Hamilton-Jacobi equation (${\partial }_{t}S+H(q,{\partial }_{q}S,t)=0$)

A principal function $S(q,t)$ obeys the Hamilton-Jacobi equation ${\partial }_{t}S+H(q,{\partial }_{q}S,t)=0$. A complete solution generates canonical momentum components $p_j={\partial }_{j}S$ and local canonical coordinates constant along motion. For a time-independent Hamiltonian, $S=W(q)-Et$ reduces the equation to $H(q,{\partial }_{q}W)=E$. Additive separation of $W$ yields conserved separation constants, subject to local regularity and independence.

A complete integral depends on as many independent parameters as configuration coordinates and supplies a locally invertible canonical transformation to constants of motion. For a type-one generating function $G(q,Q,t)$, require a nondegenerate mixed-derivative matrix in the relevant local chart. Solving $G_t+H(q,G_q,t)=0$ makes the transformed Hamiltonian zero, so both new canonical coordinates and new canonical momenta are constant. An arbitrary parameter-deficient solution does not ensure such a transformation.