06 May 2026

Hamilton-Jacobi Equation and Harmonic Oscillator

Hamilton-Jacobi equation, principal function, characteristic function, and harmonic oscillator example.

msc semester-i classical-mechanics hamilton-jacobi action

The Hamilton-Jacobi method is based on a simple idea: instead of solving directly for $q(t)$ and $p(t)$, try to find a single function whose derivatives generate the motion. This function is Hamilton’s principal function $S$.

The method connects three viewpoints: the action integral, canonical transformations, and Hamiltonian dynamics. It is especially useful when a problem can be separated into simpler parts.

Principal function

Along the true path,

\[dS=\sum_i p_i\,dq_i-H\,dt.\]

Comparing with

\[dS=\sum_i\frac{\partial S}{\partial q_i}dq_i+\frac{\partial S}{\partial t}dt,\]

gives

\[p_i=\frac{\partial S}{\partial q_i}, \qquad \frac{\partial S}{\partial t}=-H.\]

Therefore

\[\boxed{ H\left(q_i,\frac{\partial S}{\partial q_i},t\right) +\frac{\partial S}{\partial t}=0. }\]

This is the Hamilton-Jacobi equation.

Thus the momentum is obtained from the spatial derivative of $S$, while the energy information is contained in its time derivative.

Time-independent case

If $H$ is independent of time, write

\[S(q_i,t)=W(q_i)-Et.\]

Then

\[\boxed{ H\left(q_i,\frac{\partial W}{\partial q_i}\right)=E. }\]

Here $W$ is Hamilton’s characteristic function.

This separation is used when the Hamiltonian has no explicit time dependence. The constant $E$ is then the conserved energy.

Constants of integration

A complete integral for $n$ degrees of freedom contains $n$ constants $\alpha_i$:

\[S=S(q_i,\alpha_i,t).\]

The remaining constants are obtained from

\[\boxed{ \frac{\partial S}{\partial\alpha_i}=\beta_i. }\]

The constants $\alpha_i$ and $\beta_i$ determine the motion.

Example: harmonic oscillator

For

\[H=\frac{p^2}{2m}+\frac12m\omega^2q^2,\]

the time-independent Hamilton-Jacobi equation is

\[\frac{1}{2m}\left(\frac{dW}{dq}\right)^2 +\frac12m\omega^2q^2=E.\]

Thus

\[\frac{dW}{dq}=\sqrt{2mE-m^2\omega^2q^2}.\]

The characteristic function is

\[W(q,E)=\int\sqrt{2mE-m^2\omega^2q^2}\,dq.\]

Using

\[\frac{\partial S}{\partial E}=\frac{\partial W}{\partial E}-t=\beta,\]

one obtains the familiar oscillator motion

\[q=A\sin(\omega t+\phi),\]

where

\[A=\sqrt{\frac{2E}{m\omega^2}}.\]

Example: free particle

For a free particle,

\[H=\frac{p^2}{2m}.\]

The Hamilton-Jacobi equation is

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

For a time-independent Hamiltonian, put

\[S=W(x)-Et.\]

Then

\[\frac{1}{2m}\left(\frac{dW}{dx}\right)^2=E.\]

Thus

\[\frac{dW}{dx}=\sqrt{2mE}=p,\]

and

\[S=px-\frac{p^2}{2m}t.\]

Using

\[\frac{\partial S}{\partial p}=x-\frac{p}{m}t=\beta,\]

we obtain

\[x=\frac{p}{m}t+\beta,\]

which is uniform motion.

Main points

• $S$ generates momenta through $p_i=\partial S/\partial q_i$.
• The Hamilton-Jacobi equation is first order in $S$.
• For time-independent systems, $S=W-Et$.
• A complete integral gives constants of motion.
• The harmonic oscillator is recovered from the HJ equation.

Practice questions

1. Derive the Hamilton-Jacobi equation from $dS=p\,dq-H\,dt$.
2. Distinguish $S$ and $W$.
3. Solve the Hamilton-Jacobi equation for a free particle.
4. Set up the HJ equation for the harmonic oscillator.
5. Explain the role of $\partial S/\partial\alpha_i=\beta_i$.