[NOTES/CM-11002] Jacobi's Complete Integral

Jacobi's complete integral

Notation : For a classical path in phase space from \(t_0\) to \(t\), we use the notation \(q_0=q(t_0), p_0=p(t_0), q=q(t), p=p(t).\)

As a first step to introduce the Jacobi's complete integral, we seek a type I canonical transformation \[(q,p) \longrightarrow (\alpha, \beta)\] where \((\alpha,\beta)\) are a new set of variables which remain constant in time. To achieve this, we demand that the new Hamiltonian \(K\) \begin{equation}\label{EQ06} K = H(q, p) + \pp[S]{t}. \end{equation} be zero. The Hamiltonian equations then imply that the new coordinates and momenta \(\alpha, \beta\) remain constants. With \(K\) as in \eqRef{EQ06} and \(p\to \pp[S]{q}\), \(K=0\) becomes the Hamilton Jacobi equation \begin{equation}\label{HJE} \pp[S]{t} + H\Big(q, \pp[S]{q},t\Big) =0. \end{equation} As the next step, we note that equation \eqref{HJE} has \(n + 1\) first order derivatives, one w.r.t. variables \(q, t\) each. The solution will, therefore, have \(n + 1\) constants. Given a solution \(S\), \(S + C\) is also a solution. Thus one of the constants in the solution of Hamilton Jacobi equation is additive. A solution of the Hamilton Jacobi equation, which is written in terms of \(n\) non-additive constants, is known as  Jacobi’s complete integral

Relation with Hamilton's principle function

We will use \(S_H\) and \(S_J\) to distinguish between Hamilton’s principal function \(S_H(q,t;q_0,t_0)\) and Jacobi’s complete integral \(S_J(q,\alpha,t)\), respectively. In order to relate the Jacobi's complete integral with the Hamilton's principal function, we note that Hamilton's principal function \(S_J\) generates a type I canonical transformation from \(q, t\) to the initial values \(q_0,t_0\). We will now show that relation between the Hamilton's principal function and Jacobi's complete integral is given by
\begin{equation} \label{EQ08A} S_J (q, \alpha, t) - S_J (q_0 , \alpha, t_0) =S_H (q, t; q_0t_0) \end{equation} The functions appearing in the left hand side generate canonical transformations as follows.
\begin{eqnarray} \text{CT1} &:& q,p \stackrel{ S_J (q, \alpha, t)}{\longrightarrow} \alpha, \beta \\ \text{CT2} &:& q_0, p_0 \stackrel{S_J (q_0 , \alpha, t_0)}{ \longrightarrow} \alpha, \beta \end{eqnarray}
The inverse transformation \(\alpha, \beta \longrightarrow q_0, p_0\) is generated by \(-S_J (q_0 , \alpha, t_0)\). The transformation CT1 followed by inverse of CT2, is just the transformation \begin{equation}\label{EQ10} \text{CT12} : q, p \rightarrow q_0, p_0\end{equation} Hamilton's principal function appearing in the right hand side of \eqref{EQ08A} is the generator of this transformation CT12. Also the left hand side of the \eqref{EQ08A} becomes the generator of transformation, CT12, if \(\alpha \) is determined by requiring \begin{equation} \pp[S_J (q, \alpha, t)]{\alpha_k} -\pp[S_J (q_0, \alpha, t_0)]{\alpha_k} =0. \end{equation} Thus subject to the above condition on \(\alpha\), the left hand side of \eqref{EQ08A} coincides with the Hamilton's principal function.