Definition:Canonical Variable

Definition
Let $J[...y_i...]$ be a functional of the form


 * $\displaystyle J[...y_i...]=\int_{x_0}^{x_1}F\left({x, ...y_i..., ...y_i'...}\right)\mathrm{d}{x},~i=\left({1, ..., n}\right)$

Consider the variables ${x, y_1, ..., y_n, y_1', ..., y_n', F}$.

Now, make a transformation


 * $F_{y_i'}=p_i$.

Then the new variables ${x, y_1, ..., y_n, p_1, ..., p_n,}$ $H$ corresponding to $J[...y_i...]$ are called the canonical variables.