[NOTES/CM-10007] Generator of a Canonical Transformation

Canonical Transformation
The phase space formulation of classical dynamics offers the possibility of canonical transformations (CT) from an old set \(\{(q_k, p_k),k=1,2,\ldots\}\) to a new set \(\{(Q_k, P_k), k=1,2,\ldots\}\) \begin{equation} Q_k=Q_k(q,p,t), \qquad P_k=P_k(q,p,t).\, \end{equation} which are functions of old variables \((q_k, p_k)\).

A transformation \((q_k, p_k) \to (Q_k, P_k)\) is called canonical if equations of motion in terms of new variables \(Q_k,P_k\) are of the Hamiltonian form. In other words one should be able to find a function \(K(q,p,t\) such that the equations of motion (EOM) \begin{equation} \dot{Q}_k = \pp[K]{P_k}, \qquad \dot{P}_k = - \pp[K]{Q_k} \end{equation} are equivalent to the EOM in terms of the original variables \(q_k, p_k\). In other words, {\it i.e.} the equations
\begin{eqnarray} \text{Old EOM } &\qquad&\dot{q}_k = \pp[H]{p_k}, \qquad \dot{p}_k = - \pp[H]{q_k}\\ \text{New EOM } &\qquad& \dot{Q}_k = \pp[K]{P_k}, \qquad \dot{P}_k = - \pp[K]{Q_k}. \end{eqnarray}
should be equivalent. Also this equivalence should hold for arbitrary Hamiltonian function \(H(q,p)\).

Finite and infinitesimal CT
The new variables may depend on some continuous parameters. For example, new variables may be related to the old variables by a rotation of axes. In such a case it may be sufficient to discuss infinitesimal transformations, {\it e.g.} rotations by infinitesimal angles. Thus we speak of {\tt infinitesimal CT} when CT depends on a continuous parameter having an infinitesimal value.

A canonical transformation will be called a {\tt finite CT} if is it not an infinitesimal transformation. So following two possibilities arise

1. CT does not depends on a continuous parameter, An example of this case is \(Q=p, P=-q\);
2. CT depends on a continuous parameter having a finite value, for example a rotation by an angle \(\pi/3\). 

Action Principle
The Hamiltonian equations of motion can be derived from an action principle. Define action \begin{equation} S = \int_1^2 (\sum_k p_k \dot{q}_k - H(q,p,t) ) dt. \end{equation} The path in phase followed by the system is the one for which the action \(S\) is extremum under infinitesimal variations of the phase space variables keep the generalized coordinates \(q_k\) fixed at the end points.

The action in terms of new variables \(Q_k,P_k\) and new Hamiltonian \(K(Q,P,t)\) \begin{equation} \underline{S} = \int_1^2 (\sum_k P_k \dot{Q}_k - K(Q,P,t) ) dt. \end{equation} should also give the same equations of motion. Obviously it is sufficient to demand that the integrands appearing in \(S\) and \(\underline{S}\) differ by a total time derivative, {\it i.e.}
\begin{equation}\Label{EQ06} \sum_k p_k \dot{q}_k - H(q,p,t)dt =\sum_k P_k \dot{Q}_k - K(Q,P,t) + \dd[F]{t}. \end{equation}
For later use, we write this equation will be rewritten as
\begin{equation}\Label{EQ07}
\sum_k p_k {dq}_k - H(q,p,t) dt =\sum_k P_k {dQ}_k - K(Q,P,t) dt +dF.
\end{equation}
The function \(F\) is called generating function of canonical transformation.