How to find out whether a transformation is a canonical transformation? We had a couple of examples where we were supposed to calculate the Canonical Transformation (CT), but we never actually talked about a condition that decides whether a transformation is a canonical one or not. 
Let me give you an example: We had the transformation: $$P=q \cdot \cot(p), \qquad Q=\ln \left(\frac{\sin(p)}{q}\right).$$ 
How do I see whether this transformation is a canonical one or not? 
You don't have to carry out the full calculation, but maybe you can give me a hint what I need to show here?
 A: Hint: Poisson Brackets are canonical invariants, this is
$$\{F,G\}_{q,p}=\{F,G\}_{Q,P} $$
A: Another way (a practical shortcut) is to try to find a generating function.  In this case, we shall use $F_3(Q, p)$ since $Q$ and $p$ appear to be more basic variable.  The original equations are equivalent to
\begin{align}
P &= q \, \cot p
\tag{1} \\
q &= e^{-Q} \, \sin p.
\tag{2}
\end{align}
Eq. (1) is equivalent to
\begin{align}
P = e^{-Q} \, \cos p.
\tag{3}
\end{align}
Now from Eqs. (2) and (3),
we can readily verify that $F_3(Q, p) = e^{-Q} \cos p$ satisfies
\begin{align}
P = - \frac{ \partial F_3 }{ \partial Q }, \tag{4} \\
q = - \frac{ \partial F_3 }{ \partial p }. \tag{5}
\end{align}
This means for the given transformation is generated by this $F_3(Q, p)$,
and hence is canonical.
Note that the possible functional form of $F_3(Q, p)$
can be deduced from a trial-and-error approach. In this case, we actually integrated Eq. (4),
$$
F_3 = -\int P \, dQ = -\int e^{-Q} \cos p \, dQ = e^{-Q} \cos p,
$$
and then verified it satisfied Eq. (5).
A: There are three easy tests to check if a transformation is canonical. Note
that some multiplicative constants might pop up in certain textbooks, depending
on the exact definition of canonical transformation. 
Notation
Let $x = (p, q)$ be the $2n$ variables, and the transformed variables be $\tilde{x}(x) =
(\tilde{p}(p, q), \tilde{q}(p, q))$. 
The method of the symplectic jacobian
Let $J = \partial \tilde{x} /\partial x $ be the Jacobian matrix of the
transformation. Moreover, let $\mathbb{E}$ be a $2n \times 2n$ block matrix
$$
\mathbb{E} = 
\begin{pmatrix}
    0 & 1 \\
    -1 & 0
\end{pmatrix}
$$
Then the transformation is canonical if and only if
$$
J\mathbb{E}J^T = \mathbb{E}
$$
The method of Poisson brackets
The transformation is canonical if and only if the fundamental Poisson
brackets are preserved
$$
\{\tilde{p}_i, \tilde{p}_j\} = 0 \qquad
\{\tilde{q}_i, \tilde{q}_j\} = 0 \qquad
\{\tilde{q}_i, \tilde{p}_j\} = \delta_{ij}
$$
The method of the Liouville differential form
This is somewhat less practical, but I include it for completeness.
The transformation is canonical if and only if the differential form
$\sum_i p_i \mathrm{d}q_i - \sum_i \tilde{p}_i \mathrm{d}\tilde{q}_i$
is closed.
A: The answer by Enucatl is satisfying enough.
However, in the example
$$P=q \cot(p),$$
$$Q=\ln \left (\frac{\sin(p)}{q}\right),$$ given in the question, it seems there is dimensional mismatching.
The argument inside $\cot$ must be some $[p/(p_o)]$ where $p_o$ has dimensions of momentum and the argument of the logarithm must be
$$q_o \frac{\sin(p/p'_o)}{q},$$
$p'_o$ neednot be equal to $p_o$. Even if P and Q do not have dimensions of momentum and length respectively it may not matter (well known as per any general case of a canonical transformation).
I am curious to know if the operations for dimensional matching implied (like the fashionable(which I don't like) way of certain books taking $c=1$ and calling the relativistic energy of a free particle $E =(m^2+p^2)^{1/2}$ instead of                  $ E = (m^2 c^4+p^2c^2)^{1/2}$ etc.).
