\begin{definition}[Definition:Canonical Variable]
Let $\map {\mathbf y} x : \R \to \R^n$, $n \in \N $ be a vector-valued function.
Let $F: \R^{2 n + 1} \to \R$ be a differentiable mapping.
Let $J \sqbrk {\mathbf y}$ be a functional of the form:
:$\d J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$
Consider the (real) variables $x, \mathbf y, \mathbf y'$, and the mapping $F$.
Make the following transformation:
:$F_{y_i'} = p_i$
where $F_x$ denotes the partial derivative of $F$ {{WRT|Differentiation}} $x$.
Let $H$ be the Hamiltonian corresponding to $J \sqbrk {\mathbf y}$.
Then the new variables $x, \mathbf y, \mathbf p$, and the mapping $H$ corresponding to $J \sqbrk {\mathbf y}$ are called '''the canonical variables'''.
\end{definition}