# Definition:Canonical Variable

## Definition

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:

$\displaystyle J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$

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 $F$ with respect to $x$.

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.

## Also known as

By analogy with mechanical problems, the variables $p_i$ are also known as momenta.