Talk:Calculus of variations/Archive 1

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 $$ \int_{x_1}^{x_2} \frac{ \frac{df_0}{dx} \frac{df_1}{dx} } {\sqrt{1 + \left(\frac{df_0}{dx}\right)^2}}dx =0, \,$$

Undefined symbol, references
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)

Comparison with functional analysis
I saw calculus of variations first, with its cryptic "$$\delta f$$" notation, then I was introduced to functional analysis, which seems to solve the same problems. Is the $$\delta f$$ notation outdated? Are the two fields different? Was "calculus of variations" a precursor to modern functional analysis? —Ben FrantzDale 03:07, 3 May 2007 (UTC)

Connection with the wave equation
the pde for an inhomogeneous medium is

rho u_tt=-div(A grad u),

where rho and A are material parameters. Only if A is constant can this pde be written in the form given in the article. Cj67 18:54, 14 June 2007 (UTC)

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))

While we're at it, "the well-known fact that electricity takes the path of least resistance" isn't true at all (see voltage division, for instance). I'm removing it. Vonspringer 04:43, 15 August 2007 (UTC)

Clarity of Euler Lagrange Section
In the section titled "Euler Lagrange", I feel the following step is unjustified: for any number ε close to 0. Therefore, the derivative of A[f0 + εf1] with respect to ε (the first variation of A) must vanish at ε=0. Thus

$$\int_{x_1}^{x_2} \frac{ f_0'(x) f_1'(x) } {\sqrt{1 + [ f_0'(x) ]^2}}dx =0, \,$$

for any choice of the function f1. We may interpret this condition as the vanishing of all directional derivatives of A[f0] in the space of differentiable functions. If we assume that f0 has two continuous derivatives (or if we consider weak derivatives), then it follows from integration by parts that,

$$\int_{x_1}^{x_2} f_1(x) \frac{d}{dx}\left[ \frac{ f_0'(x) } {\sqrt{1 + [ f_0'(x) ]^2}} \right] \, dx =0, $$ How does this follow from integration by parts?

I feel this step requires further justification. —Preceding unsigned comment added by Danielkwalsh (talk • contribs) 01:51, 19 September 2007 (UTC)

The intermediate step: $$\left[\frac{ f_1(x) f_0'(x) } {\sqrt{1 + [ f_0'(x) ]^2}}\right]_{x_1}^{x_2} - \int_{x_1}^{x_2} f_1(x) \frac{d}{dx}\left[ \frac{ f_0'(x) } {\sqrt{1 + [ f_0'(x) ]^2}}\right] \, dx =0 $$ Since f1 is a differentiable function that vanishes at the endpoints, the first term in the left hand side above vanishes. In a related manner, the qualification that is given, i.e. "for any choice of f1 with two derivatives that vanishes at the endpoints of the interval" is somewhat ambiguous, since it could be mistakenly taken as meaning that the derivatives vanish, rather than the function itself. A better wording would be: "for any choice of a twice-differentiable f1 that vanishes at the endpoints of the interval". —Preceding unsigned comment added by 128.83.162.226 (talk) 16:32, 20 September 2007 (UTC)

Thanks. I did some research and came to the same conclusion. I went ahead and added the intermediate step, and changed the ambiguous language. The vanishing term that is evaluated from $$x_1$$ to $$x_2$$ is an important step; it should be described. Danielkwalsh 20:13, 21 September 2007 (UTC)

Linguistic ambiguity
Does "any function with at least one derivative that vanishes at the endpoints" refer to a vanishing derivative or a vanishin function ("vanish" = "apporach zero"?)? 82.181.95.21 10:00, 16 June 2007 (UTC)


 * the function should be zero at the endpoints (technically, its extension should be zero, which is equivalent to the limit being zero as you approach the endpoints).Cj67 14:26, 21 June 2007 (UTC)


 * I find the wording "vanish" very ambiguous also. It took be a while of googling to understand that it means "being or approaching zero". JimQ (talk) 12:45, 8 May 2008 (UTC)

Frechet derivative
There should be a point in this article where the variation of a functional $$J$$ is treated as the Frechet derivative of $$J$$ as a map from whatever Banach space of functions we're talking about (presumably $$C^k(\Omega)$$ for some open set $$\Omega \subseteq \mathbb{R}^n$$) to $$\mathbb{R}$$. Ideally this would go in the section in which the Euler-Lagrange equation is derived but it seems like adding it on at the end would be a goofy exposition, what with the example derivation of the Euler-Lagrange equation for minimizing distance coming first. So implementing this change would necessitate a small rewrite. —Preceding unsigned comment added by Compsonheir (talk • contribs) 18:11, 5 September 2008 (UTC)

Weak and strong extrema
I think that the definition is not clear enough. TomyDuby (talk) 19:54, 8 October 2008 (UTC)

I agree. I think this is true of the whole article. The 2nd paragraph of the article doesn't tie into the 1st, and the 3rd, while interesting, talks about functions of several variables, not functions of functions. Everything afterward, starting with extrema is completely without context. Are they examples? essential elements? or completely unrelated factoids? 130.76.32.23 (talk) 17:41, 14 October 2008 (UTC)

I edited the section on weak and strong extrema and fixed some errors that were there before. Is it better now? Compsonheir (talk) 01:01, 18 November 2008 (UTC)


 * Thanks for your excellent edit!


 * I still wonder if (1) the article needs a definition of weak and strong extrema, and (2) needs it at such a prominent location as chapter 1. Gelfand and Fomin's book, for example, does not mention these terms.


 * TomyDuby (talk) 05:33, 19 November 2008 (UTC)


 * Gelfand and Fomin's book defines weak and strong extrema in chapter 1, at least in the Dover edition that I have, and chapter 6 is all about sufficient conditions for a functional to have a strong extremum. I agree that this section could profitably be moved elsewhere in the article. My biggest complaint is that the derivation of the Euler-Lagrange equations is poor and the information is disorganized. The E-L equations should be derived in the general case with examples for a few specific functionals, and there should be some mention of calculus on maps between Banach spaces somewhere. There shouldn't be an entire section devoted to the Beltrami identity if the derivation of that identity is contained in its own article as well. Compsonheir (talk) 04:43, 20 November 2008 (UTC)


 * The definitions of weak and strong extrema do not really seem to make sense here. Given the definitions in the article, every weak extrema is actually a strong extrema, not the other way around. 138.89.251.227 (talk) 23:01, 1 January 2010 (UTC)

History
The emergence of the calculus of variations is one of the more fascinating facets of the history of mathematics. How come this long article has nary a peep on its origins and growth? --Vaughan Pratt (talk) 00:25, 10 September 2009 (UTC)
 * It does now. Noodle snacks (talk) 12:22, 3 June 2010 (UTC)

Missing stuff
Could add some stuff on: Noodle snacks (talk) 12:22, 3 June 2010 (UTC)
 * Generalisations:
 * Functionals with higher order derivatives
 * Parametric form
 * Natural boundary conditions
 * etc
 * Subsidiary Conditions
 * Algebraic constraints
 * Differential equation subsidiary conditions
 * etc
 * Direct methods

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.


 * You're right: the modern definition of action as the integral of the Lagrangian does not conform to Hamilton's original usage. (In addition to the references you cited, see, e.g., Kibble and Berkshire, Classical Mechanics, 5th edition, p. 63.)  I'll rephrase the article to avoid ascribing this definition to Hamilton. Tpudlik (talk) 22:41, 7 January 2011 (UTC)

Dot
Is there anywhere in Wikipedia where the meaning of the dot in $$\|\cdot\|_V$$ is explained? --Bob K31416 (talk) 14:34, 7 December 2011 (UTC)
 * I added an explanation in this article, and I added a general explanation of this type of use of dot to the dot section of List of mathematical symbols. --Bob K31416 (talk) 12:56, 17 December 2011 (UTC)

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)


 * Oppose: I think the two concepts are different. One is a physics principle, the other is a mathematical method. Sorry to respond 5 years later. David Spector (talk) 01:44, 30 January 2012 (UTC)

Notation Explanation
Can someone explain the expression $$\frac{dL}{df'}$$ that appears near the end of the section on the Euler-Lagrange equation, or provide a link to an explanation? I've never encountered an expression like this, and I don't see how it follows from what precedes it in the article. —Preceding unsigned comment added by Mathmoose (talk • contribs) 01:37, 16 February 2009 (UTC)


 * This is sort of an abuse of notation. We want to extremize the quantity$$\int_a^bL(f(x), f'(x), x)dx$$ with respect to the unknown function $$f$$. $$L$$ is a function of 3 variables, so we can write $$L = L(q, p, x)$$, but we always evaluate it at $$q = f(x), p = f'(x)$$. By $$\frac{\partial L}{\partial f'}$$, what they mean is $$\frac{\partial L}{\partial p}(f(x), f'(x), x)$$, but this is a little cumbersome, hence the shorthand. It's just the partial derivative of L in the second variable, evaluated at $$f'$$. This confused me when I saw it too. Unfortunately this whole subject is full of conflicting and often nonsensical notation conventions. Compsonheir (talk) 00:20, 15 April 2009 (UTC)


 * Perhaps something like this comment could be added to the article. This might help (or at least console) the reader a bit. David Spector (talk) 01:49, 30 January 2012 (UTC)

Calculus of variations a Field of Calculus?
Is this "really" true? "Calculus" really isn't a field per say anymore is it? More or less a tool. Calculus of variations is probably better described as a mix of functional analysis and optimization, right!? — Preceding unsigned comment added by 99.149.190.128 (talk) 00:57, 25 November 2012 (UTC)
 * "Many problems of mathematical physics are connected with the calculus of variations, one of the central fields of analysis." [ ]
 * Per your comment and the above quote, I just made a change in the first sentence of the lead. --Bob K31416 (talk) 16:10, 26 November 2012 (UTC)

Variational method
The usage of "Variational method" is up for discussion, see talk:Variational method -- 65.92.180.137 (talk) 01:06, 25 March 2013 (UTC)

"Extensions?"
I recently saw a lecture discussing the Calculus of Variations with the following special cases:

1) The end points are fixed (the article discusses this nicely)

2) There are free initial and free final time/independent variable

3) There are free final state and free final time/independent variable

4) The final time/independent variable is free

5) The final state is free

6) The final state and final time/independent variable is constrained to lie on a given curve.

All of these follow from the general condition: Euler-Langrange Equation + Transversality Condition = 0. The boundary value section of the article appears to discuss some, but not all of these. I was wondering if it would be possible to incorporate these other cases?

I think the lecture may have watched was http://www.youtube.com/watch?v=twaCptP4drM. Mouse7mouse9 19:34, 16 April 2013 (UTC) — Preceding unsigned comment added by Mouse7mouse9 (talk • contribs)

Redirect hatnote (Variational method)
As of 2013-04-18, Variational method redirects here, so I have added a hatnote referring to Variational method (quantum mechanics). I believe this is in accordance with the principle of least astonishment: Physics undergraduates are taught something called the 'Variational method' in their quantum mechanics classes, but if they put the term into Wikipedia, they are currently sent to a page that is superficially quite unrelated. (An application of the variational method in quantum mechanics does not usually involve extremizing any functionals, which is how this article defines it. There is some further discussion at this talk page.) I agree that the hatnote will only be relevant to a small minority of those who will end up at this page. My preferred solution would be to have a proper explanation of how these terms and techniques actually relate to each other, but until one is written, this seems like a reasonable temporary arrangement. Stevvers (talk) 06:31, 19 April 2013 (UTC)

Simplifying the example slightly
In the example, when we reach


 * Substituting for $L$ and taking the partial derivative,


 * $$ \frac{d}{dx} \ \frac{ f'(x) } {\sqrt{1 + [ f'(x) ]^2}} \ = 0 \, . $$

an alternative is to reason that therefore


 * $$ \frac{ f'(x) } {\sqrt{1 + [ f'(x) ]^2}}$$

is constant, i.e. is independent of x. Trivial algebra (which need not be included in an article at this level) shows that f'(x) is therefore also constant, so that the trajectory has constant slope, a straight line. It's a small point: but using calculus only where the CoV itself demands it makes the article's main points easier to isolate.

Nunibad (talk) 03:47, 21 April 2013 (UTC)
 * Please note the beginning sentence of the example, "In order to illustrate this process, consider the problem of finding the extremum [sic] $y = f (x) ,$ which is the shortest curve that connects two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2}).$" The example problem wasn't to show that it was a straight line, but to find the extremum extremal function $y = f (x).$ --Bob K31416 (talk) 02:00, 11 May 2013 (UTC)
 * Actually, it should be "extremal function" instead of "extremum", so I'll make that change. --Bob K31416 (talk) 02:05, 11 May 2013 (UTC)
 * I also rewrote the ending to clarify that the problem of finding the extremal function was solved. --Bob K31416 (talk) 02:21, 11 May 2013 (UTC)

More on clarity of Euler-Lagrange section
In that section we have presently the following:
 * Since the functional $J[ f ]$ has a minimum for $f = f_{0} ,$ the function $&Phi;(&epsilon;)$ has a minimum at $&epsilon; = 0$ and thus,
 * $$ \Phi'(0) \equiv \left.\frac{d\Phi}{d\epsilon}\right|_{\epsilon = 0} = \int_{x_1}^{x_2} \left.\frac{dL}{d\epsilon}\right|_{\epsilon = 0} dx = 0 \, . $$


 * Taking the total derivative of $&epsilon;&Phi;&prime;(0)$ where $J$ and $&delta;J$ are functions of $&epsilon;$ but $L[x, f, f &prime;] ,$ is not,

\frac{dL}{d\epsilon}=\frac{\partial L}{\partial f}\frac{df}{d\epsilon} + \frac{\partial L}{\partial f'}\frac{df'}{d\epsilon} $$


 * and since $f = f_{0} + &epsilon; &eta;$  and  $f &prime; = f_{0}&prime; + &epsilon; &eta;&prime;$,



\frac{dL}{d\epsilon}=\frac{\partial L}{\partial f}\eta + \frac{\partial L}{\partial f'}\eta' $$.


 * and since $&epsilon;$  and  $x$,



\frac{dL}{d\epsilon}=\frac{\partial L}{\partial f}\eta + \frac{\partial L}{\partial f'}\eta' $$.


 * Therefore,

\begin{align} \int_{x_1}^{x_2} \left.\frac{dL}{d\epsilon}\right|_{\epsilon = 0} dx & = \int_{x_1}^{x_2} \left(\frac{\partial L}{\partial f} \eta + \frac{\partial L}{\partial f'} \eta'\right)\, dx  \end{align} $$

In the introductory sentence, our attention is focussed on the fact that the function $df /d&epsilon; = &eta;$ is to have a minimum for $df &prime;/d&epsilon; = &eta;'$; but later (near the end of the quoted section) we see that

\frac{dL}{d\epsilon}=\frac{\partial L}{\partial f}\eta + \frac{\partial L}{\partial f'}\eta' $$, that is that the expression in which $df /d&epsilon; = &eta;$ is to be set to 0 is not in fact a function of $df &prime;/d&epsilon; = &eta;'$.

Comments?

Nunibad (talk) 03:25, 22 April 2013 (UTC)
 * I changed notation and then made this edit to take care of your point. --Bob K31416 (talk) 03:55, 11 May 2013 (UTC)
 * I think the notation inconsistently switches between y and f in the current version. --Memming (talk) 15:56, 12 June 2013 (UTC)
 * Could you direct me to exactly where you think that is occurring? --Bob K31416 (talk) 02:26, 16 June 2013 (UTC)

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) I also found the examples to be clear and useful. Infact - i think the examples were more useful than this site - I know that this site is supposed to be an encyclopedia - but that being said - is it not possible to have an example aproach to explaining mathematical points rather than the current formal/theoretical maths approach? — Preceding unsigned comment added by 212.49.88.110 (talk) 18:00, 24 August 2013 (UTC)


 * An example is given in this article. For more examples, the above mentioned link is the fourth one in the external links section. Having many examples in the text would be more appropriate for a textbook instead of an encyclopedia article that is supposed to summarize the subject. --Bob K31416 (talk) 18:42, 25 August 2013 (UTC)