Definition:Canonical Variable

Definition
Let $J \left[{\ldots, y_i, \ldots}\right]$ be a functional of the form:


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

Consider the variables $x, y_1, \ldots, y_n, y_1', \ldots, y_n', F$.

Now, make a transformation:


 * $F_{y_i'} = p_i$

Let $H$ be the Hamiltonian corresponding to $J \left[{\ldots, y_i, \ldots}\right]$.

The new variables $x, y_1, \ldots, y_n, p_1, \ldots, p_n, H$ corresponding to $J \left[{\ldots, y_i, \ldots}\right]$ are called the canonical variables.