[NOTES/CM-11004] Hamilton Jacobi Equation

Hamilton Jacobi equation

For a system with \(N\) degrees of freedom, the Euler Lagrange equations are \(N\) second order differential equations. The solution will involve \(2N\) constants of integration. Therefore, we can write
\begin{equation}\label{EQ05}
q = q(c, t), \qquad q_0=q(c,t_0),
\end{equation}
where \(q_0\) are the coordinates at initial time \(t_0\). The action integral
\begin{equation}\label{EQ01}
S =\int_{t_0}^{t} \sum_k p_k \dot q_k - H(q, p,t) dt
\end{equation}
can be be evaluated along a classical path and will be a function of the constants of integration and times \(t_0, t\) \[ S= S(c, t,t_0).\] Suppose we solve \eqref{EQ05} and write the constants of integration as functions \(c=c(q_0, t_0, q,t)\), we will get the action functional as a function of \(q_0,t,q,t\):
\begin{equation}
S\equiv S(q_0,t_0,q,t). \label{EQ02}
\end{equation}
We emphasize that \eqref{EQ02} is the action functional along a classical trajectory with generalized coordinates having values  \(q_0, q\) at initial and final times. This function \(S\) is called Hamilton's principle function. It is a type 1 generator of canonical transformation \[ (q,p) \rightarrow (q_0,p_0)\]. and satisfies
\begin{eqnarray}\label{EQ03}
p=\pp[S]{q}, \qquad p_0=- \pp[S]{q_0}
\end{eqnarray}
and
\begin{equation}\label{EQ04}
\pp[S]{t} + H(q,p) =0.
\end{equation}
Substituting for \(p\) from \eqRef{EQ03} in \eqRef{EQ04}, we see that Hamilton's principle function satisfies
\begin{equation}\label{EQ17A}
\boxed{\pp[S]{t} + H\Big(q,\pp[S]{q},t\Big)=0.}
\end{equation}
This equation is known as the  time dependent Hamilton Jacobi equation. Knowledge of the solution of the Hamilton Jacobi equation gives the solution of the equations of motion as the momenta can be computed using \eqref{EQ03}.

Hamilton's characteristic function
When the Hamiltonian is independent of time, time dependence can be separated by writing \(S=W(q)-Et\), where \(W\) is independent of time. Substituting \(S=W-Et\) in \eqref{EQ17A}, we get
\begin{equation}\label{EQ18}
\boxed{H\Big(q,\pp[W]{q}\Big)-E=0}.
\end{equation}
This equation is called  time independent Hamilton-Jacobi equation and \(W\) is known as {\tt Hamilton's characteristic function.}