\frac{\partial S}{\partial {\alpha }_i}=\frac{{\partial }^{2}S}{\partial {\alpha }_i\partial q^j}{\dot{q}}^j+\frac{{\partial }^{2}S}{\partial {\alpha }_i\partial t}.$$

Now differentiate the HJ equation $H(q,{\nabla }_{q}S,t)+{\partial }_{t}S=0$ with respect to ${\alpha }_i$:

$$\frac{\partial H}{\partial p_j}\frac{{\partial }^{2}S}{\partial q^j\partial {\alpha }_i}+\frac{{\partial }^{2}S}{\partial t\partial {\alpha }_i}=0.$$

Comparing the two expressions and using the non-degeneracy of ${\partial }^{2}S/\partial q\partial \alpha$, conclude ${\dot{q}}^j=\partial H/\partial p_j$ -- the first Hamilton equation.

For the second equation, differentiate $p_i=\partial S/\partial q^i$ in time:

$${\dot{p}}_i=\frac{{\partial }^{2}S}{\partial q^i\partial q^j}{\dot{q}}^j+\frac{{\partial }^{2}S}{\partial q^i\partial t}.$$

Differentiate the HJ equation with respect to $q^i$:

$$\frac{\partial H}{\partial q^i}+\frac{\partial H}{\partial p_j}\frac{{\partial }^{2}S}{\partial q^j\partial q^i}+\frac{{\partial }^{2}S}{\partial t\partial q^i}=0.$$

Substituting ${\dot{q}}^j=\partial H/\partial p_j$ and $p_i=\partial S/\partial q^i$:

$${\dot{p}}_i=-\frac{\partial H}{\partial q^i}.$$

This is the second Hamilton equation. $\square$

Proposition 2 (Eikonal equation as short-wavelength limit). The eikonal equation $|\nabla \varphi {|}^2=n^2(\mathbf{x})$ is the leading-order equation obtained from the Helmholtz equation ${\nabla }^2\psi +k^2{n}^2\psi =0$ in the limit $k\to \infty$ with the ansatz $\psi =A{e}^{ik\varphi }$.

Proof. Computing ${\nabla }^2\psi =\nabla \cdot \nabla (A{e}^{ik\varphi })$:

$${\nabla }^2\psi =({\nabla }^{2}A+2ik\nabla A\cdot \nabla \varphi +ikA{\nabla }^2\varphi -k^{2}A|\nabla \varphi {|}^2)e^{ik\varphi }.$$

Substituting into the Helmholtz equation and dividing by $e^{ik\varphi }$:

$${\nabla }^{2}A+2ik\nabla A\cdot \nabla \varphi +ikA{\nabla }^2\varphi +k^{2}A(n^2-|\nabla \varphi {|}^2)=0.$$

The terms are of order $k^2$, $k^1$, and $k^0$. The $O(k^2)$ equation is:

$$k^{2}A(n^2-|\nabla \varphi {|}^2)=0,\text{giving}|\nabla \varphi {|}^2=n^2.$$

The $O(k)$ equation is the transport equation $2\nabla \varphi \cdot \nabla A+A{\nabla }^2\varphi =0$, which is a conservation law for the energy flux along the rays. The $O(1)$ equation contains diffraction corrections. $\square$

Proposition 3 (Bohr-Sommerfeld with Maslov correction). For a one-dimensional potential with two smooth turning points $q_1<q_2$, the semiclassical quantisation condition is $\oint pdq=(n+1/2)h$ where the Maslov correction $1/2$ arises from the phase shift $\pi /4$ at each turning point.

Proof (sketch). In the classically allowed region $q_1<q<q_2$, the WKB wave function is $\psi \sim C/\sqrt{p}\cos ({\int }_{q_1}^{q}pd{q}^{\prime }/\hslash -\pi /4)$. The $-\pi /4$ phase comes from matching the oscillatory WKB solution to the exponentially decaying Airy function solution in the forbidden region $q>q_2$. Near $q=q_1$, the same matching gives $\psi \sim C^{\prime }/\sqrt{p}\cos ({\int }_{q_1}^{q}pd{q}^{\prime }/\hslash +\pi /4)$ with $+\pi /4$. Consistency of these two expressions requires ${\int }_{q_1}^{q_2}pdq/\hslash +\pi /4={\int }_{q_1}^{q_2}pdq/\hslash -\pi /4+n\pi$, giving the single-valuedness condition $\oint pdq=2{\int }_{q_1}^{q_2}pdq=(n+1/2)h$. The factor of 2 in the $1/2$ correction comes from the two turning points, each contributing $\pi /4$ of phase. $\square$

Connections Master

• Canonical transformations 09.05.02 are generated by $F_2(q,P,t)=S(q,\alpha ,t)$ with $P=\alpha$; the Hamilton-Jacobi equation is the condition that the new Hamiltonian $K=0$, making all new variables constant and solving the system by quadrature.

• Symplectic structure 09.04.05 provides the geometric foundation for the Hamilton-Jacobi theory: the graph of $p={\nabla }_{q}S$ is a Lagrangian submanifold of the symplectic phase space, and the HJ equation is the condition that this Lagrangian submanifold is invariant under the Hamiltonian flow.

• Action-angle variables 09.06.01 are obtained by separation of variables in the Hamilton-Jacobi equation for integrable systems; the action variables $J_i$ are the ${\alpha }_i$ constants that appear in the separated solution, and the angle variables are the conjugate ${\beta }^i$.

• The Schrodinger equation 12.02.01 is the second-order PDE whose $\hslash \to 0$ limit is the Hamilton-Jacobi equation; the WKB construction makes this precise and identifies the HJ equation as the leading-order phase equation.

• Geometric optics 10.08.04 derives the eikonal equation from the wave equation (and ultimately from Maxwell's equations) in the short-wavelength limit; this is the electromagnetic analogue of the WKB approximation, and the optico-mechanical analogy is the historical origin of the Hamilton-Jacobi equation.

• Path integrals 12.10.01 express the quantum propagator as a sum over classical paths weighted by $\exp (iS/\hslash )$; in the stationary-phase approximation, the dominant contribution comes from the classical trajectory that satisfies the HJ equation.

Historical and philosophical context Master

Hamilton discovered what is now called the Hamilton-Jacobi equation in 1827 while studying geometric optics, not mechanics. His Theory of Systems of Rays (1827) introduced the eikonal equation for light and the associated ray equations. Hamilton showed that a single scalar function -- the eikonal -- determines all the rays in an optical system, and that the ray directions are perpendicular to the surfaces of constant eikonal. In his Supplement to the Theory of Systems of Rays (1830) and On a General Method in Dynamics (1834-1835), he recognised that the same mathematics applies to particle mechanics: the action plays the role of the optical path length, particle trajectories are the "rays" of mechanics, and the surfaces of constant action are the "wavefronts." This optico-mechanical analogy was the first bridge between mechanics and wave theory.

Jacobi developed the separation-of-variables technique in his Vorlesungen uber Dynamik (lectures 1842-43, published posthumously 1866), showing that the Hamilton-Jacobi equation is separable precisely when the Hamiltonian system is integrable in the Liouville sense. Jacobi proved the theorem that a complete solution of the HJ equation generates the general solution of Hamilton's equations. The Hamilton-Jacobi theorem, though named for both, is in its proof technique due primarily to Jacobi.

The analogy lay dormant until de Broglie revived it in 1924. In his PhD thesis Recherches sur la theorie des quanta, de Broglie argued that if light (a wave) has particle-like properties (photons), then particles should have wave-like properties. He proposed the relation $p=h/\lambda$ (momentum equals Planck's constant divided by wavelength) by analogy with the photon relation $E=h\nu$, and identified the phase of the matter wave with the action divided by $\hslash$. De Broglie's insight was that the Hamilton-Jacobi equation describes the propagation of wavefronts of a matter wave, and that classical trajectories are the rays of this wave.

Schrodinger seized upon de Broglie's idea in 1926. In Quantisierung als Eigenwertproblem (Annalen der Physik 79, 361-376, 1926), Schrodinger started from the Hamilton-Jacobi equation and asked: what second-order wave equation has the HJ equation as its short-wavelength (eikonal) limit? By analogy with the derivation of the eikonal equation from the wave equation in optics, Schrodinger wrote down the equation that bears his name. The Hamilton-Jacobi equation is the eikonal equation of the Schrodinger equation, and the action $S$ is the eikonal (phase function) of the quantum wave function. This is the precise content of the statement that quantum mechanics reduces to classical mechanics as $\hslash \to 0$.

The WKB method was developed independently by Wentzel, Kramers, and Brillouin in 1926 to compute approximate quantum-mechanical wave functions. The connection to the Hamilton-Jacobi equation was noted immediately -- the leading-order WKB phase satisfies the HJ equation. The Maslov index was introduced by Victor Maslov in his 1965 monograph Theorie des perturbations et methodes asymptotiques, with the rigorous framework developed by Hormander (1971) in the context of Fourier integral operators. The catastrophe-theoretic classification of caustics was developed by Arnold (1972), Berry and Upstill (1980), and Kravtsov and Orlov (1990).

The philosophical content is that classical mechanics and wave optics share the same mathematical structure -- the eikonal / Hamilton-Jacobi equation -- because both are the short-wavelength limits of deeper wave theories (quantum mechanics and electromagnetic waves, respectively). The Hamilton-Jacobi equation is not merely a reformulation of Newton's laws; it is the common skeleton of all ray-based descriptions of wave propagation.

Bibliography Master

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• Schrodinger, E., "Quantisierung als Eigenwertproblem," Ann. Phys. 79 (1926), 361-376; 79, 489-527; 80, 437-490; 81, 109-139.

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