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:


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

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