||Message]
Category:
$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}} \newcommand{\Label}[1]{\label{#1}} \newcommand{\EqRef}[1]{Eq.\eqref{#1}} \newcommand{\eqRef}[1]{\eqref{#1}}$
Canonical transformation is defined in three different,equivalent ways.
Definition 1
Let \(q_k,p_k \) be a set of canonical variables obeying Hamiltonian equations of motion.
\begin{equation}\Label{EQ03B}
\dot q_k = \pp[H]{p_k}, \qquad \dot p_k =- \pp[H]{q_k}.
\end{equation}
A transformation \(q_k,p_k \to Q_k,P_k \)
\begin{equation}\Label{EQ03}
Q_k= Q_k(q,p), \qquad P_k=P_k(q,p)
\end{equation}
is called canonical if there exists a function \(K(Q,P,t)\) such that the equations of motion can have the Hamiltonian from in the new variables \(Q_k,P_k\). This means that there exists a function \(K(Q,P)\) such that the equations
\begin{equation}\Label{EQ03A}
\dot Q_k = \pp[K]{P_k}, \qquad \dot P_k =- \pp[K]{Q_k}.\end{equation}
are equivalent to Eqs.\eqref{EQ03B}. Moreover this property must be independent of the Hamiltonian \(H\) in \eqref{EQ03B}.
Definition 2
A transformation \eqref{EQ03} is called canonical if the difference
\begin{equation}
\sum_k p_k dq_k - \sum_k P_k dQ_k
\end{equation}
is a differential of some function \(F\):
\begin{equation}
\sum_k p_k dq_k - \sum_k P_k dQ_k = dF
\end{equation}
Definition 3
The coordinates and momenta obey the Poisson bracket relations \begin{equation}\label{EQ07}
\{q_k,q_\ell\}_\text{PB} =0, \quad \{q_k,p_\ell\}_\text{PB}
=\delta_{k\ell},\quad \{p_k,p_\ell\}_\text{PB} =0,
\end{equation}
A transformation \eqref{EQ03} is called canonical if it preserves the Poisson brackets. This means the functions \( Q_k(q,p,t), P_\ell(q,p,t)\) should obey the Poisson bracket relations
\begin{equation}
\{Q_k,Q_\ell\}_\text{PB} =0, \quad \{Q_k,P_\ell\}_\text{PB}
=\delta_{k\ell},\quad \{P_k,P_\ell\}_\text{PB} =0,
\end{equation}
In fact for a canonical transformation the Poisson bracket of \(q_k(Q,P,t),p_\ell(Q,P,t)\), computed using \(Q,P\) as canonical variables, should also be same as in \eqref{EQ07}.
Remark :It can be shown that all the three definitions given above are equivalent.
