General Variation of Integral Functional/Dependent on N Functions/Canonical Variables
Jump to navigation
Jump to search
This article, or a section of it, needs explaining, namely:
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.
This theorem requires a proof..
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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:
| \(\displaystyle \bigvalueat {\delta x} {x \mathop = x_j}\) | \(=\) | \(\displaystyle \delta x_j\) | |||||||||||
| \(\displaystyle \bigvalueat {\delta y_i} {x \mathop = x_j}\) | \(=\) | \(\displaystyle \delta_i^j\) | |||||||||||
| \(\displaystyle j\) | \(=\) | \(\displaystyle \tuple {0, 1}\) |
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
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 3.13$: Derivation of the Basic Formula
