The Hamilton–Jacobi equation

Use a type-2 generating function S whose new momenta are constants denoted α. Setting the transformed Hamiltonian K to zero gives a first-order partial differential equation for S. Old momentum is the position gradient of S, and the new constant coordinates are derivatives with respect to α. A complete integral contains n independent α parameters for n degrees of freedom, with suitable nondegeneracy so the transformation actually provides the needed initial data locally.

The phrase principal function is sometimes used for several closely related action functions. A complete integral parameterized by constants is not identical to the action between a fixed initial endpoint and a variable final endpoint, although both can satisfy Hamilton–Jacobi equations in appropriate domains. Keep track of which endpoint or parameter is fixed before differentiating. Without those distinctions, one can confuse the initial momentum label with a spatial coordinate or mistake one selected family of paths for all solutions.

For a free particle, choose S linear in position with slope α. The Hamilton–Jacobi equation then fixes the time-dependent term. Differentiating with respect to position says that α is the constant momentum. Differentiating with respect to α gives a constant β that is the initial position at the chosen time origin. Solving that algebraic relation for q reconstructs the familiar straight trajectory. The method has moved integration into a generating function rather than introduced a new force law.

When the Hamiltonian is autonomous, a common separation is $S=W(q,\alpha)-Et$. The reduced function W satisfies a time-independent equation involving energy E. This does not mean that W alone records the entire physical time dependence; the omitted linear time term and the derivatives with respect to the constants are what recover timing. At turning points, branches of the momentum can meet and a single expression for W may need to be replaced by several local branches.

Fix a free particle at the origin at time zero and ask for the action to reach q at positive time t. The classical velocity is q divided by t, so its action is a quadratic function of q. Its position derivative gives the final momentum. Its time derivative, with the final position held fixed, is minus the final energy. This is an especially clear case where the meaning of held fixed matters: following one trajectory while differentiating would be a different operation.

In more complicated systems, different classical paths can reach the same endpoint. Their actions define separate branches, and caustics can make a single-valued local gradient description fail. Hamilton–Jacobi theory therefore supplies local structure and powerful separability tools without guaranteeing a global smooth solution. Its resemblance to short-wavelength wave phases is meaningful, but this classical equation alone does not include wave interference, tunneling probabilities, or quantum measurement. Those require additional physical structure and controlled approximation.