Talk:Calculus of variations

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Needs more prose to accompany equations Tompw 17:54, 7 October 2006 (UTC)

The symbol L appears halfway through the discussion and is not defined. Is it the same as A, or something different?

Can someone give some books on Calculus of Variations?

"Calculus of Variations" by I M Gelfund and S V Fomin is a good book on this subject. Wilmot1 12:55, 18 February 2007 (UTC)
L is a functional, and A is the integral of L. Initially, the article defines A as the integral of a specific functional, but this is just an example. It switches to L to indicate that the same procedure works for other functionals.
The letters L and A were probably used because they correspond to the Lagrangian and the Action in Lagrangian Mechanics.
A and L are not the letter normally used in the literature S and F are more usual. However some authors use other letter. Wilmot1 12:55, 18 February 2007 (UTC)


Contents

[edit] Vote for new external link

Here is my site with calculus of variations example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith

http://www.exampleproblems.com/wiki/index.php/Calculus_of_Variations


If you are looking to extend the subject beyond its applications and look closely at the mathematical formalism try

Introduction to the Calculus of Variations by Bernard Dacorogna Imperial College Press 2004 ISBN: 186094499X

I found your examples clearly presented and useful.Dogchaser 09:02, 20 February 2007 (UTC)

[edit] Opening Paragraphs

The second paragraph says "The preceding examples have all involved unknown functions of a single variable, which may be identified with a time variable." I don't believe this is true. For example, in mechanics and optimal control, the functions can (and generally do) depend on space and time. (Cj67 17:44, 26 June 2006 (UTC))

[edit] notation bug in section on Fermat's principle in three dimensions

Is the x in P = \frac{n(x) \dot X}{\sqrt{\dot X \cdot \dot X} }.\, meant to be X? -- njh 10:02, 11 July 2006 (UTC)

Shouldn't \int_{x_1}^{x_2} \frac{ \frac{df_0}{dx} \frac{df_1}{dx} } {\sqrt{1 + \left(\frac{df_0}{dx}\right)^2}} =0, \, be \int_{x_1}^{x_2} \frac{ \frac{df_0}{dx} \frac{df_1}{dx} } {\sqrt{1 + \left(\frac{df_0}{dx}\right)^2}}dx =0, \,

[edit] Merge article with Variational Principle page

It seems like there is redundant content between this page and the variational principle page. I think they should be merged under the title of "calculus of variations", and have the article "Variational Principle" redirected to "Calculus of Variations." On a different note, does anyone know enough to write about about computational methods in the calculus of variations? I don't, but I think it's important. --69.180.18.247 14:48, 4 September 2006 (UTC)

[edit] action principle

The text says that Hamilton defines integral of T - V as the action. Pardon my nitpicking, but that seems not to be true. Charles Fox: Introduction to the Calculus of Variations (1963 printing, reprinted by Dover) says that the action is the integral of T.

Also if you look at the Feynman Lectures on Physics Volume II, chapter on The Principle of Least Action, he remarks that he (Feynman) calls the integral of T - V the action, but actually pedants call it Hamilton's first principle function. Historically something less convenient was first named the action. But Feynman hates to give a lecture on the principle of least Hamilton's-first-principle-function. Also, more and more people are calling integral T - V the action, and if you join them, soon EVERYBODY will be calling the more useful thing the action.

Point is, not Hamilton's definition, but common mid-20th century usage.

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