General Variation of Integral Functional/Dependent on N Functions/Canonical Variables

Theorem
Let $\delta J$ be a general variation of integral functional dependent on n functions.

Suppose a following coordinate transformation is done:


 * $\set {x, \ldots y_i, \ldots, \ldots, y_i', \ldots, F} \to \set {x, \ldots, y_i, \ldots, \ldots p', \ldots, H}, i = \left({1, \ldots, n}\right)$

Then, in canonical variables:


 * $\displaystyle \delta J = \int_{x_0}^{x_1} \sum_{i \mathop = 1}^n \paren {F_{y_i} - \dfrac {\d {p_i} } {\d x} } \map {h_i} x \rd x + \intlimits {\sum_{i \mathop = 1}^n p_i \delta y_i - H \delta x} {x \mathop = x_0} {x \mathop = x_1}$

where:
 * $\delta x \rvert_{x \mathop = x_j} = \delta x_j$
 * $\delta y_i \rvert_{x \mathop = x_j} = \delta_i^j$
 * $j = \tuple {0, 1}$