Definition:Canonical Variable

Definition
Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$ 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 $x, \mathbf y, \mathbf y', F$.

Make the following 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.