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:

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

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:

\(\ds \bigvalueat {\delta x} {x \mathop = x_j}\)

\(=\)

\(\ds \delta x_j\)

\(\ds \bigvalueat {\delta y_i} {x \mathop = x_j}\)

\(=\)

\(\ds \delta_i^j\)

\(\ds j\)

\(=\)

\(\ds \tuple {0, 1}\)

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 3.13$: Derivation of the Basic Formula