Same maths, different names · Audit 03
One first-order partial differential equation for a single scalar function is the equation of motion of classical mechanics, the equation of light rays, the phase equation of short-wave asymptotics across optics, acoustics, seismology and quantum mechanics, the equation of optimal control and, paired with continuity, the equation of optimal transport. This audit records who uses it, what each field calls it, when it is exact and when it is the first term of something larger, and what happens at the places where it breaks.
In plain terms
Throw a ball and it follows one path. Shine a torch and the light follows one path, a ray. In the 1830s William Rowan Hamilton noticed that both paths can be found the same way: not by following the ball or the ray step by step, but by writing down a single function of position, a kind of running total of “cost” accumulated in getting there, and then reading the direction of travel off the way that function changes from place to place. For light the running total is the time taken; for the ball it is a quantity called the action. The equation that the running total obeys is the same in both cases. Hamilton worked it out for light first and said plainly, when he turned to mechanics, that he was reusing the optical idea.
Since then the same equation has turned up wherever something moves along a path that a rule picks out. Sound in the sea and earthquake waves in rock follow rays that obey it, with travel time as the running total; seismologists use it to locate earthquakes. Light in Einstein’s curved spacetime obeys it, with the ray now a geodesic. A slowly changing train of ocean waves obeys it, with the crests as the level surfaces. A rocket steering to use the least fuel obeys it, with the running total now the best achievable cost, and engineers call it Bellman’s equation. Moving a pile of sand into a hole at least effort obeys it. And computer programs that track a spreading fire front, or the boundary of a tumour in a scan, solve it directly.
Two things make the audit worth doing. The first is that in some of these fields the equation is exactly true and in others it is the first term of an approximation: light really is a wave, and rays are what waves look like when the wavelength is much smaller than anything they meet. The ray picture fails where a whole family of rays folds over on itself, at caustics, the bright curves at the bottom of a swimming pool or inside a coffee cup, and there the wave has to be brought back. In quantum mechanics the very same thing happens: the classical path is the ray, the wavefunction is the wave, and the equation on this page is what the previous audit’s equation becomes when the one quantum term is thrown away. The second thing is historical. The analogy between light and matter was not discovered by comparing two finished theories. It was the heuristic Schrödinger used to write the wave equation in the first place; the proof that the wave equation contains the classical one, under stated conditions, came afterwards and is a real result. This page keeps the two apart: how the connection was found, and what has since been shown.
The rest of the page is a ledger. It lists the fields, what each calls the equation, when each first had it, what each assumes, and a verdict: exact when the equation is exactly equivalent to the row’s governing equations, limited when it is the leading term of a named approximation to them, analogy when only some features line up. Every claim carries a note on how far the source was actually read.
What is being audited
Take a mechanical system with coordinates q and Hamiltonian H(q, p, t). The Hamilton–Jacobi equation is a first-order partial differential equation for a single scalar function S(q, t):
time-independent form: H(q, ∇W) = E, S = W − Et
for H = |p|²/2m + V(q): |∇W|² = 2m(E − V)
eikonal form, for light in a medium of refractive index n(x): |∇𝒮|² = n(x)²
and its relation to a wave: ψ = A eiS/ħ, with A > 0 and S real, in the scalar Schrödinger equation gives, exactly and locally on a zero-free region where a smooth branch of the phase is chosen,
∂tS + |∇S|²/2m + V = (ħ²/2m) ∇²A/A (= −Q), ∂tA² + ∇·(A² ∇S/m) = 0
The equation has a distinctive geometry. Its solutions are surfaces of constant S, and its characteristics, the curves along which the PDE reduces to ordinary differential equations, are exactly Hamilton’s equations of motion: q̇ = ∂H/∂p, ṗ = −∂H/∂q. Trajectories are the rays of the wavefronts S = const, and the momentum at each point is the gradient of S. Given a complete integral, a solution depending on as many constants α as there are coordinates with det(∂²S/∂q∂α) ≠ 0, Jacobi’s theorem recovers every trajectory of the system by differentiation, which is why the PDE and the ODEs are two forms of one thing rather than two theories. Such a solution always exists locally, by the method of characteristics; separation of variables is one way to construct one, not a condition for its existence. In optics the same construction runs from Fermat’s principle: the optical path length 𝒮 obeys the eikonal equation, its gradient is the ray direction times n, and the rays are the characteristics.
The last two lines of the box are the ones that matter for the series. Substituting a wave of positive amplitude A and real phase S/ħ into the scalar Schrödinger equation gives two real equations with no approximation at all, locally on a zero-free region with a chosen smooth branch of the phase (a nowhere-vanishing ψ does not guarantee a globally single-valued real S around a non-contractible loop, as Audit 02 records); they are the Madelung pair of Audit 02, with the quantum potential written as the right-hand side. The classical Hamilton–Jacobi equation is what remains when that term is dropped, and the continuity equation then becomes the transport equation for the leading amplitude. Whether the term may be dropped is the whole question, and the explicit factor ħ² does not settle it: ∇²A/A can grow as ħ shrinks. The condition is that the amplitude vary slowly on the scale of the local wavelength, ħ|∇²A/A| ≪ |∇S|²/ħ, which is a condition on the solution, not on the constant. Where it holds, the leading amplitude is carried along the rays; where a family of rays folds, that leading amplitude is generally singular and the condition fails. The exact continuity equation, by contrast, predicts no divergence on its own; the singularity belongs to the approximation.
Three conditions travel with the identity, and each row of the ledger is read against them.
First, the equation is exact only for systems whose governing equations are themselves the characteristic equations: Hamiltonian mechanics, geodesic motion, and, in a different genre, optimisation problems whose value function is the running cost. For wave systems it is the leading term of an expansion in the ratio of wavelength to the scale on which the medium varies. That is a different logical status, and the verdict column tracks it.
Second, a smooth single-valued solution exists only until the family of rays folds. A caustic is where the projection of that family from phase space to physical space loses rank; it is not the same thing as two rays happening to cross, which can occur without any singularity. At a caustic the leading ray amplitude is generally singular and a single smooth phase ceases to suffice; the underlying wave, where there is one, shows interference and a finite peak instead. Where there is no wave, as in mechanics or control, the mathematics supplies other solution concepts: the multivalued phase can be kept as a Lagrangian manifold in phase space, or a weak notion of solution, the viscosity solution, can be used, which is single-valued and is the right object when the problem is a minimisation. The ledger says which each field uses.
Third, the analogy between rays and trajectories has a known history and a separate mathematical status. Hamilton built mechanics on optics; de Broglie and Schrödinger built wave mechanics on Hamilton’s analogy; semiclassical analysis then proved, under stated conditions, that the wave equation contains the classical one as a limit. The history explains how the connection was found. It neither adds to nor subtracts from what the limit theorems establish, and the verdict keeps the two apart.
Two columns in the ledger do the real work. The dictionary column says what plays the role of what, so that a verdict can be checked rather than believed. The direction column says which way the implication runs: from a wave equation down to rays, from an ODE system up to a PDE, or from an optimisation principle to its value function. Across the rows, “the same equation” means the same Hamilton–Jacobi form, with the Hamiltonian and the normalisation of the scalar identified in the dictionary column; the Hamiltonians differ. The verdict judges the correspondence within the row’s stated governing equation; whether that equation describes the physical system well is a separate question, answered on the “model” line.
The ledger
Each verdict applies to the correspondence named in that row’s dictionary column, not to the field as a whole. The physical adequacy of the starting equation is stated on the “model” line and does not enter the verdict.
| Field | Name used · earliest source located | Dictionary | Direction of implication | Assumptions added or dropped · model · where it breaks | Verdict |
|---|---|---|---|---|---|
| Classical mechanics | Hamilton–Jacobi equation; Hamilton’s principal function; Jacobi’s theoremHamilton 1834, 1835 · Jacobi 1837 · Jacobi’s lectures 1842–43 (published 1866) | Principal function S ↔ action along the trajectory · ∇S ↔ momentum · characteristics of the PDE ↔ Hamilton’s equations · complete integral ↔ the full family of trajectories · surfaces S = const ↔ wavefronts, by Hamilton’s own analogy | Both directions: the PDE and the ODEs are equivalent wherever a complete integral exists and the solution is smooth. | adds a complete integral with the non-degeneracy condition det(∂²S/∂q∂α) ≠ 0, which exists locally; separability is a method of construction, not a requirement · model Hamiltonian mechanics itselfbreaks a smooth single-valued S from a given initial surface exists only until the family of trajectories folds (a caustic); global complete integrals do not exist for non-integrable systems, and the local equivalence with the ODEs is unaffected by either | exact |
| Geometrical optics | Eikonal equation; characteristic function; optical path length; Fermat’s principleHamilton 1828 (characteristic function) · Bruns 1895 (the word) · Sommerfeld & Runge 1911 (from Maxwell) · Born & Wolf, Ch. 3 | Eikonal 𝒮 ↔ optical path length ↔ Hamilton’s characteristic function · ∇𝒮 ↔ n × ray direction ↔ momentum · refractive index n ↔ √(2m(E − V)), the optical-mechanical analogy · rays ↔ trajectories · Fermat ↔ Maupertuis | Maxwell’s equations → eikonal, as the leading term in the limit λ → 0. Within ray optics itself, Fermat’s principle ↔ eikonal is exact. | adds wavelength small compared with the scale of variation of n and of the amplitude; a local plane-wave form · model Maxwell’s equations in a slowly varying isotropic mediumbreaks at caustics, where the ray amplitude diverges and the wave amplitude scales as a fractional power of wavelength (Berry & Upstill 1980); at edges and shadow boundaries, where diffraction requires additional rays (Keller 1962); wherever n varies on the scale of λ | limited |
| Quantum mechanics | WKB approximation; semiclassical limit; Van Vleck determinant; EBK quantisation; Maslov indexJeffreys 1925 · Wentzel, Kramers, Brillouin 1926 · Van Vleck 1928 · Keller 1958 · Maslov 1965 (French edn 1972) · Berry & Mount 1972 (review) | S = ħ × phase ↔ classical action, so ∇S = ħk ↔ momentum · A² ↔ the transported density; for the semiclassical propagator specifically, the amplitude contains the square root of the absolute Van Vleck determinant · the dropped term (ħ²/2m)∇²A/A = −Q ↔ minus the quantum potential of Audit 02 · turning points and caustics ↔ connection formulae and the Maslov index · single-valued ψ ↔ EBK quantisation conditions | Schrödinger → Hamilton–Jacobi with the quantum potential, plus continuity, exactly (Audit 02); → classical Hamilton–Jacobi plus transport, as the leading order in ħ. | adds amplitude slowly varying on the local de Broglie scale, ħ|∇²A/A| ≪ |∇S|²/ħ, a condition on the solution and not on ħ alone; away from turning points · model the scalar Schrödinger equationbreaks at turning points, where the momentum vanishes, and at caustics, where the leading amplitude is singular and the wavefunction is finite; the expansion is generally asymptotic and not guaranteed to converge, though particular semiclassical expressions can be exact; classically forbidden regions are not a failure, since WKB describes them with growing and decaying exponentials, but the complex WKB action there is not the real phase of the exact decomposition; for chaotic systems no global complete integral exists and quantisation needs periodic-orbit theory rather than EBK (not audited here) | limited |
| Acoustics | Ray acoustics; eikonal equation for sound; geometrical theory of diffractionPierce 1981, Ch. 8 · Keller 1962 | Travel time τ ↔ S · |∇τ| = 1/c ↔ eikonal with n = c₀/c · slowness vector ∇τ ↔ momentum · sound rays ↔ trajectories · in a moving medium, the ray Hamiltonian acquires the flow velocity, as in the acoustic metric of Audit 01 | Wave equation for sound → ray equations, as the high-frequency limit. | adds wavelength small compared with the scales of the sound-speed and flow variation; amplitude slowly varying · model linear acoustics in an inhomogeneous, possibly moving, mediumbreaks at caustics (sound channels in the ocean and atmosphere produce them routinely) and at shadow zones, where diffraction fills what ray theory leaves empty; the moving-medium form is exact only within the same high-frequency limit | limited |
| Seismology | Seismic ray theory; eikonal equation for travel time; Maslov–Chapman method; Gaussian beamsČervený 2001 (monograph; Sec. 2.4 for the eikonal) | Travel time T ↔ S · (∇T)² = 1/V² ↔ eikonal · slowness ↔ momentum · seismic rays ↔ trajectories · transport equation ↔ amplitude along a ray tube | Elastodynamic equation → eikonal and transport, as the high-frequency asymptotic solution; Červený’s preface calls the method “only approximate”. | adds high frequency relative to medium variation; separate P and S eikonals in an isotropic solid · model linear elastodynamics in a heterogeneous Earthbreaks at caustics and in shadow zones, which is why the Maslov–Chapman method and Gaussian-beam summation exist. Interfaces require reflection, transmission and mode-conversion matching (Červený, Sec. 5.3), and additional approximations are needed near caustics, critical incidence and other regions where the ray expansion becomes non-uniform | limited |
| General relativity (i) | Hamilton–Jacobi equation for geodesics; Carter’s constant; separabilityCarter 1968 (Kerr) · Misner, Thorne & Wheeler 1973 | S ↔ the action of a test particle · gμν∂μS ∂νS = −m², the covariant mass-shell Hamilton–Jacobi equation, with H = ½gμνpμpν and no stationarity assumed · characteristics ↔ geodesics · separability ↔ hidden constants of motion (Carter’s fourth constant) | Both directions: geodesic equations ↔ Hamilton–Jacobi equation, exactly, as in row 1. | adds nothing beyond a complete integral; Carter’s result is that Kerr admits one · model test-particle motion on a fixed backgroundbreaks as in row 1, at conjugate points where geodesics from a common origin cross; the equation says nothing about the background’s own dynamics (the Einstein–Hamilton–Jacobi equation for the metric itself is a separate structure, noted under extensions) | exact |
| General relativity (ii) | Geometrical optics in curved spacetime; eikonal for light and gravitational waves; Fermat’s principle in GREhlers 1967 · Isaacson 1968 · Kovner 1990, Perlick 1990 (Fermat) · Perlick 2004 (review, eq. 7) | Phase S of the electromagnetic (or gravitational) wave ↔ eikonal · gij∂iS ∂jS = 0 ↔ null eikonal · light rays ↔ null geodesics · Fermat’s principle with arrival time as the functional ↔ Maupertuis for null curves | Maxwell’s (or linearised Einstein’s) equations on a curved background → null geodesics, as the high-frequency limit. | adds wavelength small compared with the curvature scale and the scale of the wave packet; for gravitational waves, the two-scale split of Isaacson · model Maxwell or linearised gravity on a fixed backgroundbreaks at and near caustics of the null congruence, where the approximation is non-uniform, and wherever the high-frequency condition fails, which for long-wavelength gravitational waves can be far from any caustic; in lensing, critical curves in the image plane map to caustics in the source plane under the lens equation, geometrical magnification is infinite only for an ideal point source on a caustic, and multiple images by themselves imply neither a caustic nor a divergence; Fermat’s principle in the general case requires a choice of observer, not just a metric | limited |
| Dispersive waves in fluids | Kinematic wave theory; Whitham’s averaged Lagrangian; ray tracing; conservation of wave actionWhitham 1965; 1974, Ch. 11 · Bretherton & Garrett 1968 · Lighthill 1978 | Phase θ ↔ S · ∂tθ + ω(∇θ, x, t) = 0 ↔ Hamilton–Jacobi with the dispersion relation as Hamiltonian · wavenumber k = ∇θ ↔ momentum · group velocity ∂ω/∂k ↔ velocity · rays ↔ trajectories · wave action ↔ the transported density | Linear wave equation → the phase equation and the wave-action equation, for a slowly varying wavetrain. In the nonlinear case (Whitham’s averaged Lagrangian) the dispersion relation depends on amplitude and the result is a coupled modulation system, not one closed phase equation. | adds a single wavetrain whose wavenumber and amplitude vary slowly compared with the wavelength and period; linearity, for the dictionary as stated · model surface and internal gravity waves, and any dispersive system with a local dispersion relationbreaks at caustics (the group-velocity envelope of a ship wake is one; Airy’s rainbow is the optical cousin); where wavetrains cross and the single-phase assumption fails; where the medium varies on the wavelength scale; in the nonlinear case the phase equation is no longer closed; wave breaking is outside the theory altogether | limited |
| Mathematics | First-order PDE and characteristics; Lagrangian manifolds; viscosity solutions; Hopf–Lax formulaCourant & Hilbert 1962 · Arnold 1967; 1989 (Ch. 9 and Appendices 11–12) · Crandall & Lions 1983 · Crandall, Evans & Lions 1984 · Crandall, Ishii & Lions 1992 (survey) · Evans 2010 (Secs 3.3, 10.1) | The graph of ∇S ↔ a Lagrangian submanifold of phase space, which stays smooth where its projection to configuration space loses rank · singularity of the projection ↔ caustic (a fold is the simplest type, not the only one) · Maslov index ↔ the phase index accumulated through caustics · viscosity solution ↔ a single-valued continuous weak solution, defined by inequalities against smooth test functions, and equal to the vanishing-viscosity limit where that limit exists | Both directions, as theorems, about the PDE itself: what a solution is before and after the classical solution breaks down. | adds for the Lagrangian picture, smoothness in phase space; for viscosity solutions, hypotheses on the Hamiltonian and the data under which a comparison principle holds, which is what gives uniqueness · model the equation, with no physical system behind itbreaks the two solution concepts answer different questions and neither is unconditional: the Lagrangian manifold retains the multivalued phase that wave asymptotics need; the viscosity solution is the object that minimisation problems need, and its existence and uniqueness are theorems under stated hypotheses, not properties of the equation as such. Which concept applies is decided by the application | exact |
| Optimal control | Hamilton–Jacobi–Bellman equation; dynamic programming; value function; Hamilton–Jacobi–Isaacs equationBellman 1957 · Isaacs 1965 · Fleming & Soner 2006 (monograph) | Value function V(x, t), the best achievable cost-to-go ↔ S · optimised Hamiltonian H(x, ∇V) = minu[running cost + ∇V·f(x, u)] ↔ H · optimal trajectories ↔ characteristics, where V is smooth · costate of Pontryagin’s principle ↔ ∇V along an optimal path, in the smooth case | Optimisation principle (the principle of optimality) → the PDE, as a theorem in the viscosity sense under stated hypotheses; the PDE → optimal controls by feedback from ∇V only where V is differentiable, and existence of an optimal control needs assumptions of its own. | adds a cost functional and a control set; the minimisation inside the Hamiltonian; regularity and growth hypotheses for the viscosity characterisation · model deterministic control; the stochastic case adds a second-order term and leaves the genrebreaks the correspondence is with a minimum principle rather than a stationary one, so the value function is single-valued and continuous, and may be non-differentiable where optimal paths merge; a unique value function does not mean a unique optimal path, since several controls can attain the same minimum. Same Hamilton–Jacobi form; a different question and a different solution concept | exact |
| Optimal transport | Benamou–Brenier formulation; Kantorovich potential; displacement interpolationBenamou & Brenier 2000 · Villani 2009 (monograph) | Velocity potential φ of the optimal flow ↔ S · ∂tφ + |∇φ|²/2 = 0 (their eq. 11) ↔ free-particle Hamilton–Jacobi · ∂tρ + ∇·(ρ∇φ) = 0 with ρ fixed at both ends (eqs 7–8) ↔ continuity with endpoint data · displacement interpolation Xt = (1 − t)x + tT(x) (eq. 32) ↔ straight-line characteristics | Monge–Kantorovich problem with quadratic cost ↔ minimisation of kinetic action subject to continuity and endpoint densities, as a theorem; the paper’s formal optimality conditions are that the velocity is a gradient (eq. 10) and that the potential obeys the Hamilton–Jacobi equation (eq. 11), and the variational equivalence is its Proposition 1.1. The pair of equations alone does not specify the problem. | adds quadratic cost; the action-minimisation principle with prescribed initial and final densities; in the paper’s own justification (Sec. 3), bounded, compactly supported probability densities on ℝd, with broader settings needing their own finite-cost and regularity assumptions · model an optimisation problem, with no dynamics behind itbreaks nothing within the theorem. The optimal map is the gradient of a convex function, and its monotonicity means the interpolating paths of distinct points do not meet at any intermediate time, so there is no crossing to continue past; the Hamilton–Jacobi equation holds as an equality along the optimum and as an inequality in the dual problem. The authors describe the optimum as “a pressureless potential flow”; that it has the form of the Madelung pair of Audit 02 with the quantum potential and the external potential removed is a remark about form only | exact |
| Computation (fronts) | Arrival-time (eikonal) formulation of front propagation; fast marching; level-set methodOsher & Sethian 1988 · Tsitsiklis 1995 · Sethian 1996 | Arrival time T of a monotonically advancing front ↔ S · F|∇T| = 1 ↔ eikonal with n = 1/F · the front ↔ a level set of T · upwind discretisation ↔ the viscosity solution of the continuum equation | A front advancing with positive speed F(x) → the eikonal equation for its arrival time, exactly at the continuum level; fast marching computes an approximation to its viscosity solution on a grid. | adds a speed F > 0 defined at every point and a front that never retreats, so that arrival time is single-valued; for the numerical method, a grid and a causality ordering · model a geometric evolution; the front need not be a wave of anythingbreaks the verdict is for the continuum representation; the algorithm approximates it. The arrival-time formulation records first arrivals only and so cannot represent later branches or interference, which is its design. The general level-set equation of Osher and Sethian, φt + F(x, κ)|∇φ| = 0 with curvature κ, contains second derivatives (their abstract: “Hamilton–Jacobi equations with parabolic right-hand sides”) and is outside this page’s first-order scope. Tsitsiklis reached the fast-marching algorithm from control theory | exact |
The same equation also appears for the gravitational field itself (the Einstein–Hamilton–Jacobi equation), in large-deviation theory (the Freidlin–Wentzell quasi-potential) and in the calculus of variations for fields. These are noted under extensions and not separately audited.
Historical relationships
The earliest source located for this structure is Hamilton, and the analogy that this series is built to test was part of his construction from the start. Hamilton developed the characteristic function for systems of rays in 1828 and its optical partial differential equation in the supplements that followed; when he turned to dynamics in 1834 he wrote, in the introductory remarks to the first essay, that the new method “is only another form of that idea which has already been applied to optics in the Theory of systems of rays”. The 1834 and 1835 essays contain the equations for the characteristic and principal functions; Jacobi’s 1837 note and his Königsberg lectures of 1842–43 turned Hamilton’s construction into the general theory of the complete integral that bears both names, and the account of that step on this page is taken from the secondary literature rather than from Jacobi’s text. The word “eikonal” is Bruns’s, from 1895. The derivation of the eikonal from Maxwell’s equations as a short-wavelength limit, rather than from Fermat’s principle, is credited to Sommerfeld and Runge in 1911, the second author being Iris Runge, and the idea to an oral remark of Debye’s; I have the attribution at second hand.
The quantum line runs through the analogy in the forward direction. De Broglie’s 1925 thesis identifies Fermat’s principle for the phase wave with Maupertuis’s principle for the particle (secondary sources; the thesis itself was not read). Schrödinger’s second communication of 1926 opens with a section titled “Die Hamiltonsche Analogie zwischen Mechanik und Optik”, writes the Hamilton–Jacobi equation as its equation (1′), and states that Hamilton’s principle is the expression of Huygens’ principle; the wave equation is then constructed to have the Hamilton–Jacobi equation as its short-wave limit. The WKB approximation of 1926, anticipated by Jeffreys in 1925, then recovered that limit from the finished theory; Van Vleck supplied the propagator amplitude in 1928; Keller in 1958 and Maslov in 1965 supplied what happens at caustics, and Arnold in 1967 the invariant that indexes them. The order of events matters for one thing only: it says how the connection was found. The analogy came first and the wave equation was built to satisfy it; the later semiclassical results are derivations of the classical limit under stated hypotheses, and their standing as mathematics does not depend on what motivated the equation they start from.
The later arrivals came from optimisation and analysis, and I have not established how far each was independent of the mechanical tradition, so they are listed without a claim about that. Bellman’s dynamic programming of 1957 reached the same partial differential form from the principle of optimality; the name “Hamilton–Jacobi–Bellman” is later usage, and Isaacs’s 1965 book on differential games calls its version simply the “main equation” (both attributions at second hand). Crandall and Lions in 1983 gave the equation, already known to them as the Hamilton–Jacobi equation, a theory of weak solutions that survives the breakdown of the classical solution, which is what control theory needed. Osher and Sethian in 1988 built a numerical method for moving fronts on the Hamilton–Jacobi form, Tsitsiklis in 1995 reached the fast-marching algorithm from control, and Sethian named it in 1996. Benamou and Brenier in 2000 found the equation, paired with continuity, inside the Monge–Kantorovich transport problem. This is a third pattern for the series: Audit 01 was many rediscoveries of one structure, Audit 02 was one origin with many named borrowings, and this page is an origin whose author stated the analogy himself, followed by reformulations and by independently established results, in optimisation and analysis, about what a solution means once the classical one fails.
Notes by row
Classical mechanics exact
The Hamilton–Jacobi equation is exactly equivalent to Hamilton’s equations, and the equivalence runs both ways: the characteristics of the PDE are the ODEs, and a complete integral of the PDE generates every solution of the ODEs. The “exact” verdict has two qualifications that the row states. A complete integral, with the non-degeneracy condition det(∂²S/∂q∂α) ≠ 0 that Jacobi’s theorem needs, is guaranteed locally by the method of characteristics; separation of variables is the usual way of writing one down explicitly, and the celebrated applications (action-angle variables, perturbation theory) are to integrable or near-integrable systems, but neither separability nor integrability is a condition for the local equivalence. Globally, non-integrable systems have no complete integral. And a smooth single-valued S from a given initial surface exists only until the family of trajectories folds. Neither qualification touches the equivalence itself. The equation’s other role, as the origin of the optical analogy, is Hamilton’s own, and it is recorded in the history above rather than as a claim of this row.
Geometrical optics limited
Two different correspondences share the row, and the verdict applies to the one in the direction column. Within ray optics, Fermat’s principle and the eikonal equation are exactly equivalent by the same characteristic construction as in mechanics, and this is what Hamilton had in 1828; the refractive index plays the part of √(2m(E − V)), and the optical-mechanical analogy is exact at this level. The limited verdict is for the relation between ray optics and Maxwell’s equations. Born and Wolf’s Chapter 3 derives the eikonal equation from Maxwell as the leading term when the wavelength tends to zero, with the amplitude and the medium varying slowly on that scale; the equation number is not cited because the derivation was checked at record level only. The failures are the standard ones and they are physical, not technical: at a caustic the ray amplitude diverges and the true field is finite, with an intensity that scales as a fractional power of the wavelength fixed by the type of caustic (Berry and Upstill 1980, catalogued by catastrophe theory); at an edge, ray optics predicts a sharp shadow and the field diffracts, which Keller’s geometrical theory of diffraction (1962) repairs by adding diffracted rays that still obey the eikonal. The rainbow is the everyday caustic: a fold in the family of rays through a drop, with Airy’s supernumerary bows as the wave correction.
Quantum mechanics limited
This row is where the series’ two previous audits meet. The substitution ψ = A eiS/ħ, with A positive and S real, in the scalar Schrödinger equation is exact locally on a zero-free region, with a smooth branch of the phase chosen, and gives the Madelung pair of Audit 02: the Hamilton–Jacobi equation with an extra term on the right, which is minus the quantum potential, and the continuity equation for A². The WKB approximation drops that term. The justification is not the explicit ħ², since ∇²A/A can itself grow as ħ shrinks, but the condition that the amplitude vary slowly on the local de Broglie scale, which is a statement about the solution and holds or fails region by region. Where it holds, the continuity equation becomes the transport equation for the leading amplitude, and for the semiclassical propagator the amplitude contains the square root of the absolute Van Vleck determinant, with normalisation and phase factors (Berry and Mount 1972, Sec. 7.2, checked in the external review of this page; for a general solution the transported amplitude depends on its initial data and the determinant is specific to the propagator). The verdict is limited because this is in general an asymptotic expansion in ħ, not an identity, and its failures are local and well catalogued: turning points, where the momentum vanishes, and caustics, where the leading amplitude is singular and the wavefunction is not. Classically forbidden regions are not a failure. WKB describes them with growing and decaying exponentials, ψ ∝ κ−1/2 exp(−∫κ dx/ħ) with κ = √(2m(V − E)), and only the turning points that bound them need connection formulae; the complex WKB action in such a region is a different object from the real phase of the exact decomposition, which is why the two should not be confused. The connection formulae, and Maslov’s index for the phase acquired through each caustic, are the repair; Keller’s 1958 quantisation conditions, with the caustic correction built in, are the result for integrable systems. For non-integrable systems no global complete integral exists, and the semiclassical theory takes a different form (periodic-orbit sums) that this page does not audit. What the row establishes for the series is a direction: the classical equation is recovered as a limit of the quantum one, under stated conditions, and the term that is dropped is precisely the term Audit 02 found to carry the difference between a wave and a fluid.
Acoustics limited
Ray acoustics is the eikonal equation with the sound speed in the role of the inverse refractive index and travel time in the role of the eikonal; Pierce’s textbook devotes its eighth chapter to it, including propagation in moving media, where the ray Hamiltonian acquires the flow velocity in the same way the acoustic metric of Audit 01 does. The specific forms of the moving-medium eikonal in that chapter were not checked and are not quoted. The verdict is limited for the same reason as in optics: the equation is the high-frequency limit of the wave equation, and its failures are the caustics and shadow zones that underwater and atmospheric sound channels produce routinely. Keller’s geometrical theory of diffraction provides a related framework, with diffraction coefficients and boundary conditions appropriate to the acoustic problem.
Seismology limited
Seismic ray theory is the same asymptotics applied to the elastodynamic equation, with the added feature that an isotropic solid supports two wave speeds and hence two eikonal equations, one for P waves and one for S. Červený’s monograph derives the eikonal in its second chapter and says in its preface that the method is “only approximate”, being the high-frequency asymptotic solution; its later chapters, on Gaussian beams and the Maslov–Chapman method, are the field’s own repair for caustics, and their existence is the best evidence that the row is limited rather than exact. Travel-time tomography, the inversion of arrival times for Earth structure, is the eikonal equation used as a measuring instrument.
General relativity exact limited
Two rows, because two correspondences. For a test particle on a fixed background, the geodesic equations are the characteristics of the Hamilton–Jacobi equation gμν∂μS ∂νS = −m², and the equivalence is exact as in mechanics. Carter’s 1968 paper is the row’s earliest located source for the decisive application: the Kerr metric has only three obvious constants of motion, and, in the abstract’s words, “a fourth one turns out to be obtainable from the unexpected separability of the Hamilton–Jacobi equation”. That separability is what makes orbits in Kerr integrable, and it was found through the PDE, not the ODEs. The second row is the geometrical-optics limit of a wave on a curved background. Ehlers in 1967 treated the passage from wave to geometrical optics in general relativity; Isaacson in 1968 did the same for gravitational waves, showing “in exact analogy to light waves” that they travel on null geodesics of the background at high frequency; Perlick’s review gives the null eikonal as its equation (7). The verdict is limited because the wavelength must be small compared with the curvature scale, and the failures are the caustics of the null congruence. In lensing language, critical curves in the image plane map to caustics in the source plane under the lens equation; a source on a caustic can also have images away from the critical curve. An ideal point source exactly on a caustic has infinite geometrical magnification; a source of finite size regularises that within geometrical optics, and wave optics regularises it in any case. Multiple images by themselves do not imply a caustic, and Perlick’s review shows a wavefront that crosses itself without one (Sec. 2.2), so geometrical optics remains adequate across much of strong lensing. It becomes non-uniform in neighbourhoods of caustics, and it fails wherever the high-frequency condition does, which for gravitational waves of long wavelength can be well away from any caustic. Fermat’s principle in the general case (Kovner 1990; Perlick 1990) is a statement about arrival time at a chosen observer, which is why it is listed as a dictionary entry and not as a verdict.
Dispersive waves in fluids limited
For a linear wavetrain, Whitham’s theory is the Hamilton–Jacobi form with the dispersion relation ω(k, x, t) as Hamiltonian and the phase as S: the wavenumber is the gradient of the phase, the group velocity is the Hamiltonian velocity, and the rays are the characteristics. What is transported along the rays is wave action, the energy divided by the intrinsic frequency, a result of Bretherton and Garrett’s 1968 paper for moving media. The nonlinear extension, which is what Whitham’s 1965 averaged Lagrangian was built for, is different in kind: the dispersion relation then depends on the amplitude and on any mean flow, and the phase equation is no longer closed but part of a coupled modulation system. The dictionary entry is for the linear case. The verified reading here is at record level: the 1965 paper’s abstract was not accessible, and the equations are quoted as standard textbook content (Whitham’s Chapter 11; Lighthill’s treatment within his chapter on internal waves) rather than from a checked page. The verdict is limited because the theory assumes a single slowly varying wavetrain, and it fails where wavetrains cross, at caustics such as the edge of a ship’s wake, and where the medium changes on the wavelength scale. Nonlinear wave breaking is outside it entirely.
Mathematics exact
The equation’s own theory has two answers to the question of what a solution is once the classical one breaks down, and the ledger records both because the physical rows use different ones. The first keeps the multivalued phase: the solution ∇S is regarded as a surface in phase space, a Lagrangian submanifold, which remains smooth while its projection to configuration space loses rank; a caustic is a singularity of that projection, of which a fold is the simplest type, the phase becomes multivalued beyond it, and Maslov’s index records the phase a wave picks up in passing through. This is Arnold’s 1967 characteristic class, set out in the appendices to his textbook, and it is what semiclassical physics needs. The second is a weak solution: the viscosity solution of Crandall and Lions (1983; with Evans, 1984; surveyed by Crandall, Ishii and Lions 1992) is defined by inequalities tested against smooth functions touching the candidate from above and below. The vanishing-viscosity limit, adding a small diffusion and letting it go to zero, is the construction that motivated the name and produces such solutions where it converges, but it is not the definition. Existence and uniqueness are theorems under hypotheses on the Hamiltonian and the data, through a comparison principle, and not properties of the equation as such. The viscosity solution is single-valued and continuous, and may be non-differentiable; it is the right object whenever the underlying problem is a minimisation, which is why control theory and front propagation use it. Evans’s textbook presents both the Hopf–Lax formula and the viscosity theory. The verdict is exact because these are theorems about the equation. The audit’s point is that the equation does not itself select a solution concept; the problem it is asked to solve does, and different problems need different additional information.
Optimal control exact
Bellman’s dynamic programming turns the principle of optimality, that the tail of an optimal path is itself optimal, into a partial differential equation for the value function, the least cost achievable from each state. That equation is the Hamilton–Jacobi form with the Hamiltonian obtained by minimising over the control. The row is exact in the sense that, under stated hypotheses, the value function is characterised as the viscosity solution of the equation; Fleming and Soner’s monograph is the standard account, and the reason the viscosity sense is needed is that value functions are typically not differentiable where optimal paths merge. Two further statements hold only in the smooth case and are stated that way in the row: that optimal controls can be recovered by feedback from ∇V, and that Pontryagin’s costate equals ∇V along an optimal path. Existence of an optimal control at all needs assumptions of its own, and a unique value function is compatible with several optimal paths attaining the same minimum. The name “Hamilton–Jacobi–Bellman” is later than Bellman’s book, and Isaacs’s 1965 differential-games version was called by him the “main equation”; both attributions are at second hand. The row’s importance for the series is that it is the same Hamilton–Jacobi form asked a different question. Mechanics asks for a stationary action and needs the multivalued phase; control asks for a minimum and needs the single-valued value function. A physicist who meets Bellman’s equation and a control engineer who meets Hamilton’s are looking at the same Hamilton–Jacobi form with different Hamiltonians and different notions of what its solution is.
Optimal transport exact
Benamou and Brenier showed in 2000 that moving one probability density onto another at least quadratic cost is equivalent to a fluid-mechanics problem: minimise the kinetic action of a density carried by a velocity field, subject to the continuity equation and to the initial and final densities (their eqs 7–8). The optimality conditions are that the velocity is a gradient (eq. 10) and that its potential obeys the free-particle Hamilton–Jacobi equation (eq. 11), which they describe as “a pressureless potential flow”; the interpolating paths are straight lines (eq. 32). The verdict is exact because the equivalence is a theorem, with one qualification the row states: the pair of equations does not by itself specify the problem, which needs the endpoint data and the minimisation principle, and in the dual formulation the Hamilton–Jacobi relation appears as an inequality with equality along the optimum. An earlier draft of this page said the density “concentrates where characteristics would cross”, which is wrong for this problem: the optimal map is the gradient of a convex function, and the interpolating paths of distinct points never meet at an intermediate time. That the equations have the form of the Madelung pair of Audit 02 with the quantum potential and the external potential removed is a remark about form only: nothing in optimal transport is a wave.
Computation exact
Three things are kept apart in this row. First, the continuum representation: a front that advances everywhere with positive speed F has a single-valued arrival time T obeying the eikonal equation F|∇T| = 1, and that representation is exact; it is the row’s verdict. Second, the numerical method: Sethian’s fast-marching method of 1996 computes an approximation to the viscosity solution of that equation on a grid, in one pass, by ordering the points by arrival time; Tsitsiklis had reached a Dijkstra-like algorithm for the same equation from optimal control in 1995, and Sethian’s own account credits him with that priority. The approximation is not part of the verdict. Third, the general level-set method of Osher and Sethian, which embeds a moving front as the zero level of a function and evolves it by an equation that the authors’ abstract describes as resembling “Hamilton–Jacobi equations with parabolic right-hand sides”: when the speed depends on curvature the equation contains second derivatives, and it is outside the first-order scope of this page. The first-arrival restriction belongs to the arrival-time formulation, not to level-set evolution in general. What the arrival-time formulation contributes to the audit is a demonstration in code of what the mathematics row says: a formulation that records only first arrivals cannot represent later branches or interference, by design.
The dictionary
Terms that name the same object across rows, wherever the ray or classical solution is single-valued, except where an entry says otherwise.
- The scalar
- S · action, principal function (mechanics) · eikonal, characteristic function, optical path length (optics) · ħ times the phase, S = ħθ (quantum mechanics) · travel time (acoustics, seismology) · phase θ (wavetrains) · value function, cost-to-go (control) · velocity potential φ (transport), related to the Kantorovich potential f by φ0 = −f up to a constant on a unit time interval, with the convention f(x) + g(y) ≤ ½|x − y|² and T(x) = x − ∇f(x) · arrival time (computation)
- Its gradient
- ∇S · momentum · n × ray direction · ħk, the quantum action gradient, where k = ∇θ is the wave vector of the dimensionless phase · slowness vector · costate (Pontryagin), in the smooth case · transport velocity
- The Hamiltonian
- H(q, ∇S) · mechanical Hamiltonian · |∇𝒮|² − n², or ω = c|k|/n for light · dispersion relation ω(k, x) · ½gμνpμpν for geodesics · the control-minimised Hamiltonian · |∇φ|²/2 for free transport
- Characteristics
- trajectories · rays · null geodesics · group-velocity rays · optimal paths, where the value function is smooth · transport paths · the curves along which the PDE is a set of ODEs; in every row, the same construction with a different Hamiltonian
- Variational principle
- Maupertuis–Jacobi least action (fixed energy) · Fermat’s least time · geodesic extremality · Whitham’s averaged Lagrangian · Bellman’s principle of optimality · Monge’s least cost. The first three are stationary principles, under which several extremals may pass through a point; the last two are minimum principles, whose value function is single-valued even where several optimal paths attain the minimum
- Transport equation
- ∂tA² + ∇·(A²∇S/m) = 0 · the continuity equation of Audit 02, exact; the transport equation for the leading amplitude, in the asymptotic rows · the square root of the absolute Van Vleck determinant, in the propagator amplitude · conservation of wave action · amplitude along a ray tube · the continuity equation of optimal transport, with its endpoint data
- Caustic
- envelope of a ray family · focal surface · conjugate points of geodesics · a singularity of the projection of the Lagrangian manifold, of which the fold is the simplest · turning point in one dimension · in gravitational lensing, the source-plane caustic to which the image-plane critical curve maps under the lens equation · where the leading ray amplitude is generally singular and a single smooth phase ceases to suffice · not the same as two rays crossing, which can happen without any singularity
- Beyond the caustic
- different solution concepts for different problems · multivalued phase: Lagrangian manifold, Maslov index, the interference of branches (optics, quantum mechanics, wavetrains, lensing) · viscosity solution: single-valued, continuous, possibly non-differentiable, defined by test-function inequalities (control, arrival-time fronts) · no crossing arises: quadratic optimal transport, by monotonicity of the optimal map. The equation does not choose; the problem does
- The dropped term
- (ħ²/2m)∇²A/A = −Q, minus the quantum potential of Audit 02, in the quantum row · in the other wave rows the corrections to the phase equation take their own forms, involving amplitude, polarisation or mode coupling, and are not in general this expression · absent by hypothesis in mechanics, control and transport, where there is no wavelength
Verdict
Three claims are usually run together. The ledger supports the first as a set of theorems with stated conditions, supports the second as an asymptotic statement with named failures, and supports the third only in a narrower form than it is usually put.
First: the Hamilton–Jacobi form is a shared mathematical framework across mechanics, ray theory and optimisation, and its role differs between them. In mechanics and geodesic motion it describes the Hamiltonian characteristics, exactly and locally. In control it characterises the value function, under specified hypotheses and in the viscosity sense. In quadratic optimal transport it appears in the optimality conditions of an action-minimisation problem that also needs continuity and endpoint data. The theory of the equation itself, in its Lagrangian-manifold and viscosity-solution forms, is a body of theorems, each with its hypotheses. In these rows there is no approximation and the word “analogy” does not arise; the Hamilton–Jacobi form is the same, the Hamiltonians differ, and so does the question asked. The single most useful thing the ledger records is that a stationary principle and a minimum principle call for different solution concepts once the classical solution fails.
Second: for the wave systems audited here, light, sound, seismic waves, linear water wavetrains, gravitational waves and the quantum wavefunction, the same form is the phase equation of a short-wavelength asymptotics, for a suitable high-frequency branch, with further equations governing amplitude and, where relevant, polarisation and mode conversion. All of these lose the single smooth phase at the same kind of place, a caustic, where the ray family’s projection becomes singular, the leading amplitude is generally singular, and the wave gives a finite, wavelength-dependent answer. In quantum mechanics the dropped term is exactly the quantum potential of Audit 02; in the other rows the corrections take their own forms. Ray trajectories and travel times probe the properties encoded in the ray Hamiltonian, as seismic tomography demonstrates; the approximation’s scale assumptions determine its domain of validity.
Third: that the optical-mechanical analogy shows mechanics to be “really” wave mechanics, or that either theory has been derived from the other without restriction. What is established is narrower and firmer. The analogy was Hamilton’s own reading of his equation and the heuristic from which de Broglie and Schrödinger built the wave equation; that is history, and it explains how the connection was found. Separately, semiclassical analysis derives the classical Hamilton–Jacobi dynamics from the Schrödinger equation under stated conditions, and that is mathematics, whose standing does not depend on the history. What is not established is a derivation of every aspect of classical physics from quantum mechanics, or the reverse, or any inference from the analogy to a physical wave or medium beneath mechanics. These correspondences establish precise relationships under stated conditions; their origins explain how some were found, and neither invalidate the controlled derivations nor identify an underlying physical medium.
What this means for a hydrodynamic theory of quantum mechanics or gravity. This is the calibration audit, and it calibrates in two directions. Toward Audit 01: for a medium satisfying Gordon’s assumptions, light rays follow the null geodesics of the associated optical metric. That establishes a correspondence for ray propagation, and no more: it is for light, not for a massive test particle; ray trajectories alone leave the metric’s conformal factor undetermined (Barceló, Liberati and Visser 2011, Sec. 3.1.1); and it says nothing about gravitational dynamics or about agreement between all observables. A proposed medium theory has therefore to be tested by comparing its complete predictions with those of a specified gravitational theory, wherever they differ. Toward Audit 02: the classical Hamilton–Jacobi equation plus continuity is the Madelung pair with Q removed, so any hydrodynamic reading of quantum mechanics must reproduce not only the classical rays but the interference beyond the caustics, and that interference depends on the relative phases and amplitudes of the branches and their dynamics, with the winding conditions of Audit 02 as one necessary ingredient among several. The debt is the same as before; this page has only located where it comes due.
What this page does not claim
It does not claim that ray optics, ray acoustics or seismic ray theory are exact; they are leading-order asymptotics, and their own literatures say so. It does not claim that semiclassical analysis derives all of classical mechanics from quantum mechanics; it recovers the classical Hamilton–Jacobi dynamics as a limit under stated conditions, with local failures at turning points and caustics. Nor does it claim the reverse, that the history of the analogy weakens those derivations. It does not claim that the optical-mechanical analogy is evidence for a wave interpretation of anything; it records that the analogy was the heuristic from which the wave equation was built. It does not claim that Bellman’s equation or the Benamou–Brenier system have physical content; they are the same Hamilton–Jacobi form asked an optimisation question, and the identification of the transport pair with the Madelung pair is a remark about form. It does not claim that every wave equation has an eikonal limit of the same kind, or that the correction to it is always the quantum potential; the wave rows are the ones audited, and their corrections differ. It does not claim that a caustic is wherever two rays cross; it is where the ray family’s projection becomes singular. It does not claim that a global complete integral exists for non-integrable systems; it does not, and that is a statement about one global generating function, not about Hamilton–Jacobi or phase-space methods as such. It does not audit the semiclassical theory of chaotic systems, the Einstein–Hamilton–Jacobi equation, large-deviation theory or the field-theoretic Hamilton–Jacobi equation, all of which are noted below. And it does not resolve the question of which solution concept applies beyond a caustic; it records that the equation does not decide and the problem does.
Extensions of the structure, and what is known about each
Every entry below was searched before being written.
Next in the series
Audit 10 is frozen-in transport: the statement that a field is carried by a flow as if painted on it, which appears as Alfvén’s theorem in magnetohydrodynamics, Kelvin’s circulation theorem and Helmholtz’s vortex theorems in fluids, and the Lie-transport of forms in geometry. It is the first audit in the series whose structure is a conservation law rather than an equation of motion.
References
Bibliographic details for every entry (authors, title, year, journal, volume, page) were verified against the publisher’s record or an equivalent index. The flag beside each says how far the content itself was consulted. Where a claim in the text rests on a specific passage, the section, equation or page is given in the text.
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Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics, 2nd ed. Springer (GTM 60). Ch. 9 (Sec. 47, Hamilton–Jacobi method); Appendices 11 (short-wave asymptotics) and 12 (Lagrangian singularities). record (table of contents)
Barceló, C., Liberati, S. & Visser, M. (2011). Analogue gravity. Living Rev. Relativ. 14, 3. secondary (Sec. 3.1.1 on the conformal factor, as cited in review)
Bellman, R. (1957). Dynamic Programming. Princeton University Press. record
Benamou, J.-D. & Brenier, Y. (2000). A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84, 375–393. full text (eqs 7–8, 10–11, 32)
Berry, M. V. & Mount, K. E. (1972). Semiclassical approximations in wave mechanics. Rep. Prog. Phys. 35, 315–397. secondary (Sec. 3.1 on forbidden regions and Sec. 7.2 on the propagator amplitude, checked in the external review of this page, not by me)
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Born, M. & Wolf, E. (1999). Principles of Optics, 7th (expanded) ed. Cambridge University Press. Ch. 3, Foundations of geometrical optics. record
Bretherton, F. P. & Garrett, C. J. R. (1968). Wavetrains in inhomogeneous moving media. Proc. R. Soc. Lond. A 302, 529–554. record
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Carter, B. (1968). Global structure of the Kerr family of gravitational fields. Phys. Rev. 174, 1559–1571. abstract
Červený, V. (2001). Seismic Ray Theory. Cambridge University Press. full text (preface and introduction; Sec. 2.4 heading; Sec. 5.3 on reflection and transmission, from the table of contents)
Courant, R. & Hilbert, D. (1962). Methods of Mathematical Physics, Vol. II: Partial Differential Equations. Interscience. record
Crandall, M. G. & Lions, P.-L. (1983). Viscosity solutions of Hamilton–Jacobi equations. Trans. Amer. Math. Soc. 277, 1–42. record
Crandall, M. G., Evans, L. C. & Lions, P.-L. (1984). Some properties of viscosity solutions of Hamilton–Jacobi equations. Trans. Amer. Math. Soc. 282, 487–502. record
Crandall, M. G., Ishii, H. & Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27, 1–67. full text (Sec. 2, Definition 2.2 and Remark 2.3)
de Broglie, L. (1925). Recherches sur la théorie des quanta. Ann. Phys. (Paris) 10e sér., 3, 22–128. secondary
Ehlers, J. (1967). Zum Übergang von der Wellenoptik zur geometrischen Optik in der allgemeinen Relativitätstheorie. Z. Naturforsch. 22a, 1328–1332. record
Evans, L. C. (2010). Partial Differential Equations, 2nd ed. American Mathematical Society (GSM 19). Secs 3.3 and 10.1. secondary (table of contents)
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Goldstein, H. (1980). Classical Mechanics, 2nd ed. Addison-Wesley. Sec. 10-8, Hamilton–Jacobi theory, geometrical optics and wave mechanics. secondary (the section is absent from the 3rd edition of 2002)
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Hamilton, W. R. (1834). On a general method in dynamics. Phil. Trans. R. Soc. 124, 247–308. full text (introductory remarks; equations F and G)
Hamilton, W. R. (1835). Second essay on a general method in dynamics. Phil. Trans. R. Soc. 125, 95–144. record
Isaacs, R. (1965). Differential Games. Wiley. secondary
Isaacson, R. A. (1968). Gravitational radiation in the limit of high frequency. I. The linear approximation and geometrical optics. Phys. Rev. 166, 1263–1271. abstract
Jacobi, C. G. J. (1837). Note sur l’intégration des équations différentielles de la dynamique. C. R. Acad. Sci. Paris 5, 61–67. secondary
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Change log
0.4 · 13 Sep 2026 Final local edits after a third external review: the remaining “medium invisible in the ray limit” sentence in the verdict replaced by a statement of what rays probe and of the approximation’s scale assumptions; Keller’s geometrical theory of diffraction described as a related framework with problem-specific coefficients rather than as applying unchanged; the seismology row’s interface statement corrected, since interfaces call for reflection, transmission and mode-conversion matching rather than marking a failure of ray theory. Verified items moved out of “Still unverified”. Reviewer’s assessment: publishable with the declared source limitations. Published.
0.3 · 13 Sep 2026 Correction pass after a second external review. Propagator amplitude stated as the square root of the absolute Van Vleck determinant. Lensing: critical curves map to caustics under the lens equation; infinite magnification confined to an ideal point source; “fails only at the caustics” replaced by non-uniformity near caustics and failure wherever the high-frequency condition fails, which for gravitational waves can be far from a caustic. Gordon paragraph rewritten: a ray-propagation correspondence for light, conformal factor undetermined by rays, no claim about where a medium theory’s predictions must differ. Exact decomposition stated locally on a zero-free region with a chosen phase branch; “asymptotic, not convergent” softened; “kinks” replaced by possible non-differentiability; non-integrability confined to the global complete integral; Kantorovich and velocity potentials related explicitly; k and ħk distinguished. Benamou–Brenier assumptions made concrete and Proposition 1.1 named. Crandall, Ishii and Lions Definition 2.2 read directly; Berry and Mount provenance recorded as the reviewer’s. Barceló, Liberati and Visser 2011 added. Still draft.
0.2 · 13 Sep 2026 Revised after external review. Equation box corrected (the Hamiltonian with V stated; the exact decomposition scoped to real A > 0 and real S away from zeros; the right-hand side identified as −Q); quantum dictionary corrected (S = ħ × phase; the Van Vleck determinant confined to the propagator); classically forbidden regions removed from the list of WKB failures and the neglect of the quantum term stated as a condition on the solution rather than on ħ. Caustics distinguished from ray crossings throughout, with the lensing row corrected (source-plane caustics, image-plane critical curves, multiple images not implying divergence). Optimal-transport row rewritten: the pair alone does not specify the problem; paths do not cross by monotonicity of the optimal map; “concentrates where characteristics cross” withdrawn; equation numbers added. Mathematics and control rows rewritten: viscosity solutions defined by test functions, existence and uniqueness conditional on a comparison principle, “keeps one branch” withdrawn, feedback and costate statements confined to the smooth case. Computation row narrowed to the arrival-time eikonal, with the curvature-dependent level-set equation noted as second order. Complete integrals: non-degeneracy condition stated, separability no longer implied as a requirement. “Every wave equation” replaced by the audited scope; nonlinear Whitham theory separated from the linear dictionary; “the same equation” read as “the same Hamilton–Jacobi form”. Verdict and final note rewritten: the semiclassical limit is a derivation under conditions regardless of the history; rays measure a medium’s large-scale structure; Gordon’s metric is for light; winding conditions are one ingredient of interference. “One origin and one author” and “three independent arrivals” withdrawn. Fraser and Nakane 2023 and Crandall, Ishii and Lions 1992 added. Still draft.
0.1 · 12 Sep 2026 First draft on the series template. Sources verified before drafting; four pinpoints corrected during verification (Goldstein’s optics section is in the 2nd edition, not the 3rd; Arnold’s caustic material is in Appendices 11–12, not Sec. 46; Sommerfeld’s co-author was Iris Runge; Peres 1962 did not name the Einstein–Hamilton–Jacobi equation) and one claim withdrawn before it was written (that Lighthill has a chapter on ray theory).
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