[NOTES/CM-11001] Hamilton's Principal Function

Hamiltonian equations of motion
For a system with \(n\) degrees of freedom, the Hamiltonian equations of motion, \begin{equation}\label{EQ02} \dot{q}_k = \pp[H]{q_k}, \qquad \qquad \dot{p}_k = - \pp[H]{q_k},\quad k=1,2,\ldots , \end{equation} are \(2n\) first order differential equations and the solution will be in terms of \(2n\) arbitrary constant \(c_k , k = 1,\ldots, n.\) \begin{equation}\label{EQ03} q_k = f_k (t, c),\qquad p_k = g_k (t, c), ~~k = 1,\ldots, n. \end{equation} where \(f, g\) are some functions of \(t\) and constants \(c\). Also \(\dot{q}\)'s are now determined as function of \(t\) and constants \(c\).

Hamilton's principal function
The equations of motion are obtained by demanding that the variation of action integral \begin{equation}\label{EQ01}S[q(t)] = \int_{t_0}^{t} L(q, \dot q, t)dt.\end{equation} be zero when the end points are fixed. Consider two points \((t_0 , t )\) on the trajectory with values of generalized coordinates \(q_0, q\) respectively. Then \begin{equation}\label{EQ04} q_0 = f (t_0, c), \qquad q = f (t, c). \end{equation} We assume that the \(2n\) equations \eqref{EQ04} can be solved for the \(2n\) constants and that the constants \(c\) can be expressed in terms of the coordinate values \(q_0, q\) at the end points. The action integral \eqRef{EQ01} computed along the trajectory, will turn out to be a function of \(q,t;q_0,t_0\) The action,the right hand side of \eqref{EQ01}, can therefore be expressed in terms of \(q_0, t_0, q, t\). Thus we write \begin{equation}\label{EQ05} S \equiv S(q,t;q_0,t_0), \end{equation} and \(S(q,t;q_0,t_0)\) will be called Hamilton’s principal function. General variations, i.e. variations without keeping end points fixed, of \eqref{EQ01} around the classical path give rise to change in action integral depending on the end points only. This variation is known to be given by \begin{eqnarray}\label{EQ06} \Delta S &=& S(q + \Delta q, t + \Delta t, q_0 + \Delta q_0, t 0 + \Delta t_0)\\ &=& (p\Delta q-H\Delta t)-(p_0\Delta q_0-H\Delta t_0 ). \label{EQ07} \end{eqnarray} where \(p, p_0\) are generalized momenta at times \(t, t_0\). The equations \eqref{EQ07} imply that \begin{equation}\label{EQ08} p=\pp[S]{q}, \qquad p_0 =-\pp[S]{q_0}. \end{equation}The above equations show that Hamilton’s principal function is a generator of type I canonical transformation from \((q, p)\) st time \(t\) to the coordinates \((q_0 , p_0 )\) at iniial time \(t_0\). We also note that \eqref{EQ07} implies that \begin{equation}\label{EQ09} \pp[S]{t}=-H(q,p,t), \qquad \pp[S]{t_0} = H(q_0,p_0,t_0). \end{equation}

To summarize, the Hamilton’s principal function can be written down if solution of the equations of motion is known. Conversely, if the Hamilton’s principal function is known, one can obtain the solution \(q = q(t), p = p(t)\).