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:


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

Then, in canonical variables:


 * $\displaystyle \delta J = \int_{x_0}^{x_1} \sum_{i \mathop = 1}^n \left({F_{y_i} - \dfrac {\d {p_i} } {\d x} }\right) h_i \left({x}\right) \rd x + \left({\sum_{i \mathop = 1}^n p_i \delta y_i - H \delta x}\right) \Bigg\rvert_{x \mathop = x_0}^{x \mathop = x_1}$

where $\delta x \rvert_{x \mathop = x_j} = \delta x_j, \delta y_i \rvert_{x \mathop = x_j} = \delta _i^j, j = \left({0, 1}\right)$