Hamilton-Jacobi Equation

In the winter of 1925–26, Erwin Schrödinger went looking for a wave equation and started from a piece of pure classical mechanics: the Hamilton-Jacobi equation. That is how central this object is — it is simultaneously the endpoint of the canonical-transformation program and the doorstep of quantum theory. The idea behind it is the boldest move in mechanics. Instead of integrating $2n$ coupled equations of motion, demand a canonical transformation so radical that the new Hamiltonian vanishes identically. In the new coordinates nothing moves: every $Q_i$ and every $P_i$ is frozen. All the dynamics gets compressed into the transformation itself, and the condition that such a transformation exist is a single first-order partial differential equation for its generating function $S$ — the Hamilton-Jacobi equation. The audacity pays twice. Practically, a complete integral $S(q,\alpha,t)$ turns the entire solution of the mechanics problem into differentiation and algebra: the $n$ constants $\alpha_i$ are the new momenta, their conjugates $\beta_i=\partial S/\partial\alpha_i$ are the other $n$ constants, and no further integration is ever needed. Conceptually, the level surfaces $S=\text{const}$ propagate through configuration space like wavefronts, with trajectories threading them as rays — Hamilton's optical-mechanical analogy, the very bridge Schrödinger crossed. You arrive here with the type-2 generating machinery and Hamilton's principal function already in hand; this article welds them into the equation itself, proves the recipe extracts the true motion, and solves the harmonic oscillator without ever touching its equation of motion.

Key takeaways

The Hamilton-Jacobi (HJ) equation is the single first-order nonlinear PDE $H\big(q,\frac{\partial S}{\partial q},t\big)+\frac{\partial S}{\partial t}=0$ for a generating function $S(q,t)$ that transforms the dynamics into one in which all new coordinates and momenta are constants.
You need a complete integral — a solution carrying $n$ nontrivial constants $\alpha_i$ with $\det(\partial^2S/\partial q_i\partial\alpha_j)\neq0$ — not the general solution; the constants become the new momenta.
Jacobi's theorem guarantees the recipe: setting $\beta_i=\partial S/\partial\alpha_i$ and $p_i=\partial S/\partial q_i$ reproduces exactly the trajectories of Hamilton's equations, by algebra alone.
Along the actual motion $S$ is the classical action $\int L\,\mathrm{d}t$ ; its level surfaces move through configuration space as wavefronts with trajectories as rays.
When $H$ is time-independent, $S=W(q,\alpha)-Et$ splits off the time; the reduced equation for Hamilton's characteristic function $W$ and its separable solutions are the gateway to action-angle variables and integrability.
Key idea: find a canonical transformation that makes the new Hamiltonian zero. The generating function $S$ that does this satisfies $H(q,\partial S/\partial q,t)+\partial S/\partial t=0$ ; its solution turns the equations of motion into trivial statements that the new coordinates and momenta are constant.

Where this lives today

The HJ equation never retired; it changed industries. Its modern descendant, the Hamilton-Jacobi-Bellman equation of optimal control, is the same "one function generates all trajectories" idea with the action replaced by a cost-to-go: the value function of an optimal-control problem obeys a first-order nonlinear PDE that is formally an HJ equation with a minimization inside the Hamiltonian. Every trajectory-optimization stack — rocket-landing guidance, robot motion planning, autonomous-vehicle planners, and the continuous-time limits of reinforcement learning — is solving a Hamilton-Jacobi problem, and the 2020s wave of neural PDE solvers has made "learning the value function" synonymous with learning $S$ . Level-set methods in computational geometry propagate interfaces with HJ equations too: the same wavefronts, now tracking tumor boundaries in medical imaging or flame fronts in engines. In physics proper, the equation remains the semiclassical workhorse. Attosecond spectroscopy and strong-field ionization are analyzed with trajectories carrying eikonal phases $e^{iS/\hbar}$ ; gravitational-wave source modeling uses HJ separability of geodesics in the Kerr geometry (Carter's constant is a separation constant of exactly the kind this chapter studies); and in 2026 the semiclassical propagators inside molecular-dynamics codes still evaluate $S$ along classical paths to interpolate quantum amplitudes between them.

The physical picture

Picture the function $S(q,t)$ painted over configuration space, and watch its level surfaces $S=\text{const}$ . They form a family of moving fronts, and the momentum at any point is the gradient $p=\partial S/\partial q$ — perpendicular to the local front. Particle trajectories are the rays that pierce the fronts at right angles, exactly as light rays pierce optical wavefronts. The HJ equation is nothing but the propagation law of these mechanical wavefronts. This is not a metaphor bolted on afterwards: Hamilton derived the mechanics and the optics from the same variational structure in the 1830s, and Schrödinger ran the analogy in reverse to guess what "mechanical wave optics" should look like.

Figure

Wavefronts of constant S sweep through configuration space and trajectories thread them as perpendicular rays

The second picture is bookkeeping. Any mechanics problem hides $2n$ constants of integration — the initial conditions. Hamilton-Jacobi theory is an accounting scheme that makes all $2n$ explicit coordinates of a frozen phase space: choosing a complete integral packages $n$ of them as the new momenta $\alpha_i$ , and differentiation produces the other $n$ as $\beta_i=\partial S/\partial\alpha_i$ . The flow that looks complicated in $(q,p)$ becomes a single motionless point in $(\beta,\alpha)$ :

Figure

The canonical transformation generated by S freezes the flow — swirling phase curves become a single stationary point

Be honest about when this wins. Trading $2n$ ordinary differential equations for one nonlinear PDE is, in general, a bad trade — PDEs are harder. The method earns its keep in the cases where the PDE falls apart additively, $S=\sum_k S_k(q_k)$ , each piece solvable by a one-dimensional integral. That separability is the hidden hypothesis behind every HJ success story, is governed by its own theory (the Staeckel conditions, next in this chapter), and is far from generic — it is, in fact, the classical definition of an integrable system.

The equation and its machinery

The derivation is a three-line argument sitting on the type-2 generating machinery you already own. Recall from the canon of generating-function theory that a type-2 function $F_2(q,P,t)$ generates a canonical transformation through $p_i=\partial F_2/\partial q_i$ , $Q_i=\partial F_2/\partial P_i$ , and transforms the Hamiltonian by $K=H+\partial F_2/\partial t$ . Write $S$ for the $F_2$ we seek and demand the most extreme simplification imaginable — $K\equiv0$ . Substituting the generating relations into that demand yields the master equation:

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 \qquad (1)

where: $H$ — the Hamiltonian, with every momentum $p_i$ replaced by $\partial S/\partial q_i$ ; $S(q,t)$ — Hamilton's principal function, the unknown. Read its anatomy: one equation, first order, in the $n+1$ variables $(q,t)$ , and nonlinear whenever $H$ is quadratic in the momenta — for $H=p^2/2m+V$ it reads $\frac{1}{2m}(\partial S/\partial q)^2+V+\partial S/\partial t=0$ , with the characteristic squared gradient of an eikonal equation. Nothing about the system was assumed: every Hamiltonian has an HJ equation. What is special is when it can be solved. What counts as "solved" is the next subtlety, and it is where most first encounters go wrong. A first-order PDE has a general solution involving an arbitrary function — useless here. What the method needs is a complete integral:

S=S(q_1,\dots,q_n,\alpha_1,\dots,\alpha_n,t)+A,\qquad \det\!\left(\frac{\partial^2 S}{\partial q_i\,\partial\alpha_j}\right)\neq 0 \qquad (2)

where: $\alpha_1,\dots,\alpha_n$ — $n$ independent nontrivial constants of integration; $A$ — an additive constant, physically irrelevant since only derivatives of $S$ enter the theory. The determinant condition is not pedantry: it certifies that the $\alpha_i$ genuinely deform the solution in $n$ independent directions, and it is exactly the invertibility you need in the next step. The $\alpha_i$ are then identified with the new momenta $P_i$ — a free choice, made once, that fixes the transformation. With a complete integral in hand, the motion is extracted with no further integration:

\beta_i=\frac{\partial S}{\partial \alpha_i},\qquad p_i=\frac{\partial S}{\partial q_i} \qquad (3)

where: $\beta_i$ — the new constant coordinates $Q_i$ , fixed by initial conditions along with the $\alpha_i$ . The first set of $n$ equations contains $t$ explicitly and — thanks to the determinant condition in (2) — can be inverted for $q_i(t;\alpha,\beta)$ ; the second set then evaluates the momenta along the way. That this algebra reproduces the true dynamics is Jacobi's theorem, proved below; it is the license for the whole enterprise. Two structural facts complete the toolkit. First, along the actual motion the principal function is the classical action: $\mathrm{d}S/\mathrm{d}t=\sum_i p_i\dot q_i-H=L$ , so $S=\int L\,\mathrm{d}t$ evaluated on the trajectory — the full story of what that means lives in the companion article on Hamilton's principal function. Second, whenever $H$ has no explicit time dependence, energy is conserved and the time variable splits off linearly:

S(q,\alpha,t)=W(q,\alpha)-Et,\qquad H\!\left(q,\frac{\partial W}{\partial q}\right)=E \qquad (4)

where: $W(q,\alpha)$ — Hamilton's characteristic function; $E$ — the conserved energy, serving as one of the $\alpha$ 's. The reduced, time-free PDE for $W$ is the time-independent HJ equation, and it is the workhorse: separation of variables, action variables $J_i=\frac{1}{2\pi}\oint(\partial W/\partial q_i)\,\mathrm{d}q_i$ , and the frequencies $\omega_i=\partial H/\partial J_i$ all grow out of it. Each of those has its own article later in this chapter; here we only note that (4) is where they plug in.

Which equation answers which question

you want	use	notes
the master PDE	$H(q,\partial S/\partial q,t)+\partial_t S=0$ , eq. (1)	valid for any $H$ , time-dependent or not
what "a solution" must mean	complete integral, eq. (2)	$n$ constants + non-degeneracy; NOT the general solution
the trajectory	$\beta_i=\partial S/\partial\alpha_i$ , invert for $q(t)$ , eq. (3)	pure algebra; Jacobi's theorem is the warrant
the momenta	$p_i=\partial S/\partial q_i$ , eq. (3)	evaluate along $q(t)$
conserved-energy systems	$S=W-Et$ , eq. (4)	hands off to the time-independent HJ equation
a periodic degree of freedom	$J_i=\frac{1}{2\pi}\oint p_i\,\mathrm{d}q_i$	hands off to action-angle variables

Derivation AThe Hamilton-Jacobi equation from a vanishing new Hamiltonian

We derive equation (1) by demanding that a type-2 generating function produce a new Hamiltonian that is identically zero. The strategy is minimal: quote the three transformation rules that the type-2 machinery already gives us, impose $K=0$ , and substitute. The one point requiring genuine care is what the demand buys: $K=0$ must hold as an identity in the new variables, which is what freezes every $Q_i$ and $P_i$ — a weaker demand (like $K$ constant) freezes fewer things and leads to the characteristic-function variant instead.

Step 1. Let $S(q,P,t)$ be a type-2 generating function. From the canon of generating-function theory, it defines a canonical transformation through

p_i=\frac{\partial S}{\partial q_i},\qquad Q_i=\frac{\partial S}{\partial P_i},\qquad K=H+\frac{\partial S}{\partial t}

Why type 2: we want the new momenta to end up being the constants $\alpha_i$ , so the generating function should have the old coordinates and NEW momenta as its natural arguments — exactly the type-2 signature.

Step 2. Impose the design goal $K=0$ . Hamilton's equations in the new variables then read

\dot Q_i=\frac{\partial K}{\partial P_i}=0,\qquad \dot P_i=-\frac{\partial K}{\partial Q_i}=0

Why this is the boldest possible choice: every canonical variable is now a constant of the motion. The dynamics has not disappeared — it has been shoved entirely into the time dependence of the transformation generated by $S$ .

Step 3. Write out $K=0$ using the first and third rules of Step 1. Since $K=H(q,p,t)+\partial S/\partial t$ and every $p_i$ inside $H$ equals $\partial S/\partial q_i$ ,

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

Why this is now a PDE for $S$ alone: the momenta have been eliminated in favor of derivatives of the unknown function — one scalar equation in the $n+1$ independent variables $(q_1,\dots,q_n,t)$ .

Step 4. Count constants to see what a useful solution must carry. A first-order PDE in $n+1$ variables admits solutions with $n+1$ constants of integration; one is purely additive (only derivatives of $S$ appear anywhere), leaving $n$ meaningful constants $\alpha_i$ — precisely enough to serve as the new momenta $P_i$ . The non-degeneracy $\det(\partial^2S/\partial q_i\partial\alpha_j)\neq0$ makes that identification legitimate. Note: setting instead $K=E$ constant for conservative systems yields the characteristic-function route $S=W-Et$ , confirming the two standard forms are the same demand at different strengths.

Go deeper · Derivation B · 1 minJacobi's theorem — the algebraic recipe really solves the dynamics

Derivation BJacobi's theorem — the algebraic recipe really solves the dynamics

The recipe of equation (3) makes a strong claim: differentiate a complete integral, set the derivatives equal to constants, invert — and the functions $q(t)$ , $p(t)$ you obtain satisfy Hamilton's equations exactly. We prove it directly. The strategy is to differentiate the defining relations with respect to time along the candidate motion and against the HJ equation, then use the non-degeneracy condition to cancel the resulting linear systems. Watch the determinant condition do real work in both halves — without it the argument (and the method) genuinely fails.

Step 1. Differentiate the HJ equation $H(q,\partial S/\partial q,t)+\partial_t S=0$ — an identity in $(q,\alpha,t)$ — with respect to $\alpha_j$ at fixed $q,t$ :

\sum_i \frac{\partial H}{\partial p_i}\,\frac{\partial^2 S}{\partial q_i\,\partial\alpha_j}+\frac{\partial^2 S}{\partial t\,\partial\alpha_j}=0

Why $\alpha_j$ enters only through $S$ : the Hamiltonian has no idea the constants exist; they live inside the solution, so the chain rule routes every $\alpha_j$ -dependence through the second derivatives of $S$ .

Step 2. Differentiate the candidate relation $\beta_j=\partial S/\partial\alpha_j$ totally with respect to time along the motion $q(t)$ :

0=\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial S}{\partial\alpha_j}=\sum_i \frac{\partial^2 S}{\partial \alpha_j\,\partial q_i}\,\dot q_i+\frac{\partial^2 S}{\partial\alpha_j\,\partial t}

Step 3. Subtract Step 1 from Step 2. The mixed time terms cancel, leaving

\sum_i \frac{\partial^2 S}{\partial \alpha_j\,\partial q_i}\left(\dot q_i-\frac{\partial H}{\partial p_i}\right)=0 \quad\text{for } j=1,\dots,n

Why this forces the first Hamilton equation: the matrix $\partial^2S/\partial\alpha_j\partial q_i$ is nonsingular by the completeness condition (2), so the only vector it annihilates is zero — hence $\dot q_i=\partial H/\partial p_i$ for every $i$ .

Step 4. Now differentiate the HJ identity with respect to $q_j$ at fixed $\alpha,t$ :

\frac{\partial H}{\partial q_j}+\sum_i \frac{\partial H}{\partial p_i}\,\frac{\partial^2 S}{\partial q_i\,\partial q_j}+\frac{\partial^2 S}{\partial t\,\partial q_j}=0

Step 5. Differentiate the momentum relation $p_j=\partial S/\partial q_j$ totally with respect to time, using $\dot q_i=\partial H/\partial p_i$ from Step 3:

\dot p_j=\sum_i \frac{\partial^2 S}{\partial q_j\,\partial q_i}\,\frac{\partial H}{\partial p_i}+\frac{\partial^2 S}{\partial q_j\,\partial t}

Step 6. Compare Steps 4 and 5: the double sums are identical, so

\dot p_j=-\frac{\partial H}{\partial q_j}

which is the second Hamilton equation. The recipe is vindicated: any complete integral of one PDE encodes every trajectory of the $2n$ -dimensional flow, unpacked by differentiation and inversion alone. Both Hamilton equations emerged with the correct signs, and each half of the argument visibly consumed the non-degeneracy of $\partial^2S/\partial q\,\partial\alpha$ .

Go deeper · Derivation C · 1 minSolving the harmonic oscillator with no equation of motion

Derivation CSolving the harmonic oscillator with no equation of motion

We solve $H=\frac{p^2}{2m}+\frac12 m\omega^2q^2$ end-to-end by the HJ method: separate off time, integrate one quadrature, apply the $\beta$ -recipe, and recover the familiar sinusoid — never once writing $\ddot q=-\omega^2 q$ . The strategy showcases the standard workflow for any conserved-energy problem: $S=W-Et$ , solve for $\mathrm{d}W/\mathrm{d}q$ , and differentiate with respect to the constant before doing the harder integral — the step ordering that saves labor is half the lesson here.

Step 1. $H$ is time-independent, so write $S=W(q,E)-Et$ with the energy as the single constant $\alpha=E$ . The time-independent HJ equation reads

\frac{1}{2m}\left(\frac{\mathrm{d}W}{\mathrm{d}q}\right)^2+\frac12 m\omega^2q^2=E

Step 2. Solve algebraically for the gradient — this is the momentum function:

\frac{\mathrm{d}W}{\mathrm{d}q}=\sqrt{2mE-m^2\omega^2q^2}=p(q,E)

Why we do not integrate yet: the recipe needs $\partial S/\partial E$ , and differentiating under the integral sign first produces a much easier integral than $W$ itself (which is an area integral giving arcsine plus a square-root term).

Step 3. Apply the recipe $\beta=\partial S/\partial E=\partial W/\partial E-t$ , differentiating under the integral sign:

t+\beta=\int\frac{m\,\mathrm{d}q}{\sqrt{2mE-m^2\omega^2q^2}}

Step 4. Evaluate the quadrature. With $u=q\sqrt{m\omega^2/2E}$ the integral is a standard arcsine:

t+\beta=\frac{1}{\omega}\arcsin\!\left(q\,\sqrt{\frac{m\omega^2}{2E}}\right)

Step 5. Invert for the trajectory and read off the momentum:

q(t)=\sqrt{\frac{2E}{m\omega^2}}\,\sin\big(\omega(t+\beta)\big),\qquad p(t)=\frac{\mathrm{d}W}{\mathrm{d}q}\Big|_{q(t)}=\sqrt{2mE}\,\cos\big(\omega(t+\beta)\big)

Why this is the complete answer: amplitude fixed by the constant $E$ , phase by the constant $\beta$ — the two initial conditions in their HJ costume. The pair automatically satisfies $p=m\dot q$ , a free consistency check.

Step 6. Harvest the action variable while the machinery is warm. The orbit is the phase ellipse of area $2\pi E/\omega$ , so

J=\frac{1}{2\pi}\oint p\,\mathrm{d}q=\frac{E}{\omega}\qquad\Rightarrow\qquad H=\omega J,\quad \frac{\partial H}{\partial J}=\omega

Why this matters downstream: the frequency emerged from pure differentiation of $H(J)$ — the pattern that action-angle theory industrializes for every separable system. *The result reproduces $q=q_0\sin(\omega t+\phi)$ with $E=\tfrac12 m\omega^2q_0^2$ ; the $J=E/\omega$ check matches the known SHO phase-space area.*

A glimpse of wave mechanics

The optical analogy is quantitative, and one substitution shows it. Write a quantum wavefunction in polar form, $\psi=A\,e^{iS/\hbar}$ , insert it into the Schrödinger equation, and expand in powers of $\hbar$ : at order $\hbar^0$ the phase $S$ obeys exactly the Hamilton-Jacobi equation $\partial_t S+(\nabla S)^2/2m+V=0$ , while order $\hbar^1$ makes the amplitude $A^2$ flow like a conserved fluid with velocity $\nabla S/m$ — the WKB transport equation. Classical mechanics is the eikonal (geometrical-optics) limit of quantum mechanics, with $S/\hbar$ as the phase and trajectories as the rays. The single-valuedness of $\psi$ around a closed orbit then forces $\oint p\,\mathrm{d}q=2\pi\hbar(n+\tfrac12)$ — Bohr-Sommerfeld quantization, born directly from the HJ structure. The full derivation, the quantum potential it exposes, and its semiclassical consequences belong to the connection-to-Schrödinger article at the end of this chapter; here it is enough to see why Schrödinger reached for this equation first.

Limiting cases

Case	Result	Meaning
time-independent $H$	$S=W(q,\alpha)-Et$	time separates; solve $H(q,\partial W/\partial q)=E$ for the characteristic function
fully separable	$S=\sum_k S_k(q_k,\alpha)-Et$	each coordinate reduced to a quadrature — the integrable, Liouville-Arnold case
cyclic coordinate $q_j$	$S_j=\alpha_j q_j$	conserved $p_j=\alpha_j$ ; a cyclic coordinate's share of $S$ is linear — separation at its most trivial
free particle	$S=\alpha q-\frac{\alpha^2}{2m}t$	the complete integral is exact plane-wave phase; fronts advance at $v_{\text{front}}=E/\alpha$ , half the particle speed
eikonal limit of QM	$\psi=Ae^{iS/\hbar}$ , $\hbar\to0$	Schrödinger reduces to HJ at leading order; WKB amplitude at next order

Common trapA complete integral is not the general solution — and the general solution is useless to you

The general solution of a first-order PDE contains an arbitrary function; chasing it is both hopeless and pointless. The method requires only a complete integral: any solution with $n$ independent nontrivial constants satisfying $\det(\partial^2S/\partial q_i\partial\alpha_j)\neq0$ . Three concrete traps follow. Do not count the additive constant $A$ among your $n$ — it shifts nothing physical, and mistaking it for a real constant leaves you one short. Do not skip the determinant check: a "solution" like $S=f(q_1+\alpha_1+\alpha_2,\dots)$ with degenerate constants cannot be inverted for the motion. And do not forget that your choice of complete integral fixes what the $\alpha_i$ mean — energy in one solution, initial momentum in another; the $\beta_i$ inherit their meaning from that choice, so mixing conventions mid-problem scrambles the initial conditions.

Worked examples

1 · A stone in gravity, by wavefront (confidence builder)

A stone of mass $m=0.50\ \mathrm{kg}$ moves vertically in uniform gravity, $H=\frac{p^2}{2m}+mgz$ , with total energy $E=9.81\ \mathrm{J}$ . Solve by Hamilton-Jacobi: find $z(t)$ , the maximum height, and the speed at $z=0$ . ( $g=9.81\ \mathrm{m/s^2}$ .)

Step 1. Time-independent problem: $S=W(z,E)-Et$ with $\frac{1}{2m}(W')^2+mgz=E$ , so $W'=p(z)=\sqrt{2m(E-mgz)}$ .

Step 2. Apply the recipe before integrating: $\beta=\partial W/\partial E-t=\int\frac{m\,\mathrm{d}z}{\sqrt{2m(E-mgz)}}-t$ . The integral evaluates to $-\frac{1}{g}\sqrt{\frac{2(E-mgz)}{m}}$ , so $t+\beta=-\frac{1}{g}\sqrt{\frac{2(E-mgz)}{m}}$ .

Step 3. Invert: $E-mgz=\frac{m g^2}{2}(t+\beta)^2$ , i.e. $z(t)=\frac{E}{mg}-\frac{g}{2}(t+\beta)^2$ — the parabola of free fall, produced by algebra. The momentum follows as $p=W'|_{z(t)}=-mg(t+\beta)$ , linear in time as it must be.

Step 4. Numbers. Maximum height (where $p=0$ , at $t=-\beta$ ): $z_{\max}=\frac{E}{mg}=\frac{9.81}{0.50\times9.81}=\boxed{2.0\ \mathrm{m}}$ . Speed at the ground: $p(0)=\sqrt{2m(E-0)}=\sqrt{2\times0.50\times9.81}=3.13\ \mathrm{kg\,m/s}$ , so $v=p/m=\boxed{6.3\ \mathrm{m/s}}$ . Cross-check: $v=\sqrt{2gz_{\max}}=\sqrt{2\times9.81\times2.0}=6.26\ \mathrm{m/s}$ ✓.

2 · The quantum fingerprint of a classical calculation (exam level)

The HCl molecule vibrates like a harmonic oscillator with angular frequency $\omega=5.63\times10^{14}\ \mathrm{rad/s}$ and reduced mass $\mu=1.63\times10^{-27}\ \mathrm{kg}$ . Using the HJ solution of the oscillator, (a) find the action variable demanded by Bohr-Sommerfeld quantization $J=\hbar(n+\tfrac12)$ in the ground state, (b) the corresponding classical energy, and (c) the classical amplitude of that orbit. ( $\hbar=1.055\times10^{-34}\ \mathrm{J\,s}$ .)

Step 1. From the HJ derivation, the oscillator's action variable is $J=E/\omega$ . Bohr-Sommerfeld with $n=0$ demands $J=\hbar/2=5.27\times10^{-35}\ \mathrm{J\,s}$ .

Step 2. Energy: $E=\omega J=5.63\times10^{14}\times5.27\times10^{-35}=\boxed{2.97\times10^{-20}\ \mathrm{J}}\approx0.185\ \mathrm{eV}$ — the zero-point energy of the molecular vibration, and exactly $\tfrac12\hbar\omega$ because the semiclassical spectrum $E_n=\hbar\omega(n+\tfrac12)$ happens to be exact for the oscillator.

Step 3. Amplitude from the HJ trajectory $q(t)=\sqrt{2E/\mu\omega^2}\sin(\omega(t+\beta))$ : the prefactor is the amplitude, $q_0=\sqrt{\frac{2\times2.97\times10^{-20}}{1.63\times10^{-27}\times(5.63\times10^{14})^2}}$ . The denominator is $\mu\omega^2=1.63\times10^{-27}\times3.17\times10^{29}=517\ \mathrm{N/m}$ (a chemically sensible bond stiffness), so $q_0=\sqrt{5.94\times10^{-20}/517}=\sqrt{1.15\times10^{-22}}=\boxed{1.07\times10^{-11}\ \mathrm{m}}$ — about $0.011\ \mathrm{nm}$ , roughly 8% of the 0.127 nm bond length.

Step 4. The moral: a classical PDE, solved once, delivered the quantization-ready invariant $J$ , the correct zero-point energy scale, and the physical amplitude of a molecular vibration. This is HJ theory functioning as the load-bearing bridge between mechanics and quantum theory — the reason it is the final classical formalism before wavefunctions.

Origins · 1 minCarl Gustav Jacob Jacobi and how this was discovered

Carl Gustav Jacob Jacobi

1804–1851 · Königsberg and Berlin

Jacobi took Hamilton's optical-mechanical papers of 1834–35 — which Hamilton himself regarded chiefly as a triumph of formal optics — and turned them into a working method for dynamics. Where Hamilton required his principal function to satisfy two PDEs simultaneously, Jacobi saw that a single equation suffices, that any complete integral (not some canonical "true" solution) unlocks the motion, and he proved the theorem that bears out the recipe. His Königsberg lectures of 1842–43, published posthumously as the Vorlesungen über Dynamik, made the method a discipline: there he separated the two-center problem, invented elliptic coordinates for mechanics, and solved geodesics on the ellipsoid — a problem thought hopeless. A prodigious calculator in an era of great ones, Jacobi believed mathematics should be pursued "for the honor of the human spirit" — and his refinement of Hamilton's idea became, ninety years later, the exact scaffold on which Schrödinger hung wave mechanics.

History

1828

— Hamilton's Theory of Systems of Rays founds the characteristic-function method in optics: one function encodes every ray of an optical instrument.

1834

— Hamilton's two Essays on a General Method in Dynamics transplant the idea to mechanics, introducing the principal function satisfying a pair of PDEs.

1837

— Jacobi simplifies radically: one first-order PDE and any complete integral suffice — the Hamilton-Jacobi equation in its working form, with Jacobi's theorem as warrant.

1843

— Jacobi's Königsberg lectures (the Vorlesungen über Dynamik) separate the two-center problem in elliptic coordinates, the method's first conquest beyond known territory.

1926

— Schrödinger, seeking mechanics-as-wave-optics, builds his equation so that its eikonal limit is exactly the Hamilton-Jacobi equation — classical mechanics becomes the short-wavelength shadow of quantum theory.

Going further

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., Ch. 10 · Landau & Lifshitz, Mechanics, 3rd ed., §47-48 · Morin, Introduction to Classical Mechanics, Ch. 15 (appendix) · Taylor, Classical Mechanics, Ch. 13 · Arnold, Mathematical Methods of Classical Mechanics, §46-47 Key results to keep in mind: the HJ equation follows from demanding $K=H+\partial S/\partial t=0$ for a type-2 generating function; the SHO solution reproduces $q=q_0\sin(\omega t+\phi)$ and $J=E/\omega$ ; the eikonal expansion of the Schrödinger equation yields the HJ equation at $O(\hbar^0)$ and the continuity equation at $O(\hbar)$ ; and Bohr-Sommerfeld $\oint p\,\mathrm{d}q=2\pi\hbar(n+\tfrac12)$ gives $E_n=\hbar\omega(n+\tfrac12)$ for the oscillator. Worked-example numbers on this page ( $z_{\max}=2.0\ \mathrm{m}$ , $v=6.3\ \mathrm{m/s}$ for the stone; $E=2.97\times10^{-20}\ \mathrm{J}$ , $q_0=1.07\times10^{-11}\ \mathrm{m}$ for HCl) are illustrative arithmetic computed from the boxed formulas.

Builds on: Generating Function to a Trivial Hamiltonian · Hamilton's Principal Function · Liouville Integrable Systems Leads to: Time-Independent Hamilton-Jacobi Equation · Hamilton's Characteristic Function · Separation of Variables in Hamilton-Jacobi Theory · Action-Angle Variables · Staeckel Conditions for Separability

Isn't trading 2n ODEs for one nonlinear PDE a terrible deal? PDEs are harder than ODEs

Yes — in general it is a terrible deal, and admitting that is the beginning of wisdom about this method. A generic nonlinear first-order PDE in many variables is hopeless. The trade wins in exactly one circumstance: when the PDE separates additively, $S=\sum_k S_k(q_k)$ , so that each piece satisfies a one-variable ODE solvable by quadrature. Separability is rare and precious — it is essentially the definition of a classically integrable system, and the Staeckel conditions later in this chapter tell you when orthogonal coordinates achieve it. The honest scoreboard: for separable systems HJ is the most efficient solution technology ever built; for non-separable ones it is a conceptual (still profound) reformulation, and you integrate numerically like everyone else.

What exactly is the difference between a complete integral and the general solution of the HJ equation?

The general solution of a first-order PDE depends on an arbitrary function — an infinite-dimensional family. A complete integral depends on just $n$ constants $\alpha_i$ (plus the trivial additive one), with the non-degeneracy $\det(\partial^2S/\partial q_i\partial\alpha_j)\neq0$ . It is a much smaller object, and it is all you need: Jacobi's theorem shows every trajectory of the system is recovered from it by differentiation and inversion. In fact the general solution can be reconstructed from a complete integral by envelope constructions, so nothing is lost. The practical rule: hunt for $n$ constants, verify the determinant, stop. Demanding more generality than that is the most common way to convince yourself the method is unusable.

Do the alphas have to be the energy and things like it, or can I choose them freely?

Freer than you might think, and the freedom is the point. Any $n$ independent constants parametrizing a complete integral are legitimate new momenta — energy, an initial momentum, an angular momentum, a separation constant with no obvious name. Different complete integrals of the same HJ equation give different but equally valid frozen coordinate systems, related to each other by canonical transformations among the constants. What you must NOT do is change interpretation midstream: once you pick the complete integral, the meaning of each $\beta_i=\partial S/\partial\alpha_i$ is fixed by that choice (e.g. with $\alpha=E$ in one dimension, $\beta$ is minus the time of passage through a reference point). The action variables $J_i$ are simply the canonically best choice of $\alpha$ 's for periodic systems — best because their conjugates advance linearly in time.

Where does the HJ equation come from if I believe in the action principle rather than canonical transformations?

From differentiating the action with respect to its endpoint. Define $S(q,t)$ as the action $\int L\,\mathrm{d}t'$ evaluated along the true trajectory arriving at position $q$ at time $t$ (from fixed initial data). Endpoint variation of the action gives $\partial S/\partial q_i=p_i$ and $\partial S/\partial t=-H$ — and inserting the first into the second is literally the HJ equation: $\partial_t S+H(q,\partial S/\partial q,t)=0$ . So the equation is the statement "the action-as-function-of-endpoint has momentum for its space gradient and minus energy for its time derivative." The canonical-transformation derivation and this variational one are two roofs over the same room; the companion article on Hamilton's principal function builds the second route in full.

Why does S equal W minus Et only for time-independent Hamiltonians, and what breaks otherwise?

The ansatz $S=W(q)-Et$ solves the HJ equation only if plugging it in removes all explicit time dependence: $H(q,\partial W/\partial q)=E$ requires $H$ to contain no $t$ . If $H$ depends on time — a driven oscillator, say — energy is not conserved, no single constant $E$ exists, and time refuses to separate; you must attack the full $(n+1)$ -variable PDE or find a cleverer separation (sometimes a time-dependent canonical pre-transformation restores separability, e.g. moving to an oscillating frame). The split, when valid, has a clean reading: the stationary fronts $W=\text{const}$ sit in configuration space while the moving fronts $S=\text{const}$ sweep across them at rate $E$ — the classical shadow of a stationary quantum state's phase rotating at frequency $E/\hbar$ .

Is S a physically measurable field, or just bookkeeping?

Classically, $S$ itself is gauge-like — only its derivatives are physical ( $p=\partial S/\partial q$ , $H=-\partial_t S$ ), and the additive constant is meaningless. But differences of $S$ between paths are measurable the moment quantum mechanics enters, because $S/\hbar$ is a phase. The Aharonov-Bohm effect, matter-wave interferometry, and every semiclassical interference pattern measure $\Delta S$ between alternative classical paths in units of $\hbar$ . So the honest answer is layered: within pure classical mechanics $S$ is an extraordinarily efficient bookkeeping device; the moment $\hbar$ is switched on, it is promoted to the phase of the wavefunction, and interference makes it experimentally visible. That promotion is why the eikonal limit works and why HJ theory, uniquely among classical formalisms, survives the quantum transition intact.

The HJ equation is one equation, but Hamilton's equations are 2n. Where did the information go?

Into the function's arguments. A solution $S(q,\alpha,t)$ of the single PDE is not one function of $q$ and $t$ — it is an $n$ -parameter family of them, and the completeness condition guarantees the family is genuinely $n$ -dimensional. Each member of the family carries a Lagrangian submanifold's worth of trajectories (all initial positions with momenta $p=\partial S/\partial q$ at $t=0$ ), and sweeping the $\alpha$ 's covers all of phase space. So the counting balances: $n$ constants in the family plus $n$ constants $\beta_i$ from the extraction recipe equals the $2n$ initial conditions of the ODE system. The PDE looks smaller only because a function of $n+1$ variables is a vastly bigger object than a finite list of ODE solutions.

Why do wavefronts of S move at a different speed than the particles?

Because front speed and ray speed answer different questions. For a free particle with $S=\alpha q-\frac{\alpha^2}{2m}t$ , a surface of constant $S$ satisfies $\alpha\,\mathrm{d}q=\frac{\alpha^2}{2m}\,\mathrm{d}t$ , so the front advances at $v_{\text{front}}=\alpha/2m=v/2$ — half the particle velocity $v=\alpha/m$ . In general $v_{\text{front}}=E/|\nabla W|=E/p$ , while the particle moves at $v=p/m$ : their product is $E$ (up to the potential's offset), not their ratio one. This is precisely the phase-velocity/group-velocity split of wave optics — fronts are phase, particles are group — and de Broglie leaned on exactly this classical fact when assigning matter waves a phase velocity $E/p$ exceeding $c$ without contradiction. Seeing it appear already in pure mechanics is one of the quiet delights of HJ theory.