[NOTES/CM-10002] A Summary Finite Canonical Transformations

We have four sets of variables \(q_k,p_k,Q_k,P_k\)  with two relations among them. Thus for any canonical transformation (CT) only two of the four variables are independent. For finite canonical transformations, it turns out to be useful to regard the generating function \(F\) taking a set of two independent variable consisting of one old and one new phase space variable. This leads us to  consider following four types of canonical transformations and corresponding generating functions are denoted as \(F_k, k=1,...,4\).

It may be remarked that the above classification is not exhaustive. Besides the above four types of transformations listed above, one can also have ``transformations of mixed type". If the generating function is given, the transformation equations can be worked out and the new Hamiltonian can be found. We illustrate this for a Type 1 CT. For other types of transformations a similar approach can be used to find the transformation equations.

Type 1 transformation
We consider the transformations for which the old and new generalized coordinates \(q_k,Q_k\) are independent variables. In this case the generating function will be a function of \(q_k, Q_k\) and we write it as \(F_i(q,Q,t)\). The remaining two variables \(p_k, P_k\) and the new Hamiltonian \(K\) are given by

\begin{equation} p_k = \pp[F_1]{q_k}, \qquad P_k = - \pp[F_1]{Q_k},\qquad K = H + \pp[F_1]{t}.\label{EQ09}\end{equation}

Note that every choice of a function \(F_1(q,Q,t)\) will 'generate' a canonical transformation. These above equations determine the transformation and the new Hamiltonian \(K(Q,P,t)\) in terms of the generating function.

Type 2 transformation
\In type two transformations \(q_k, P_k\) are considered independent. The transformation equations are \begin{equation} p_k= \pp[F_2]{q_k}, \quad Q_k = \pp[F_2]{P_k},\qquad K = H + \pp[F_2]{t}. \end{equation}

Type 3 transformation
In this case \(p_k, Q_k\) form independent sets of variables and the remaining variables in terms the generating function \(F_3(p,Q)\) are \begin{equation} q_k = -\pp[F_3]{p_k}, \qquad P_k = -\pp[F_3]{Q_k},\qquad K = H + \pp[F_3]{t} \end{equation}

Type 4 transformation
The case when the variables \(p_k, P_k\) are independent, will be called type 4 transformation. In this case the remaining variables \(q_k, Q_k\) are given by \begin{equation} q_k = -\pp[F_4]{p_k}, \qquad Q_k = \pp[F_4]{P_k}, \qquad K = H + \pp[F_4]{t}. \end{equation}

Note that in all the four cases we have \(K-H=\pp[F_k]{t}\).