# Definition:Canonical Variable

## Definition

Let $\mathbf y=\langle y_i\rangle_{1\le i\le n}$ be a vector-valued function.

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 variables $x,\mathbf y,\mathbf y',F$.

Now, make a transformation:

$F_{y_i'}=p_i$

Let $H$ be the Hamiltonian corresponding to $J\sqbrk{\mathbf y}$.

The new variables $x,\mathbf y,\mathbf p,H$ corresponding to $J\sqbrk{\mathbf y}$ are called the canonical variables.

## Also known as

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