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

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Theorem

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

Suppose a following coordinate transformation is done:

$\set {x, \ldots y_i, \ldots, \ldots, y_i', \ldots, F} \to \set {x, \ldots, y_i, \ldots, \ldots p', \ldots, H}, i = \tuple {1, \ldots, n}$

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:

\(\displaystyle \bigvalueat {\delta x} {x \mathop = x_j}\) \(=\) \(\displaystyle \delta x_j\)
\(\displaystyle \bigvalueat {\delta y_i} {x \mathop = x_j}\) \(=\) \(\displaystyle \delta_i^j\)
\(\displaystyle j\) \(=\) \(\displaystyle \tuple {0, 1}\)



Proof


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