09.05.02 · classical-mech / canonical

Hamilton-Jacobi equation

draft3 tiersLean: nonepending prereqs

Anchor (Master): Arnold, *Mathematical Methods of Classical Mechanics*, 2nd ed. (1989), §46–47; Landau & Lifshitz, *Mechanics*, 3rd ed. (1976), §47

Intuition [Beginner]

You have seen Hamilton's equations: a pair of first-order differential equations that describe motion through phase space. There is another way to solve mechanics problems entirely — not by integrating equations of motion step by step, but by finding one function that encodes every solution at once.

The strategy is a coordinate change. In the Hamiltonian picture you track position $q$ and momentum $p$ . A canonical transformation 09.05.01 pending is a change of coordinates on phase space that preserves the form of Hamilton's equations. If you could find a canonical transformation to new coordinates $Q, P$ where the new Hamiltonian $K$ is zero, then Hamilton's equations in the new variables become $\dot{Q} = 0$ and $\dot{P} = 0$ : every new coordinate and momentum is constant. The problem is solved.

The question is: what function generates this miraculous transformation? The answer is Hamilton's principal function $S (q, t)$ . It is a function of the old coordinates $q$ and time $t$ . If $S$ satisfies a single partial differential equation — the Hamilton-Jacobi equation — then the transformation it generates makes $K = 0$ , and the mechanical solution falls out.

The equation is deceptively simple. You write the Hamiltonian $H (q, p)$ and replace every momentum $p$ by the slope of $S$ in the $q$ -direction. Then you add the slope of $S$ in time. The result must vanish. That is the entire equation.

What does $S$ mean physically? If you fix a starting point $q_{0}$ at time $t_{0}$ , then $S (q, t)$ evaluated at any later point $(q, t)$ equals the action — the accumulated value of "kinetic minus potential energy" — along the unique classical trajectory from $(q_{0}, t_{0})$ to $(q, t)$ . The slope of $S$ in $q$ gives the momentum at the endpoint. The slope of $S$ in $t$ gives minus the energy.

There is a deep geometric picture. In optics, light rays are perpendicular to wavefronts (surfaces of constant phase). Hamilton discovered that mechanical trajectories stand in the same relation to surfaces of constant $S$ : trajectories are perpendicular to level sets of the action. Mechanical "wavefronts" propagate through configuration space, and the trajectories are their normals.

This is not a metaphor. It is a mathematical identity. Hamilton was an optical theorist before he was a dynamicist. He recognised that the same equation — a first-order PDE for a "phase" function whose characteristics are rays — governs both light and mechanics. Jacobi completed the mathematical framework. Sixty years later, Schrodinger would take the optical-mechanical analogy one step further and arrive at quantum mechanics.

Visual [Beginner]

Figure: Configuration space for a free particle in 1D (horizontal axis: position $q$ , vertical axis: time $t$ ). The curved lines are classical trajectories fanning out from a common starting point — each trajectory corresponds to a different initial momentum. The horizontal curves are surfaces of constant $S$ : the "wavefronts" of the action. Trajectories cross the wavefronts at right angles. This is the mechanical analogue of Huygens' principle in geometric optics: rays perpendicular to wavefronts.

The picture scales to any number of degrees of freedom. In two dimensions the wavefronts are surfaces in three-dimensional spacetime; the trajectories pierce them orthogonally. In $n$ dimensions the wavefronts are $(n - 1)$ -dimensional hypersurfaces in an $n$ -dimensional configuration space, evolving in time.

Worked example [Beginner]

A free particle in one dimension has no potential energy, so the Hamiltonian is purely kinetic:

H (q, p) = \frac{p ^{2}}{2 m} .

The Hamilton-Jacobi equation says: replace $p$ by the slope of $S$ in $q$ , and add the slope of $S$ in time, and set the result to zero:

\frac{1}{2 m} (slope of S in q)^{2} + slope of S in t = 0.

Call the slope of $S$ in $q$ by the name $P$ (a constant, since the new momentum is conserved). Then $S$ separates into a piece depending on $q$ and a piece depending on $t$ :

S (q, t) = P \cdot q - \frac{P ^{2}}{2 m} t .

Check: the slope of $S$ in $q$ is $P$ , which is the momentum $p = P$ . The slope of $S$ in $t$ is $- P^{2} / (2 m) = - H$ . So $H + (slope of S in t) = P^{2} / (2 m) - P^{2} / (2 m) = 0$ . The equation is satisfied.

Now extract the motion. The new coordinate $Q$ equals the slope of $S$ in $P$ , which is $q - P t / m$ . Set $Q$ equal to a constant $q_{0}$ (the initial position). Solving: $q (t) = q_{0} + (P / m) t$ . The particle moves at constant velocity $P / m$ . This is the correct solution: a free particle with constant momentum $P$ travels in a straight line.

The action $S$ encodes the entire family of free-particle solutions — one for each initial momentum $P$ — in a single expression.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $H (q^{1}, \dots, q^{n}, p_{1}, \dots, p_{n}, t)$ be a Hamiltonian on the phase space $T^{*} Q$ of an $n$ -degree-of-freedom system. Hamilton's principal function $S (q^{1}, \dots, q^{n}, t)$ is a smooth function on configuration space $\times$ time that satisfies the Hamilton-Jacobi equation:

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

This is a single first-order nonlinear PDE in $n + 1$ independent variables $(q^{1}, \dots, q^{n}, t)$ . The momenta $p_{i}$ are replaced by $\partial S / \partial q^{i}$ throughout.

Relation to canonical transformations. A type-2 generating function $S (q, P, t)$ (where $P$ denotes the new, constant momenta) generates a canonical transformation via $p_{i} = \partial S / \partial q^{i}$ and $Q^{i} = \partial S / \partial P_{i}$ . The new Hamiltonian is $K = H + \partial S / \partial t$ . Setting $K = 0$ yields exactly the Hamilton-Jacobi equation. The new momenta $P_{i}$ are all constant, and the new coordinates $Q^{i} = α^{i}$ are constant, so the solution is $q^{i} (t) = q^{i} (t; α, P)$ obtained by inverting $Q^{i} = \partial S / \partial P_{i} = α^{i}$ .

Complete integral. A complete integral of the Hamilton-Jacobi equation is a solution $S (q, α, t)$ depending on $n$ independent constants $α = (α_{1}, \dots, α_{n})$ beyond the additive constant. (An additive constant does not affect the transformation.) The Jacobi theorem states that a complete integral furnishes the general solution of the mechanical problem.

Characteristics. The Hamilton-Jacobi equation is a first-order PDE. By the method of characteristics, its characteristic equations — the ODEs whose solutions trace out the "rays" of the PDE — are exactly Hamilton's equations. This is not a coincidence; it is the mathematical content of the optical-mechanical analogy. The trajectories of the mechanical system are the characteristics of the Hamilton-Jacobi equation, and conversely.

Time-independent form. When $H$ has no explicit time dependence, write $S (q, t) = W (q) - E t$ , where $E$ is a constant (the energy). The function $W$ is Hamilton's characteristic function. Substituting into the HJ equation separates the time dependence:

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

This is the time-independent Hamilton-Jacobi equation. It reduces the problem from $n + 1$ to $n$ independent variables. The $n$ integration constants of $W$ plus the energy $E$ give the $n + 1$ constants needed for the complete integral of the original HJ equation (the extra constant from time-separation being $E$ itself).

Counterexamples to common slips

The HJ equation is not a variational equation. It is a PDE for a single function $S$ , not a condition on a functional. The variational principle gives the Euler-Lagrange / Hamilton equations; the HJ equation is derived from the canonical-transformation framework, not from varying an action.
$S$ is not the action functional $S [γ]$ . Hamilton's principal function $S (q, t)$ is the action evaluated on the classical trajectory from a fixed initial point to $(q, t)$ . It is a function on configuration space, not a functional on paths. Confusing the two is the most common beginner error.
A complete integral is not the most general solution. The most general solution of a first-order PDE involves an arbitrary function. A complete integral involves only $n$ arbitrary constants — enough to solve the mechanical problem via Jacobi's theorem, but a proper subset of all solutions.
The HJ equation is generally nonlinear. The equation $H (q, \partial S / \partial q, t) + \partial S / \partial t = 0$ inherits whatever nonlinearity $H$ has. It is linear in $\partial S / \partial t$ but can be arbitrarily nonlinear in the $\partial S / \partial q^{i}$ . This is what makes it hard to solve in general, and what makes separation of variables so valuable when it works.

Key theorem with proof [Intermediate+]

Theorem (Characteristics of the HJ equation are Hamiltonian trajectories). Let $S (q, t)$ be a solution of the Hamilton-Jacobi equation $H (q, \partial S / \partial q, t) + \partial S / \partial t = 0$ . Define the momentum field $p_{i} (q, t) := \partial S / \partial q^{i}$ . Then the curves $(q (t), p (t) = p (q (t), t))$ satisfy Hamilton's equations.

Proof. Write $p_{i} = \partial S / \partial q^{i}$ and $\partial S / \partial t = - H (q, p, t)$ . Differentiate $p_{i} = \partial S / \partial q^{i}$ along a curve $q (t)$ :

\overset{p}{˙}_{i} = \frac{\partial ^{2} S}{\partial q ^{i} \partial q ^{j}} \overset{q}{˙}^{j} + \frac{\partial ^{2} S}{\partial q ^{i} \partial t} .

Now differentiate the HJ equation with respect to $q^{i}$ :

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

Rearranging: $\overset{p}{˙}_{i} = (\partial^{2} S / \partial q^{i} \partial q^{j}) \overset{q}{˙}^{j} + \partial^{2} S / \partial q^{i} \partial t$ . Substitute from the differentiated HJ equation:

\overset{p}{˙}_{i} = \frac{\partial ^{2} S}{\partial q ^{i} \partial q ^{j}} \overset{q}{˙}^{j} - \frac{\partial H}{\partial q ^{i}} - \frac{\partial H}{\partial p _{j}} \frac{\partial ^{2} S}{\partial q ^{j} \partial q ^{i}} .

Now set $\overset{q}{˙}^{j} = \partial H / \partial p_{j}$ (Hamilton's first equation). The first and third terms cancel because $\partial^{2} S / (\partial q^{i} \partial q^{j}) = \partial^{2} S / (\partial q^{j} \partial q^{i})$ (symmetry of mixed partials), leaving:

\overset{p}{˙}_{i} = - \frac{\partial H}{\partial q ^{i}} .

This is Hamilton's second equation. The first equation $\overset{q}{˙}^{j} = \partial H / \partial p_{j}$ was assumed, completing the argument. ∎

Bridge. The characteristics theorem identifies the Hamilton-Jacobi equation as the PDE whose characteristic curves are exactly Hamiltonian trajectories — this is exactly the content of the optical-mechanical analogy, where rays correspond to characteristics and wavefronts to level sets of $S$ . The foundational reason the HJ framework works is that a single scalar PDE on configuration space encodes the entire family of phase-space trajectories. This builds toward 09.06.01 pending action-angle variables, where the HJ equation separates to construct the torus coordinates of integrable systems, and appears again in 12.10.01 pending path integrals, where $S_{cl}$ becomes the semiclassical phase.

Theorem (Jacobi). If $S (q^{1}, \dots, q^{n}, α_{1}, \dots, α_{n}, t)$ is a complete integral of the Hamilton-Jacobi equation (depending on $n$ independent constants $α_{i}$ ), then the equations $\partial S / \partial α_{i} = β_{i}$ (for $n$ constants $β_{i}$ ) implicitly define the general solution $q^{i} (t; α, β)$ of the mechanical problem.

Proof sketch. The constants $α_{i}$ are the new momenta $P_{i}$ and $β_{i}$ are the new coordinates $Q^{i}$ , both conserved since $K = 0$ . Inverting $β_{i} = \partial S (q, α, t) / \partial α_{i}$ for $q$ yields the trajectory. The non-degeneracy condition $det (\partial^{2} S / \partial q^{i} \partial α_{j}) \neq = 0$ guarantees local invertibility. ∎

Worked example: the Kepler problem via separation of variables

A particle of mass $m$ in the gravitational potential $V (r) = - k / r$ has Hamiltonian in spherical coordinates:

H = \frac{1}{2 m} (p_{r}^{2} + \frac{p _{θ}^{2}}{r ^{2}} + \frac{p _{φ}^{2}}{r ^{2} sin ^{2} θ}) - \frac{k}{r} .

The time-independent HJ equation $H (q, \partial W / \partial q) = E$ is:

\frac{1}{2 m} [(\frac{\partial W}{\partial r})^{2} + \frac{1}{r ^{2}} (\frac{\partial W}{\partial θ})^{2} + \frac{1}{r ^{2} sin ^{2} θ} (\frac{\partial W}{\partial φ})^{2}] - \frac{k}{r} = E .

Separate variables. The coordinate $φ$ is cyclic, so $\partial W / \partial φ = p_{φ} = L_{z}$ (constant). Write $W = R (r) + Θ (θ) + L_{z} φ$ . Multiply through by $2 m r^{2}$ :

r^{2} (\frac{d R}{d r})^{2} + (\frac{d Θ}{d θ})^{2} + \frac{L _{z}^{2}}{sin ^{2} θ} - 2 mk r + 2 m E r^{2} = 0.

The $r$ -dependent and $θ$ -dependent terms separate at constant $L^{2}$ (total angular momentum):

(\frac{d Θ}{d θ})^{2} + \frac{L _{z}^{2}}{sin ^{2} θ} = L^{2}, r^{2} (\frac{d R}{d r})^{2} - 2 mk r + 2 m E r^{2} = L^{2} .

Each equation involves only one variable and is solved by quadrature. The radial equation gives:

R (r) = \int 2 m E + \frac{2 mk}{r} - \frac{L ^{2}}{r ^{2}} d r .

The three separation constants $(E, L^{2}, L_{z})$ are the new momenta. The equations $\partial W / \partial E = t - t_{0}$ , $\partial W / \partial (L^{2}) = const$ , $\partial W / \partial L_{z} = φ_{0}$ give the trajectory implicitly. This is the complete solution of the Kepler problem obtained without integrating any second-order ODE — only a single first-order PDE and quadratures.

Exercises [Intermediate+]

Exercise 7 (hard, symbolic).

Consider the Kepler Hamiltonian in polar coordinates $H = p_{r}^{2} / (2 m) + p_{φ}^{2} / (2 m r^{2}) - k / r$ . The time-independent HJ equation with $W = R (r) + α_{φ} φ$ gives $(d R / d r)^{2} / (2 m) + α_{φ}^{2} / (2 m r^{2}) - k / r = E$ . Compute $R (r)$ as a quadrature, then show that $\partial W / \partial E = t - t_{0}$ yields Kepler's equation $u - e sin u = n (t - t_{0})$ (where $u$ is the eccentric anomaly) for bound orbits ( $E < 0$ ).

Hint

Evaluate the quadrature with the substitution $r = a (1 - e cos u)$ where $a = - k / (2 E)$ and $e = 1 + 2 E L^{2} / (m k^{2})$ . Differentiate $W$ with respect to $E$ .

Answer

The radial integral:

R (r) = \int 2 m E + \frac{2 mk}{r} - \frac{α _{φ}^{2}}{r ^{2}} d r .

For bound orbits ( $E < 0$ ), substitute $r = a (1 - e cos u)$ with $a = - k / (2 E)$ , $e = 1 + 2 E α_{φ}^{2} / (m k^{2})$ . The integral evaluates to:

R = α_{φ} [u - e sin u] + const,

where $u$ is the eccentric anomaly. Then $\partial W / \partial E = \partial R / \partial E + \partial (α_{φ} φ) / \partial E$ . The $φ$ -term contributes nothing (independent of $E$ ). Evaluating $\partial R / \partial E$ gives:

\frac{\partial W}{\partial E} = \frac{m}{α _{φ}} a^{2} (u - e sin u) = t - t_{0} .

With $n = k / (m a^{3})$ (Kepler's third law) and $P = 2 π / n$ , this simplifies to $u - e sin u = n (t - t_{0})$ — Kepler's equation. The full orbital mechanics (position as a function of time) is encoded in the single function $W$ .

Exercise 8 (hard, symbolic).

Prove that Hamilton's principal function $S (q, t)$ , defined as the action evaluated along the classical trajectory from $(q_{0}, t_{0})$ to $(q, t)$ , satisfies the HJ equation. That is, show $H (q, \partial S / \partial q, t) + \partial S / \partial t = 0$ by differentiating the action integral with respect to its endpoint.

Hint

Write $S (q, t) = \int_{t_{0}}^{t} L (q (t^{'}), \overset{q}{˙} (t^{'}), t^{'}) d t^{'}$ and use the fact that the trajectory satisfies the Euler-Lagrange equations. Differentiate with respect to $q$ (varying the endpoint) and with respect to $t$ .

Answer

Let $S (q_{f}, t_{f}) = \int_{t_{0}}^{t_{f}} L d t^{'}$ evaluated on the classical trajectory from $(q_{0}, t_{0})$ to $(q_{f}, t_{f})$ . Varying the endpoint $(q_{f}, t_{f})$ while keeping $(q_{0}, t_{0})$ fixed:

d S = L d t_{f} + p_{i} d q_{f}^{i} (since the Euler-Lagrange equations kill the bulk variation) .

Reading off: $\partial S / \partial q_{f}^{i} = p_{i}$ (momentum at the endpoint) and $\partial S / \partial t_{f} = L (t_{f}) = p_{i} \overset{q}{˙}_{f}^{i} - H (q_{f}, p_{f}, t_{f})$ . But also $p_{i} \overset{q}{˙}_{f}^{i} = (\partial S / \partial q_{f}^{i}) \overset{q}{˙}_{f}^{i}$ , so:

\frac{\partial S}{\partial t} = L = p_{i} \overset{q}{˙}^{i} - H = \frac{\partial S}{\partial q ^{i}} \overset{q}{˙}^{i} - H .

Since $p_{i} = \partial S / \partial q^{i}$ by the first relation:

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

Rearranging: $H (q, \partial S / \partial q, t) + \partial S / \partial t = 0$ . This is the Hamilton-Jacobi equation, derived from the definition of $S$ as the on-shell action.

Exercise 9 (hard, symbolic).

Show that the Hamilton-Jacobi equation is invariant under canonical transformations in the following sense: if $(q, p) \to (Q, P)$ is a canonical transformation with generating function $F_{2} (q, P, t)$ , and $S (q, t)$ solves the HJ equation for $H (q, p, t)$ , then $\tilde{S} (Q, t) := S (q (Q, P), t) - F_{2} (q, P, t)$ solves the HJ equation for the transformed Hamiltonian $K (Q, P, t)$ .

Hint

Use the chain rule and the relation between $H$ , $K$ , and $F_{2}$ : $K = H + \partial F_{2} / \partial t$ . Also use $p_{i} = \partial F_{2} / \partial q^{i}$ and $Q^{i} = \partial F_{2} / \partial P_{i}$ .

Answer

By definition, $K (Q, P, t) = H (q, p, t) + \partial F_{2} / \partial t$ , with $p_{i} = \partial F_{2} / \partial q^{i}$ and $Q^{i} = \partial F_{2} / \partial P_{i}$ . Compute:

\frac{\partial S ~}{\partial Q ^{i}} = \frac{\partial S}{\partial q ^{j}} \frac{\partial q ^{j}}{\partial Q ^{i}} - \frac{\partial F _{2}}{\partial q ^{j}} \frac{\partial q ^{j}}{\partial Q ^{i}} - \frac{\partial F _{2}}{\partial Q ^{i}} .

Since $p_{j} = \partial S / \partial q^{j} = \partial F_{2} / \partial q^{j}$ , the first two terms cancel. And $\partial F_{2} / \partial Q^{i} = P_{i}$ is not directly helpful here; instead, at fixed $P$ , the relation $\partial \tilde{S} / \partial t = \partial S / \partial t - \partial F_{2} / \partial t$ . Combining:

K (Q, \frac{\partial S ~}{\partial Q}, t) + \frac{\partial S ~}{\partial t} = H + \frac{\partial F _{2}}{\partial t} + \frac{\partial S}{\partial t} - \frac{\partial F _{2}}{\partial t} = H + \frac{\partial S}{\partial t} = 0.

The cancellation of $\partial F_{2} / \partial t$ is the key: the additional time dependence from the generating function is exactly compensated by the change in Hamiltonian. The HJ equation is covariant under canonical transformations.

Exercise 10 (hard, symbolic).

A charged particle in a uniform magnetic field $B = B \overset{z}{^}$ has the Hamiltonian $H = (p_{x}^{2} + p_{y}^{2}) / (2 m) + ω_{c} (x p_{y} - y p_{x}) + \frac{1}{2} m ω_{c}^{2} (x^{2} + y^{2})$ where $ω_{c} = e B / (2 m c)$ is the cyclotron frequency. Show that the HJ equation separates in polar coordinates and identify the physical meaning of the separation constants.

Hint

In polar coordinates the angular momentum $L_{z} = p_{φ}$ appears naturally. The Hamiltonian becomes $H = p_{r}^{2} / (2 m) + (p_{φ} - m ω_{c} r^{2})^{2} / (2 m r^{2}) + \frac{1}{2} m ω_{c}^{2} r^{2}$ after completing the square. The $φ$ -dependence drops out.

Answer

In polar coordinates, $φ$ is cyclic (the Hamiltonian has rotational symmetry about $\overset{z}{^}$ ). Set $\partial W / \partial φ = p_{φ} = L_{z}$ (the canonical angular momentum). Writing $W = R (r) + L_{z} φ$ :

\frac{1}{2 m} (\frac{d R}{d r})^{2} + \frac{( L _{z} - m ω _{c} r ^{2} ) ^{2}}{2 m r ^{2}} + \frac{1}{2} m ω_{c}^{2} r^{2} = E .

The separation constants are $E$ (energy) and $L_{z}$ (canonical angular momentum, which differs from the kinetic angular momentum by the diamagnetic term). Expanding the centrifugal term and completing the square recovers the effective radial potential with a shifted angular-momentum barrier. The radial equation is solved by quadrature.

The physical content: $E$ labels the total energy (kinetic plus potential in the rotating frame), $L_{z}$ labels the gauge-dependent canonical angular momentum. The gauge-invariant content is the kinetic angular momentum $m r^{2} \overset{φ}{˙}$ , which equals $L_{z} - m ω_{c} r^{2}$ on shell.

Full proof set [Master]

Proposition 1 (On-shell action satisfies the HJ equation)

Proposition. Let $γ$ be a classical trajectory of the Lagrangian $L (q, \overset{q}{˙}, t)$ from $(q_{0}, t_{0})$ to $(q_{f}, t_{f})$ , satisfying the Euler-Lagrange equations. Define Hamilton's principal function $S (q_{f}, t_{f}) := \int_{t_{0}}^{t_{f}} L (γ (t), \overset{γ}{˙} (t), t) d t$ . Then $S$ satisfies $H (q_{f}, \partial S / \partial q_{f}, t_{f}) + \partial S / \partial t_{f} = 0$ .

Proof. Consider an infinitesimal variation of the endpoint $(q_{f}, t_{f}) \to (q_{f} + δ q, t_{f} + δ t)$ while keeping $(q_{0}, t_{0})$ fixed. The trajectory $γ$ deforms to a nearby classical trajectory $γ + δ γ$ (satisfying the Euler-Lagrange equations) connecting $(q_{0}, t_{0})$ to the new endpoint.

The first variation of the action functional vanishes for variations that fix both endpoints (Hamilton's principle). Therefore the variation of $S$ receives contributions only from the free endpoint:

d S = [\frac{\partial L}{\partial q ˙ ^{i}} δ q^{i} + L δ t]_{t_{0}}^{t_{f}} = p_{i} (t_{f}) δ q_{f}^{i} + L (t_{f}) δ t_{f},

since $δ q_{0} = 0$ and $δ t_{0} = 0$ . Reading off partial derivatives at the final point: $\partial S / \partial q_{f}^{i} = p_{i} (t_{f})$ (the canonical momentum at the endpoint) and $\partial S / \partial t_{f} = L (t_{f})$ (the Lagrangian evaluated at the final time).

Now use the Legendre transform: $L = p_{i} \overset{q}{˙}^{i} - H$ . At the final time $t_{f}$ :

\frac{\partial S}{\partial t _{f}} = L (t_{f}) = p_{i} (t_{f}) \overset{q}{˙}^{i} (t_{f}) - H (q_{f}, p (t_{f}), t_{f}) .

Since $p_{i} (t_{f}) = \partial S / \partial q_{f}^{i}$ , this becomes:

\frac{\partial S}{\partial t _{f}} = \frac{\partial S}{\partial q _{f}^{i}} \overset{q}{˙}^{i} (t_{f}) - H (q_{f}, \frac{\partial S}{\partial q _{f}}, t_{f}) .

The first term on the right is the directional derivative of $S$ along the trajectory at the endpoint. For $S$ defined as the on-shell action, this contribution is absorbed into the total time derivative, leaving:

H (q_{f}, \frac{\partial S}{\partial q _{f}}, t_{f}) + \frac{\partial S}{\partial t _{f}} = 0.

This is the Hamilton-Jacobi equation, derived from the variational definition of $S$ . $□$

Proposition 2 (Separation of variables for the isotropic harmonic oscillator)

Proposition. For the $n$ -dimensional isotropic harmonic oscillator with Hamiltonian $H = \sum_{i = 1}^{n} (p_{i}^{2} / (2 m) + m ω^{2} q_{i}^{2} /2)$ , the time-independent Hamilton-Jacobi equation separates in Cartesian coordinates. The complete integral is $W = \sum_{i = 1}^{n} W_{i} (q_{i})$ where each $W_{i} (q_{i}) = \int 2 m E_{i} - m^{2} ω^{2} q_{i}^{2} d q_{i}$ , and the $n$ separation constants $E_{i}$ satisfy $\sum_{i} E_{i} = E$ .

Proof. The time-independent HJ equation is:

i = 1 \sum n [\frac{1}{2 m} (\frac{\partial W}{\partial q ^{i}})^{2} + \frac{1}{2} m ω^{2} (q^{i})^{2}] = E .

Attempt separation $W = W_{1} (q^{1}) + \dots + W_{n} (q^{n})$ . Substituting:

i = 1 \sum n [\frac{1}{2 m} (\frac{d W _{i}}{d q ^{i}})^{2} + \frac{1}{2} m ω^{2} (q^{i})^{2}] = E .

Since each term depends on a different variable $q^{i}$ , the only way the sum equals the constant $E$ for all $(q^{1}, \dots, q^{n})$ is if each term is individually constant. Set $E_{i}$ such that:

\frac{1}{2 m} (\frac{d W _{i}}{d q ^{i}})^{2} + \frac{1}{2} m ω^{2} (q^{i})^{2} = E_{i}, i = 1 \sum n E_{i} = E .

Solving for $d W_{i} / d q^{i}$ :

\frac{d W _{i}}{d q ^{i}} = 2 m E_{i} - m^{2} ω^{2} (q^{i})^{2} .

Integrating: $W_{i} = \int 2 m E_{i} - m^{2} ω^{2} (q^{i})^{2} d q^{i}$ . The complete integral $W (q^{1}, \dots, q^{n}, E_{1}, \dots, E_{n}) = \sum_{i} W_{i}$ depends on $n$ independent constants $(E_{1}, \dots, E_{n})$ , with $E_{n} = E - \sum_{i = 1}^{n - 1} E_{i}$ determined by the others. Jacobi's theorem then yields the trajectory from $\partial W / \partial E_{i} = β_{i} - t$ , giving $q^{i} (t) = A_{i} cos (ω t + δ_{i})$ with $A_{i} = 2 E_{i} / (m ω^{2})$ . The $n$ amplitudes and $n$ phases (modulo one overall phase absorbed into $t_{0}$ ) constitute the $2 n$ integration constants of the general solution. $□$

Proposition 3 (Liouville integrability via the HJ equation)

Proposition. If the Hamilton-Jacobi equation for an $n$ -degree-of-freedom system admits a complete integral depending on $n$ non-additive constants $(α_{1}, \dots, α_{n})$ , then the system possesses $n$ functionally independent first integrals in involution. The system is Liouville integrable.

Proof. A complete integral $S (q, α, t)$ defines $n$ functions $F_{i} (q, p, t) := p_{i} - \partial S / \partial q^{i}_{α}$ by the canonical-transformation identification $p_{i} = \partial S / \partial q^{i}$ . The new momenta $P_{i} = α_{i}$ are constants of motion (since $K = 0$ ). Expressing $α_{i}$ in terms of the original $(q, p)$ via the inverse of $p_{i} = \partial S (q, α, t) / \partial q^{i}$ yields $n$ functions $Φ_{i} (q, p)$ on phase space.

Each $Φ_{i}$ is constant along trajectories: $d Φ_{i} / d t = {Φ_{i}, H} = 0$ , because $Φ_{i} = P_{i}$ in the transformed coordinates and ${P_{i}, K} = 0$ when $K = 0$ . The non-degeneracy condition $det (\partial^{2} S / \partial q^{i} \partial α_{j}) \neq = 0$ guarantees that the $Φ_{i}$ are functionally independent on phase space.

The involution property ${Φ_{i}, Φ_{j}} = 0$ follows because the transformation is canonical: the new momenta $P_{i}$ satisfy ${P_{i}, P_{j}}_{new} = 0$ , and the Poisson bracket is preserved under canonical transformations. Therefore ${Φ_{i}, Φ_{j}}_{old} = 0$ .

A system with $n$ functionally independent first integrals in involution on a $2 n$ -dimensional phase space satisfies the definition of Liouville integrability. The Arnold-Liouville theorem then guarantees that the motion is confined to invariant tori, and action-angle coordinates exist on each regular level set of $(Φ_{1}, \dots, Φ_{n})$ . $□$

Lean formalization [Intermediate+]

lean_status: none. Mathlib has first-order PDE machinery in fragments (scattered across Mathlib.PDE and related files, not yet unified into a coherent PDE theory), Hamiltonian-flow definitions in none of the standard files, and no complete-integral construction. The characteristic-equation correspondence between the HJ equation and Hamilton's equations is a natural formalisation target once the cotangent-bundle symplectic structure and Hamiltonian vector field are in place. Specific missing pieces: a definition of HamiltonJacobiEquation as a bundled first-order PDE on Q × ℝ, the complete-integral type class with its non-degeneracy condition det (∂²S / ∂q ∂α) ≠ 0, the Jacobi theorem stating that complete integrals solve the mechanical problem via quadrature, and the separation-of-variables machinery for natural Hamiltonians with the Stackel condition. Each of these would require as prerequisites a formalisation of the Legendre transform, the canonical-transformation generating-function calculus, and the method of characteristics for first-order PDEs. This unit ships without a lean_module.

Advanced results [Master]

The Hamilton-Jacobi equation on manifolds

On a configuration manifold $Q$ with Hamiltonian $H : T^{*} Q \times R \to R$ , the Hamilton-Jacobi equation is a PDE for $S : Q \times R \to R$ :

H (q, d S_{q}, t) + \frac{\partial S}{\partial t} = 0,

where $d S_{q} \in T_{q}^{*} Q$ is the exterior derivative of $S$ restricted to the $Q$ -direction. The momentum field $p (q, t) := d S_{q}$ is a section of $T^{*} Q$ . The characteristics of this PDE are the integral curves of the Hamiltonian vector field $X_{H}$ on $T^{*} Q$ . The solution surface $Σ = {(q, d S_{q}, t)} \subset T^{*} Q \times R$ is a Lagrangian submanifold 05.05.01 of the extended phase space with its natural contact structure.

Generating-function interpretation

Hamilton's principal function $S$ is a type-2 generating function that produces the time- $t$ map of the Hamiltonian flow. Specifically, if $ϕ_{t} : T^{*} Q \to T^{*} Q$ is the Hamiltonian flow, then $S (q, t) = \int_{t_{0}}^{t} L \circ γ d t^{'}$ evaluated on the trajectory from $(q_{0}, t_{0})$ to $(q, t)$ satisfies $p (t) = \partial S / \partial q$ and $- p_{0} = \partial S / \partial q_{0}$ , so $S$ generates the canonical transformation $ϕ_{t}$ . The HJ equation is the statement that the time-1 map of the flow is generated by a function satisfying this PDE.

The eikonal equation and geometric optics

For a particle of mass $m$ in potential $V (q)$ , the time-independent HJ equation at energy $E$ is:

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

which rearranges to $∣\nabla W ∣ = 2 m (E - V (q))$ . This is formally identical to the eikonal equation of geometric optics: $∣\nabla ψ ∣ = n (x) / ψ$ , where $n (x)$ is the refractive index. The identification is $n (x) \propto E - V (x)$ : regions of high potential correspond to regions of low refractive index, and mechanical "light rays" (trajectories) bend toward regions of low potential, just as optical rays bend toward regions of high refractive index.

Hamilton's original insight (1834) was to read this backwards: just as Fermat's principle (shortest optical path) produces ray optics from the eikonal equation, Maupertuis' principle (least action) produces mechanics from the HJ equation. The wave-particle duality of optics has its exact counterpart in mechanics; Schrodinger's 1926 contribution was to take the wave picture seriously and write down the wave equation whose short-wavelength limit is the eikonal equation — that wave equation is the Schrodinger equation.

Separation of variables and the Stackel condition

The HJ equation separates in coordinates $(q^{1}, \dots, q^{n})$ when a complete integral can be written as $W = W_{1} (q^{1}) + \dots + W_{n} (q^{n})$ . This is possible when the Hamiltonian has the Stackel form: there exists an invertible matrix $Φ (q)$ and a vector of separation constants $α$ such that the Hamiltonian can be decomposed into a sum $H = \sum_{i} f_{i} (q^{i}, p_{i}, α)$ where each $f_{i}$ depends on only one pair $(q^{i}, p_{i})$ . The Stackel condition characterises all separable coordinate systems for natural Hamiltonians $H = g^{ij} p_{i} p_{j} /2 + V (q)$ on a Riemannian manifold $(Q, g)$ : separation occurs when the Hamilton-Jacobi equation admits $n$ quadratic first integrals, and the metric $g$ belongs to the Stackel class. This is the geometric classification of when the HJ method works in full.

The WKB approximation and the quantum-classical bridge

Write the Schrodinger wavefunction as $ψ (q, t) = A (q, t) exp (i S (q, t) /ℏ)$ and substitute into the time-dependent Schrodinger equation $i ℏ \partial ψ / \partial t = - (ℏ^{2} /2 m) \nabla^{2} ψ + V ψ$ . Separating real and imaginary parts at leading order in $ℏ$ yields:

\frac{( \nabla S ) ^{2}}{2 m} + V + \frac{\partial S}{\partial t} = O (ℏ) .

As $ℏ \to 0$ , this becomes the Hamilton-Jacobi equation. The next-order correction gives a transport equation for the amplitude $A$ that is equivalent to Liouville's theorem for the classical probability density. The WKB expansion is thus an asymptotic series in $ℏ$ whose leading term is classical mechanics via the HJ equation. The quantum-classical correspondence is exact at this order: classical mechanics is the short-wavelength (eikonal) limit of quantum mechanics, and the HJ equation is the bridge equation.

Maupertuis' principle

The principle of least action in the Maupertuis form states: among all paths $q (t)$ with fixed energy $E$ connecting $q_{0}$ to $q_{f}$ , the physical trajectory makes the abbreviated action $W = \int p_{i} d q^{i}$ stationary. The function $W (q)$ appearing in the time-independent HJ equation is exactly this abbreviated action, regarded as a function of the endpoint. This gives a variational characterisation of $W$ that is independent of the canonical-transformation derivation: $W$ is the action functional of Maupertuis' principle, and the HJ equation is its Hamilton-Jacobi PDE.

Caustics and the breakdown of the HJ solution

Hamilton's principal function $S (q, t)$ is defined by the on-shell action along a unique classical trajectory from $(q_{0}, t_{0})$ to $(q, t)$ . When multiple trajectories reach the same point $(q, t)$ , the function $S$ becomes multi-valued. This happens at caustics — envelopes of the family of classical trajectories where neighbouring rays focus.

At a caustic, the mapping $(q_{0}, P) \mapsto q (t; q_{0}, P)$ from initial data to endpoint fails to be locally invertible: the Jacobian $det (\partial q^{i} / \partial P_{j})$ vanishes. Equivalently, the non-degeneracy condition in Jacobi's theorem, $det (\partial^{2} S / \partial q^{i} \partial α_{j}) \neq = 0$ , breaks down. The generating-function interpretation of $S$ fails at caustics because the type-2 generating function cannot describe a canonical transformation at points where the coordinate change has vanishing determinant.

The physical signature of a caustic is infinite classical density: the momentum field $p = \partial S / \partial q$ develops a singularity, and the Liouville density (which is proportional to $∣ det (\partial q / \partial P) ∣^{- 1}$ ) diverges. In the WKB picture 12.10.01 pending, this corresponds to the breakdown of the leading-order semiclassical approximation — the amplitude $A$ diverges and the next-order transport equation ceases to normalise. The resolution requires the uniform approximation (Ludwig 1966, Kravtsov 1968) or, equivalently, the full quantum wavefunction, which remains finite and develops Airy-function diffraction patterns near caustics. Caustics connect the HJ framework to singularity theory: the generic caustic types are classified by the Arnol'd-Thom catastrophe hierarchy (fold, cusp, swallowtail, ...).

Hamilton-Jacobi theory and the Kolmogorov-Arnol'd-Moser theorem

For a near-integrable Hamiltonian $H = H_{0} (I) + ε H_{1} (I, θ)$ with $(I, θ)$ action-angle variables of the integrable part $H_{0}$ , the HJ equation becomes:

H_{0} (\frac{\partial S}{\partial θ}) + ε H_{1} (θ, \frac{\partial S}{\partial θ}) + \frac{\partial S}{\partial t} = 0.

For $ε = 0$ the complete integral is $S = \sum_{i} I_{i} θ^{i} - H_{0} (I) t$ . For $ε \neq = 0$ , one seeks a canonical transformation $(I, θ) \to (J, ϕ)$ that removes the angle dependence order by order in $ε$ . The generating function $S (θ, J, t)$ satisfies the perturbed HJ equation, and the perturbation series is a formal power series in $ε$ .

The KAM theorem (Kolmogorov 1954, Arnol'd 1963, Moser 1962) addresses what happens to this construction: for sufficiently small $ε$ and Diophantine frequency vectors $ω = \partial H_{0} / \partial I$ , the invariant tori survive as perturbed Lagrangian tori. The generating function converges on these surviving tori. For resonant frequencies (rational ratios), the tori break up into island chains and stochastic layers. The HJ equation thus provides the perturbation-theoretic framework within which KAM theory operates: the question of whether the perturbed HJ equation has a complete integral is answered negatively in general (Poincare proved the series diverges for generic perturbations), but KAM identifies the subset of phase space where the integral exists in a suitably weakened sense.

Synthesis. The Hamilton-Jacobi equation is the foundational reason that classical mechanics admits a wave-optics reformulation: the central insight is that a single scalar PDE on configuration space encodes the full phase-space dynamics through its characteristics. This is exactly the structure that identifies particle trajectories with wavefront normals, and the bridge is between the Lagrangian variational principle and the first-order PDE for the generating function. Putting these together, the HJ framework generalises across manifolds, connects to integrability via the Stackel condition and the Arnold-Liouville theorem, and builds toward 09.06.01 pending action-angle variables and 12.10.01 pending the WKB limit of quantum mechanics. The pattern recurs in 13.01.01 pending geodesic optics, where the HJ equation for the geodesic Hamiltonian gives the eikonal equation of geometric optics on curved spacetime. The quantum-classical bridge passes through the HJ equation as the short-wavelength limit of the Schrodinger equation, and the KAM perturbation theory operates within the HJ generating-function framework to characterise the survival of invariant tori in near-integrable systems.

Connections [Master]

09.04.02 pending Hamilton's equations. The HJ equation is the PDE whose characteristics are solutions of Hamilton's equations. Every solution of Hamilton's equations lies on a characteristic of the HJ equation, and every characteristic gives a solution. This is the PDE-ODE duality at the heart of the optical-mechanical analogy.
09.05.01 pending Canonical transformations. The HJ equation is the equation satisfied by the generating function of the canonical transformation that reduces $H$ to zero. The entire theory of canonical transformations — symplectomorphisms of $T^{*} Q$ — is the ambient framework.
09.06.01 pending Action-angle variables. For integrable systems, the action variables $J_{i}$ are defined as loop integrals of $p_{i} d q^{i}$ on the invariant tori, and the angle variables $θ^{i}$ are conjugate. The action variables arise as the separation constants when the HJ equation separates; the construction of action-angle coordinates is a direct application of the HJ method.
12.10.01 pending Path integrals. In the semiclassical limit $ℏ \to 0$ , the Feynman path integral is dominated by the classical trajectory. The stationary-phase approximation yields $ψ \sim A exp (i S_{cl} /ℏ)$ where $S_{cl}$ is Hamilton's principal function — the solution of the HJ equation. The connection from the quantum side to the classical HJ equation passes through the path-integral formulation.
13.01.01 pending General relativity and null geodesics. The HJ equation for the geodesic Hamiltonian $H = g^{ij} p_{i} p_{j} /2$ on a Riemannian manifold $(M, g)$ gives $∣\nabla W ∣_{g}^{2} = 2 E$ , which is the eikonal equation for the metric $g$ . In Lorentzian signature, null geodesics satisfy the HJ equation at $E = 0$ : the eikonal equation for light propagation in curved spacetime.
02.12.01 Phase space, vector fields, integral curves. The characteristic equations of the HJ equation define a vector field (the Hamiltonian vector field $X_{H}$ ) on phase space; the integral curves of $X_{H}$ are the classical trajectories. The phase-space picture is the natural habitat for the HJ theory.

Historical & philosophical context [Master]

William Rowan Hamilton published "On a general method in dynamics" in the Philosophical Transactions of the Royal Society in 1834 ^{[Hamilton 1834]}. He had spent the preceding decade developing a unified mathematical treatment of geometrical optics, in which the eikonal equation governs the propagation of wavefronts and its characteristics are the light rays. His 1834 paper transplanted the entire optical framework to mechanics: the action $S$ plays the role of the optical path length (the eikonal), mechanical trajectories are the "rays," and the HJ equation is the mechanical eikonal equation.

Carl Gustav Jacob Jacobi recognised the power and depth of Hamilton's construction and developed it into a systematic integration method in his Vorlesungen uber Dynamik (lectures of 1842-43, published posthumously in 1866 ^{[Jacobi 1866]}). Jacobi's contribution was the complete-integral theorem: a complete integral of the HJ equation suffices to solve the mechanical problem without integrating any differential equation beyond quadratures. He also developed the separation-of-variables technique and applied it to central-force problems and the geodesics of ellipsoids. The theory as presented today — the PDE, the complete integral, the separation method — is due to Jacobi as much as to Hamilton.

The optical-mechanical analogy lay dormant as a mathematical curiosity until 1926, when Erwin Schrodinger, influenced by de Broglie's matter waves and by the recognition that the HJ equation is the short-wavelength limit of a wave equation, wrote down the equation that bears his name. The Schrodinger equation is to the HJ equation what the wave equation is to the eikonal equation: the finite-wavelength completion. Hamilton's 1834 insight — that mechanics and optics share the same mathematical structure — was, with the addition of a single postulate (the wavelength $λ = h / p$ ), the seed of quantum mechanics.

The philosophical significance is that the HJ equation reveals classical mechanics to be a "ray optics" approximation to a deeper wave theory. Deterministic particle trajectories are the characteristics of a wave-like PDE. The classical-quantum transition is not a replacement of one framework by another but a passage from the short-wavelength limit to the full wave picture — just as ray optics gives way to wave optics when apertures become comparable to the wavelength.

Bibliography [Master]

Hamilton, W. R., "On a general method in dynamics," Phil. Trans. Roy. Soc. 124 (1834), 247–308.
Hamilton, W. R., "Second essay on a general method in dynamics," Phil. Trans. Roy. Soc. 125 (1835), 95–144.
Jacobi, C. G. J., Vorlesungen uber Dynamik (lectures of 1842-43, published 1866), ed. A. Clebsch.
Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd ed. (Springer GTM 60, 1989), §46–47.
Landau, L. D. & Lifshitz, E. M., Mechanics, 3rd ed. (Course of Theoretical Physics Vol. 1, Pergamon, 1976), §46–47.
Goldstein, H., Poole, C. P. & Safko, J., Classical Mechanics, 3rd ed. (Pearson, 2002), Ch. 10.
Susskind, L. & Hrabovsky, G., The Theoretical Minimum: Classical Mechanics (Basic Books, 2014), Lecture 9–10.
Tong, D., Classical Dynamics (DAMTP Cambridge lecture notes), §4 "Hamilton-Jacobi theory."
Abraham, R. & Marsden, J. E., Foundations of Mechanics, 2nd ed. (Addison-Wesley, 1978), §5.2.
Marsden, J. E. & Ratiu, T. S., Introduction to Mechanics and Symmetry, 2nd ed. (Springer TAM 17, 1999).
Schrodinger, E., "Quantisierung als Eigenwertproblem," Annalen der Physik 79 (1926), 361–376.
Lanczos, C., The Variational Principles of Mechanics, 4th ed. (Dover, 1986), Ch. VIII.
Synge, J. L., Classical Dynamics (Handbuch der Physik III/1, Springer, 1960).
Courant, R. & Hilbert, D., Methods of Mathematical Physics, Vol. II (Interscience, 1962), Ch. II (characteristics of first-order PDEs).
Stackel, P., "Uber die Integration der Hamilton-Jacobischen Differentialgleichung mittels Separation der Variabeln," Habilitationsschrift, Halle (1891).

Wave 2 physics unit, produced 2026-05-18. All three cross-domain hooks_out targets are proposed; no QM/GR/action-angle seed unit yet exists to receive confirmed promotion. Status remains draft pending Tyler's review and the §11 Next-Actions retro per PHYSICS_PLAN.

Prerequisites

09.05.01 pending
09.04.02 pending
09.04.01 pending
02.12.01 pending

Used in

09.06.01
09.08.01

Tier anchors

beginner: Susskind & Hrabovsky, *The Theoretical Minimum: Classical Mechanics* (2014), Lecture 9–10
intermediate: Goldstein, *Classical Mechanics* 3e, Ch. 10
master: Arnold, *Mathematical Methods of Classical Mechanics*, 2nd ed. (1989), §46–47; Landau & Lifshitz, *Mechanics*, 3rd ed. (1976), §47

References

tong
raw/pdfs/dynamics/four.pdf · §4 Hamilton-Jacobi theory — the action as generating function, complete integrals, separation of variables
TODO_REF
Goldstein, Poole & Safako — Classical Mechanics, 3rd ed. (Pearson, 2002) · Ch. 10 The Hamilton-Jacobi Theory
TODO_REF
Hamilton — On a general method in dynamics, Phil. Trans. Roy. Soc. 124 (1834) · 247-308; originator paper
TODO_REF
Arnold — Mathematical Methods of Classical Mechanics, 2nd ed. (Springer GTM 60, 1989) · §46 Huygens' principle; §47 The Hamilton-Jacobi method
TODO_REF
Landau & Lifshitz — Mechanics, 3rd ed. (Course of Theoretical Physics Vol. 1, Pergamon, 1976) · §46 The Hamilton-Jacobi equation; §47 Separation of variables

Reviewer

Tyler (pending external classical-mechanics reviewer per PHYSICS_PLAN §6)

Estimated time

beginner: 15m
intermediate: 40m
master: 55m