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$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}}\newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}\newcommand{\Label}[1]{\label{#1}}\newcommand{\eqRef}[1]{\eqref[#1]}$
A canonical transformation is a change of variable in phase space such that the equations of motion in the new variables are of the Hamiltonian form.
In the Lagrangian form of dynamics, the Euler Lagrange equations of motion are derived from Lagrangian \(\mathcal (q, \dot{q},t)\). There is a lot of flexibility in choice of generalized coordinates. If a change of coordinates is made from \(q_k\to Q_k= Q_k(q)\),the equations of motion in the new coordinates coincide with Euler Lagrange equations of motion, with a new Lagrangian \(\mathcal L^\prime(Q,\dot{Q},t)\), obtained from \(\mathcal L (q, \dot{q},t)\) by changing the variables from \(q_k\) to \(Q_k\): \begin{equation} \mathcal L^\prime(Q,\dot{Q},t)\equiv\mathcal L(q, \dot{q},t). \end{equation} In the phase space formulation, the state of a system is described by generalized coordinates and momenta \((q_k,p_k), k=1,..,n.\) Equations of motion are given by Hamilton's equations
\begin{equation} \dot{q}_k = \pp[H]{p_k}, \qquad \dot{p}_k= - \pp[H]{q_k}.
\end{equation}
What can we say about change of variables \(q_k, p_k \to Q_k, P_k\),
\begin{equation}\label{EQ03} Q_k=Q_k(q, p, t), \qquad P_k=P_k(q, p, t),
\end{equation}
in phase space? The above transformation equations must be invertible. This means the it should be possible to solve the above equations and write the old variables \(q_k,p_k\) as functions of new variables \(Q_k,P_k\):
\begin{equation}\label{EQ03A} q_k=q_k(Q, P, t), \qquad p_k=p_k(Q, P, t), \end{equation}
For a general change of variable, such as the one in equation above, it is not necessary that new equations of motion be of the Hamiltonian form. Only those transformations are useful when the equations of motion have the Hamiltonian form \begin{equation} \dot{Q}_k = \pp[K]{P_k}, \qquad \dot{P}_k= - \pp[K]{Q_k}. \end{equation} for some function \(K(Q,P) \) of the new variables \(Q_k, P_k\). The class of transformations satisfying this requirement is called canonical transformation.
