Definition:Canonical Variable

Definition
Let $y_i$ be a real function.

Let $ \mathbf y= \langle y_i \rangle_{1 \le i \le n}$

Let $J \left[{ \mathbf y}\right]$ be a functional of the form:


 * $\displaystyle J \left[{ \mathbf y }\right] = \int_{x_0}^{x_1} F \left({x, \mathbf y, \mathbf y'}\right) \mathrm d x$

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 $J \left[{ \mathbf y}\right]$.

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

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