= Hamilton-Jacobi equation
{c}
{title2=$\partial_tS+H(q,\partial_qS,t)=0$}
A principal function $S(q,t)$ obeys the <Hamilton-Jacobi equation> $\partial_tS+H(q,\partial_qS,t)=0$. A complete solution generates <canonical momentum> components $p_j=\partial_jS$ and local canonical coordinates constant along motion. For a time-independent <Hamiltonian>, $S=W(q)-Et$ reduces the equation to $H(q,\partial_qW)=E$. Additive separation of $W$ yields conserved separation constants, subject to local regularity and independence.
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