Editing Canonical transformation (section)

==Notation==
Boldface variables such as {{math|'''q'''}} represent a list of {{mvar|N}} [[generalized coordinates]] that need not transform like a [[Vector (geometric)|vector]] under [[rotation]] and similarly {{math|'''p'''}} represents the corresponding [[generalized momentum]], e.g.,
<math display="block">\begin{align}
\mathbf{q} &\equiv \left (q_{1}, q_{2}, \ldots, q_{N-1}, q_{N} \right )\\
\mathbf{p} &\equiv \left (p_{1}, p_{2}, \ldots, p_{N-1}, p_{N} \right ).   
\end{align}</math>

A dot over a variable or list signifies the time derivative, e.g.,
<math>\dot{\mathbf{q}} \equiv \frac{d\mathbf{q}}{dt}</math>and the equalities are read to be satisfied for all coordinates, for example:<math>\dot{\mathbf{p}} = -\frac{\partial f}{\partial \mathbf{q}}\quad  \Longleftrightarrow \quad  \dot{{p_i}} = -\frac{\partial f}{\partial {q_i}} \quad (i = 1,\dots,N). </math>

The [[dot product]] notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, e.g.,
<math>\mathbf{p} \cdot \mathbf{q} \equiv \sum_{k=1}^{N} p_{k} q_{k}.</math>

The dot product (also known as an "inner product") maps the two coordinate lists into one variable representing a single numerical value. The coordinates after transformation are similarly labelled with {{math|'''Q'''}} for transformed generalized coordinates and {{math|'''P'''}} for transformed generalized momentum.