General Variation of Integral Functional/Dependent on N Functions/Canonical Variables
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the meaning of $j = \tuple {0, 1}$ in this context, and hence the meaning of $x_j$ and $\delta x_j$, and also $\delta_i^j$, all of which are obscure
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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:
- $\set {x, \ldots y_i, \ldots, \ldots, y_i', \ldots, F} \to \set {x, \ldots, y_i, \ldots, \ldots p', \ldots, H}, i = \left({1, \ldots, n}\right)$
Then, in canonical variables:
- $\displaystyle \delta J = \int_{x_0}^{x_1} \sum_{i \mathop = 1}^n \paren {F_{y_i} - \dfrac {\d {p_i} } {\d x} } \map {h_i} x \rd x + \intlimits {\sum_{i \mathop = 1}^n p_i \delta y_i - H \delta x} {x \mathop = x_0} {x \mathop = x_1}$
where:
- $\delta x \rvert_{x \mathop = x_j} = \delta x_j$
- $\delta y_i \rvert_{x \mathop = x_j} = \delta_i^j$
- $j = \tuple {0, 1}$
the meaning of $j = \tuple {0, 1}$ in this context, and hence the meaning of $x_j$ and $\delta x_j$, and also $\delta_i^j$, all of which are obscure
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
Proof
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Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 3.13$: Derivation of the Basic Formula
