[NOTES/CM-10010] Group Structure of Canonical Transformations

A canonical transformation \[ \{q_k, p_k\} \longrightarrow \{Q_k, P_k\}\] on phase space is specified by giving functions \(Q_k= Q_k(q,p,t),P_k= P_k(q,p,t)\). The canonical transformations have the following properties.

1. If \[\{q,p\} \longrightarrow \{Q,P\}\] is a canonical transformation given by \( Q_k=Q_k(q,p,t), P_k=P_k(q,p,t)\) and if \[\{Q_k,P_k\} \longrightarrow \{Q^\prime_k,P^\prime_k\}\] is another canonical transformation given by functions \[Q^\prime_k=Q^\prime_k(Q,P,t), P^\prime_k=P^\prime_k(Q,P,t) \] then the composite transformation \[\{q_k,p_k\} \longrightarrow \{Q^\prime_k, P^\prime_k\}\] is also a canonical transformation given by functions \[Q^\prime_k= Q^\prime(Q(q,p,t), P(q,p,t)),\qquad P^\prime_k= P^\prime(Q(q,p,t), P(q,p,t)).\].
2. The process of combining two canonical transformations to obtain a another canonical transformation is the same as applying two mappings in a succession. \[ f\circ (g\circ h) = (f\circ g)\circ h\] and it obeys associative property.
3. The identity transformation \(\{q,p\} \longrightarrow \{Q,P\}\), where \(Q=q, P=p \), is a canonical transformation
4. If \(\{q,p\} \longrightarrow \{Q,P\}\) is a canonical transformation, then inverse transformation \(\{Q,P\} \longrightarrow \{q,p\}\) exists and is a canonical transformation.

The above four properties imply that the set of all canonical transformations is a group.