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:


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

Then, in canonical variables:


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

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