Talk:Euler–Lagrange equation

General Comments

1. I think that section entitled "Statement" should be renamed "Definition" because it is defining the Euler-Lagrange equation. "Statement" seems kind of meaningless to me in this context.

2. I would restrict consideration in this section to Euclidean spaces and ordinary time. Tangent spaces and tangent bundles etc, should be presented later for those interested in these important but less commonly applied concepts. The section on generalization to manifolds is the obvious place for their discussion.

3. There should be some reference to the Wiki article "Action(physics)" and to the historic application of the Euler-Lagrange equation for finding extrema of the action.

4. Lx and Lv should be Lq and Lq'. x and v sound in conflict. x is used in place of t, which historically was the time variable. v is ok in itself, but in context I think it would be better to have it as q'. Dot notation is not comkpatible with the section on Generalizations. 190.160.150.252 (talk) 02:37, 5 July 2018 (UTC)

Obvious oversights

1. The section entitled "Examples" has only one example, suggesting the name "Example" would be more appropriate.

2. The definition of ds as an integral is confusing. Typically ds is a differential and appears to the right of the integral. The result of the integration is the "Action".

3. L and F are both used as names of the same thing. L should replace F.

4. L should be referred to as "the Lagrangian" as is customary, not as the "integrand function".

5. I think a less trivial example should be used -- not that I would want to go to war over it. If you add an example, retain the name "Examples" for this section. 190.160.150.252 (talk) 02:37, 5 July 2018 (UTC)

Link does not work

The site referenced at the end http://www.exampleproblems.com/ is not functional. It would be better to remove it. —Preceding unsigned comment added by 87.120.6.146 (talk) 08:37, 8 February 2008 (UTC)

Statement Section Unclear

Many of the symbols in the statement section are undefined. Particularly, I have no idea what R, X, and Y are. In fact, the entire line

$F:R\times X\times Y\ni (t,x,y)\mapsto F(t,x,y)\in R\,\!$

is very confusing. Can anyone expand it to make it clearer?--132.239.27.145 19:14, 21 September 2007 (UTC)

I think this whole line needs removing. Nothing is defined properly and it is totally useless to understanding the article. F does not need to be a function of just x,y and t either. It looks like whoever wrote this had no idea what they were talking about! Dazza79 (talk) 11:21, 12 March 2008 (UTC)

Please do not remove the line which defines the most essential object. The formulation as it was, was in principle unambiguous because the way the functional J is expressed in terms of f and f' (which is the conventional mathematical notation for the derivative of f) left no other possibility than: f maps R to X and its derivative maps R to Y. I made these things a bit more explicit in the text. Bas Michielsen (talk) 00:44, 14 March 2008 (UTC)

The set X still appears to be undefined, leaving me unable to understand what is going on in the definitions. I call this an urgent problem. Dratman (talk) 01:08, 31 March 2017 (UTC)

If people are querying about these three lines

"*${\boldsymbol {q}}$ is the function to be found:

${\begin{aligned}{\boldsymbol {q}}\colon [a,b]\subset \mathbb {R} &\to X\\t&\mapsto x={\boldsymbol {q}}(t)\end{aligned}}$"

[a, b] is the domain (set of values of t) while X the codomain (set of values of x). But yes, X needs clarification and a definition, surely just the n-dimensional set of real numbers (Rⁿ)? M∧Ŝc²ħεИ_τlk 07:20, 31 March 2017 (UTC)

In the statement section, second paragraph, smooth paths have codomain M (nowhere defined), I think it should be X, paths in the configuration space. Jesuslop (talk) — Preceding undated comment added 12:27, 2 October 2021 (UTC)

Proof Feedback

I'd appreciate feedback on this proof. Is it too long? Too technical? Not technical enough? A simple proof is very appropriate for this page, I'm just not sure if this is that proof. --Dantheox 02:36, 14 December 2005 (UTC)

How exactly do we come in Proof in $J'(0)$ from partial derivative of F with respect to $\epsilon$ to partial derivatives of F with respect to $f$ and $f'$? I don't get it.

It's a standard application of the chain rule -- you can expand out a total derivative with respect to $\epsilon$ in terms of the derivatives with respect to other quantities. --Dantheox 04:02, 7 March 2006 (UTC)

And where does the sum come from? In http://en.wikipedia.org/wiki/Chain_rule#Chain_rule_for_several_variables there is a sum, since the f is a sum of u and v. And here we just have F. I'm not very familiar with partial derivatives, just know how to compute simple ones.

The sum comes from the matrix product in the multidimensional case: Have a look at http://en.wikipedia.org/wiki/Chain_rule#The_fundamental_chain_rule. For the first example with $f,g,h$ in http://en.wikipedia.org/wiki/Chain_rule#Chain_rule_for_several_variables choose E = R, F= R², G=R; you have to regard $g,h$ as to define a single function R → R². The derivative of this function is (at each point) a linear mapping from R to R² represented by a column vector (the two entries being the derivatives of each component funtion resp.). The derivative of $f$ is (at each point) a linear map from R² to R represented by a row vector (the two entries being the partial derivatives in direction of the two coordinates). The matrix product of the two matrices gives exactly the formula in http://en.wikipedia.org/wiki/Chain_rule#Chain_rule_for_several_variables.

The proof is very hard to follow- I don't understand what everything means- for instance, why do you introduce L_x, rather than stating it explicitly? Is it dL/dq(t)?

The proof is "ontologically backwards" because its assuming that we are must find an extremum, there is no justification for such an assumption other than it produces the desired result. There is also no logical reason to even work with a Lagrangian other than it produces the desired result. The sensible proof is to start with the definition of force (i.e. Newton's second law), definition of kinetic and potential energy and take things from there. This will give you an Euler-Lagrange equation type equation in terms of kinetic energy and potential energy. The usual Euler-Lagrange equation can then be derived by combining the energies into the Lagrangian as a mathematical convenience. Hamiltons principle of stationary (not extreme or least!) action then follows from the Euler-Lagrange equation. The fact that one can get back to Euler-Lagrange if one assumes stationary action, is merely a nice mathematical excercise but is getting things backwards in terms of understaning why things are the way they are. - :P — Preceding unsigned comment added by 41.164.241.226 (talk) 13:11, 27 November 2012 (UTC)

Euler-Lagrange Methods & Lagrange multipliers

Is anybody able to give me some hints about the connection between Euler-Lagrange Methods and Lagrange multipliers? At first glance they seem to be closely related. Maybe somebody can clarify this and maybe add a short note to the articles. Cyc 12:37, 22 September 2006 (UTC)

Does summation over repeated indices need to be made explicit ?

My concern is with this section:

\partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) - ...

   where ...
   \partial\, is a vector of derivatives:
       \partial_\mu = \left(\frac{1}{c} \frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right). \,

Seems like it could be interpreted as a set of N equations (N being the dimension of the "vector"), rather that the sum over mu (do we always assume summation over repeated indices ? Apologies if this is nit-picking.

Many thanks

—Preceding unsigned comment added by JM516 (talk • contribs) 23:03, 4 October 2007 (UTC)

In quantum mechanics, the sum is implicit due to Einstein summation notation which is taken as read. — ras52 (talk) 14:41, 5 December 2007 (UTC)

You are correct. Summation over repeated indices (one 'upper' and one 'lower' index anyway) is always implied. Furthermore, the E-L equation is not just one equation but a set of N equations, where N is the number of degrees of freedom of the system in question (so here it would be the dimension of the vector). This should be made more clear in the article Dazza79 (talk) 11:10, 12 March 2008 (UTC)

Not a mathematics encyclopedia

You all do realize that Wikipedia is not meant to be an academic encyclopedia, rather a popular encyclopedia - right? Why then have you all made the effort to write an article on a mathematical formula which only somebody with significant master of post-Calc mathematics has any chance of comprehending? Remember, one very strong indicator of how well you understand a subject is how simply you can explain it.-66.213.90.2 (talk) 17:24, 7 September 2008 (UTC)

Thank you for your suggestion. When you feel an article needs improvement, please feel free to make those changes. Wikipedia is a wiki, so anyone can edit almost any article by simply following the edit this page link at the top. The Wikipedia community encourages you to be bold in updating pages. Don't worry too much about making honest mistakes — they're likely to be found and corrected quickly. If you're not sure how editing works, check out how to edit a page, or use the sandbox to try out your editing skills. New contributors are always welcome. You don't even need to log in (although there are many reasons why you might want to). --A r m y 1 9 8 7 ! ! ! 20:18, 7 September 2008 (UTC)

I do not share your optimism that errors will get corrected quickly. Unfortunately, there are many articles which become inconsistent because people are changing notations into ones which they think more "standard" but forgetting to do this consistently. I just undid some anonymous and unmotivated edits of this E-L article. Somebody also thought it necessary to change J into S and f into q but did not do this everywhere. I do not know what we do should do with that kind of things. Bas Michielsen (talk) 23:15, 7 September 2008 (UTC)

(That was just the {{sofixit}} template... --A r m y 1 9 8 7 ! ! ! 08:18, 8 September 2008 (UTC))

Not right! Wikipedia does not at all claim to be a "popular encyclopedia" in the sense that every article should be understandable by a layman. I think, that the first purpose of any encyclopedia remains to provide correct information. There may be many reasons why an article is difficult to understand for some particular person. It is too easy to simply blame the author. You write an article on the Euler-Lagrange equation for someone who decided to read on that particular subject, it is not more than reasonable, then, to expect that this reader is familiar with at least modern analysis. The advantage of wikipedia is that articles have associated discussion pages where you can ask for clarifications. In my opinion, therefore, the best way to go is to use the discussion page to help improve the article. Bas Michielsen (talk) 22:24, 7 September 2008 (UTC)

That is not entirely correct (see WP:MTAA), but I agree that this is a somewhat technical topic that is hard to explain to laymen. But on the other hand, that doesn't mean that we shouldn't even try. For example, the lead section of the featured article Laplace-Runge-Lenz vector does try to explain what that stuff is to the general audience. Maybe I'll try to write a reasonably accessible introduction to this article while keeping the technical details intact, if/when I have enough spare time. --A r m y 1 9 8 7 ! ! ! 23:05, 7 September 2008 (UTC)

Thanks for your quick response! I scanned the link you suggested, although it says "widest possible audience" it also recognises that this does not imply just everybody. Indeed, improving the introduction, perhaps by adding an example from elementary mechanics the details of which can be given in the core of the article, seems the best solution. I think, we should also eliminate the use of the word "formula" where "equation" is meant. Bas Michielsen (talk) 23:29, 7 September 2008 (UTC)

Here's an example from the intro, "extremize a given cost functional." Now, I'm willing to bet that with a little effort, you could rewrite this in a much more comprehensible way. As it is, I *think* I understand what it's saying, but I'm not sure. The first pupose of an encyclopedia is not to provide correct information, but to inform the reasder. A reader isn't informed by reading something that is incomprehensible. An article which the general audience finds incomprehensible does a poor job of informing the general audience.-198.97.67.56 (talk) 13:50, 9 September 2008 (UTC)

Your comment would be much more helpful if you made clear what your interpretation of that fragment is and why you are not sure whether your interpretation is correct. By the way, I think that providing correct information is part of informing the reader (correctly). So if the information provided is correct, we have already made some progress haven't we? Bas Michielsen (talk) 14:32, 9 September 2008 (UTC)

As I said, the first purpose of an encycolopedia is to inform the reader. Information must be accurate and comprehensible, being either of those two alone is not sufficient. So, yes, providing correct information is part of informing the reader and you all have done a lot of great work on that half of the problem. I just want to encourage you all to work a bit more on the other half of the problem. I'd do it myself, but I can't really write it to be more comprehensible when I don't know what you are trying to say in the first place. I *think* what you are trying to say by "extremize a given cost functional" is "find the best (that is, optimal) values to input into a problem".-198.97.67.57 (talk) 18:52, 9 September 2008 (UTC)

Thank you very much for this comment. I will give it a try. So below you find an attempt to be as explicit as possible while remaining as faithful as possible to what actually is at stake.

Many mathematical modelling problems, in particular in theoretical physics but also in other application domains, can be formulated as problems of selecting elements from a set of functions such that a functional, which associates a number to any element of the set of functions, reaches an extremal value, i.e. a maximal or a minimal value. When this functional has the particular form of an integral of a Lagrangian function composed with a function of the function set, a necessary condition for the functional to reach an extremal value at a specific function is that this specific function satisfy the Euler-Lagrange differential equation. So it appears that functions can be selected by formulating an optimisation (minimalisation or maximalisation) problem on a set of functions and, in certain cases, solving such optimisation problems can be reduced to solving differential equations. It also happens that the differential equation is found first and the interpretation of this as the Euler-Lagrange equation of a functional optimisation problem is found much later (see for example the Korteweg-de Vries equation).

A typical example of such a modelling problem, from classical mechanics, is to find the trajectory of a planet or a comet etc. under the influence of the gravitational force attracting this body to the sun. The set of functions, in this particular case, consists of functions, say f, mapping a time coordinate smoothly to points in space, i.e. associating to a time t a point f(t) in space. The functional, in this particular case, associates a number to each such a trajectory-describing function, by integrating over time some special combination of the gravitational potential energy at the point f(t) and the kinetical energy of the body which depends on the velocity of the body and hence on the time derivative f'(t). This special combination is called the Lagrangian of the problem.

In order to save space on this discussion page, I invite you to change this text such that you feel it is more close to what you understand it tries to say (well ..., if you get such a feeling in the first place) Bas Michielsen (talk) 22:55, 9 September 2008 (UTC)

I have a background in Bayesian inference so I'm not a green horn when it comes to mathematics and when I read the new version that Bas Michelson just wrote I get the distinct feeling that he tried to make it -more- opaque. I don't think you all are getting it. Incomprehensibility is not a virtue no matter what you learned in college. If you -add- informaton and, thereby, make the content -harder- for your audience to comprehend, you have written a worse document. I don't understand what the problem is here. I can explain Bayesian inference to a sixth-grader because I understand it well enough to explain it simply. I could probably explain information theory to a high school student and she'd understand it. Why is writing clearly about the Eular-Lagrange equation so difficult?-66.213.90.2 (talk) 00:32, 10 September 2008 (UTC)

OK, if the only way you seem to be able to respond is by insulting people, I am giving up. I am just wondering how you think to ever understand the Euler-Lagrange equation if you are unable to ask pertinent questions on a text introducing the subject using only the most elementary concepts of mathematics. Bas Michielsen (talk) 07:52, 10 September 2008 (UTC)

There are many articles on mathematical subjects in Wikipedia which are written so that a general audience can understand them. Perhaps it'd help if you studied some of them.-198.97.67.56 (talk) 12:31, 10 September 2008 (UTC)

If you can find a way to explain this to a general audience, you are welcome to do so. However, this requires at least some knowlege of calculus of variations to be covered properly. You can write something to the effect of "It's an equation which is useful in physics" but that doesn't tell much of anything. As a person who had only 3 semesters of physics in college, I found the introduction readable and helpful. The new one seems even simpler. I can't imagine that it could be made any simpler.129.15.127.254 (talk) 22:30, 29 January 2009 (UTC)

I agree that this is not very consumable in the current form. The page Lagrangian mechanics is a better description of what is going on. I am reading Optimal control and estimation by Stengel and he is referring to a Euler-Lagrange equations. This will work well for systems of motion control; can the Euler-Lagrange equations be applied to a thermal system, to a economics systems? It seems they are generic minimum-maximum of an energy equations for a system with constraints and specific to a mechanical system? The math formulae have an explicit 't' implying a time evolution. The page needs some concrete cases instead of the overly abstract notation that it currently embodies. 184.146.224.118 (talk) 21:41, 1 January 2021 (UTC)

Other articles that are more enlightening, are Lagrange multiplier and Hamiltonian (control theory). 184.146.224.118 (talk) 14:28, 2 January 2021 (UTC)

"Field theory"

There was a little section on relativistic field theory that was removed, but the reference to it remains in the introduction. It seems it was removed for being too technical. Without getting too deep into the argument that every Wikipedia page should be readable to everyone, this is probably where the E-L equations get used most and what a lot of people are looking for, so I reckon the section should be reintroduced, or at least the intro should link to http://en.wikipedia.org/wiki/Classical_field_theory#Relativistic_field_theory. Thanks. 91.106.56.174 (talk) 18:11, 3 March 2011 (UTC)

I second the suggestion above. Either the field theory second be removed entirely, or reintroduced in altered form. — Preceding unsigned comment added by 41.58.61.32 (talk) 20:27, 23 November 2011 (UTC)

A copy of the original section is given below. Bbanerje (talk) 21:44, 24 November 2011 (UTC)

Copy of the section Field theory

Field theory

Field theories, both classical field theory and quantum field theory, deal with continuous coordinates, and like classical mechanics, has its own Euler–Lagrange equation of motion for a field,

$\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)-{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0.\,$ where

• $\psi \,$ is the field, and
• $\partial \,$ is a vector differential operator:

  $\partial _{\mu }=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},{\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right).\,$

Note: Not all classical fields are assumed commuting/bosonic variables, (like the Dirac field, the Weyl field, the Rarita-Schwinger field) are fermionic and so, when trying to get the field equations from the Lagrangian density, one must choose whether to use the right or the left derivative of the Lagrangian density (which is a boson) with respect to the fields and their first space-time derivatives which are fermionic/anticommuting objects. There are several examples of applying the Euler–Lagrange equation to various Lagrangians:

• Dirac equation;
• Proca equation;
• electromagnetic tensor;
• Korteweg–de Vries equation;
• quantum electrodynamics.

Locus of shortest curve

I think for any given general points A and B , the shortest length should collapse to a point or a set of points, hence the result should be of the form (x-x1)^2 + (y-y1)^2=0 , or a product of the same (the locus of the combined individual loci of 2 points)

i.e the locus of points, The function need not be an explicit function, it can be implicit and it is still real valued.

Note that if one drops the assumption made in considering a small "functional" perturbation this is indeed the result if one tries to solve the fact that:

d/dy (intergal (1+f`(x)^2)^1/2 . dx) =0

so (y-y1)=i.(x-x1) , f`(x)=i is a solution

The fact that you assume dy/dx is the distance to arrive at the formal sqrt(1+f`(x)^2) as the formula for curve length itself implies you are trying to assume the minimal distance is a line (dx^2+dy^2=ds^2). The logic seems circular. Hence I am not sure why one puts in that perturbation. -Alok 08:35, 24 May 2011 (UTC)

Please verify Several_functions_of_one_variable

Sources such as this pdf indicate that the euler-lagrange equations for several functions of one variable are:

${\begin{aligned}{\cfrac {\partial {\mathcal {L}}}{\partial f_{i}}}-{\cfrac {d}{dx}}\left({\cfrac {\partial {\mathcal {L}}}{\partial f_{i}'}}\right)=0\end{aligned}}$

I havn't been able to find the cited source for this section to see why something else is listed here.

I have gone ahead and changed this using what seems a dependable source.

differentiable

This article frequently states that a function must be differentiable. Last I heard, a function is NOT "differentiable"; it is "differentiable with respect to <a specific variable (or function)>" There are many examples here where the authors seem to think that the "with respect to X" can be neglected. It can not, unless the wrt can only be understood one way. L(q(t)) is "differentiable wrt WHAT? x? t? q(t)? something else? This needs to be fixed, imho. The instances of this sloppiness are too numerous to list, it is pervasive and detracts significantly from the utility of the article (imho)Abitslow (talk) 06:25, 20 December 2013 (UTC)

The differentiability of a properly defined function is a well-defined notion because a function is only properly defined when at least its domain and range space are given. Functions of more than one variable are usually called differentiable if all the partial derivatives are continuous and, hence, for well-defined functions there is no ambiguity in the notion of differentiability. I agree with Abitslow, that this need not be a sufficiently precise classification for all purposes. However, in the given example, L(q(t)) defines the value of the composition of the function L with the function q. This composed function is unambiguously defined when the functions L and q are (and in such a way that the domain of L is in the range space of q, of course). The differentiability of "L after q" can be inferred from the differentiability of q and L. In this example, there really is no ambiguity at all because "L after q" is a function of just one variable. Bas Michielsen (talk) 12:47, 20 December 2013 (UTC)

Notation Errors

Firstly in "Several functions of several variables" partial derivatives were replaced with ordinary derivatives. But that makes not much sense, since the functions are functions of several variables. Total derivatives don't come into question. Then the partial derivative of the functional would be a function of the single variable $x_{i}$ and $x_{i}$ would be a function of several variables. But that's not the case here.

Secondly in "Single function of two variables with higher derivatives" and the following section summation is taken over $\mu _{1},...,\mu _{i}$. Then due to the theorem of Schwarz there would be

$2{\frac {\partial ^{2}}{\partial x_{1}\partial x_{2}}}$.

So I think summation should be taken over all combinations with repetition, so that $\mu _{1}\leq ...\leq \mu _{i}$. If this is the case, the sum should be noted with

$\sum _{i=1}^{n}(-1)^{i}\sum _{\mu _{1}\leq ...\leq \mu _{i}}\ldots$.

— Preceding unsigned comment added by 188.103.118.233 (contribs)

I disagree with the partial derivatives, and the current wording of this page confuses me every time I use it as reference (that is, about three times a year). Taking the full derivative is important in most cases. Consider this Lagrangian for one function of several variables:

${\mathcal {L}}={\cfrac {1}{2}}\sum {f_{x_{i}}^{2}}\Rightarrow {\cfrac {\partial {\mathcal {L}}}{\partial f_{x_{i}}}}=f_{x_{i}}$.

That is, derivative of the Lagrangian wrt. image gradient is the gradient itself. Then, the partial derivative:

${\cfrac {\partial }{\partial x_{i}}}{\cfrac {\partial {\mathcal {L}}}{\partial f_{x_{i}}}}={\cfrac {\partial }{\partial x_{i}}}f_{x_{i}}=0$

is zero simply because $f_{x_{i}}$ and $x_{i}$ are different variables. In contrast, taking the full derivative leads to the second derivative of $f$:

${\cfrac {\mathrm {d} }{\mathrm {d} x}}{\cfrac {\partial {\mathcal {L}}}{\partial f_{x_{i}}}}={\cfrac {\mathrm {d} }{\mathrm {d} x}}f_{x_{i}}=f_{x_{i},x_{i}}$ also known as ${\cfrac {\partial ^{2}f}{{\partial x_{i}}^{2}}}$.

In my example, this ultimately leads to the equation ${\cfrac {\partial {\mathcal {L}}}{\partial f}}=\Delta f$, which is the correct result (the heat equation).

I am convinced that all the sections concerning several variables need to be fixed, and I will do so in three months.

— Emu cz (talk) 10:26, 8 April 2016 (UTC)

The original version that I added several years ago was from the book by Courant and Hilbert, two of the greatest mathematicians the world has known. My suggestion would be to track that source down and compare the current equations with those in that text. The description in the Courant-Hilbert book is quite detailed and I would trust their results over almost any other source (modulo typesetting errors). Bbanerje (talk) 22:04, 9 April 2016 (UTC)

The current notation conforms to the one by Hilbert and Courant (MMP vol. I, p. 192, Eq. 25). It feels awkward to me, but arguing with these two gentlemen is beyond the scope of this talk page ;) Anyway, I will not edit this article unless I find a super-reliable proof. — Emu cz (talk) 15:10, 11 April 2016 (UTC)

Make the scope of this article mathematics

The subsection Classical mechanics (see this version at the time of this post) has a very long-winded way of explaining how to use the EL equations to solve problems in mechanics. This is the subject of the Lagrangian mechanics article, which gives the same example more compactly and in both Cartesian and spherical coordinates. The ordering of derivatives is clear also, without textbooky over-detailed explanations "evaluate this" then "treat this as" etc. I am going to deleted this subsection.

This article is more mathematical and general, giving various extensions (e.g. higher derivatives, more variables). It would cut pointless duplication if mathematical examples are contained in this article, physics examples elsewhere. M∧Ŝc²ħεИ_τlk 17:29, 11 January 2016 (UTC)

Amended the post above after deleting the section to clarify and add links. M∧Ŝc²ħεИ_τlk 19:06, 11 January 2016 (UTC)

Manifold generalization

There's no reference on the Manifold generalization part. Everton.constantino (talk) 04:53, 16 January 2016 (UTC)

This section could use an explicit definition of F. — Preceding unsigned comment added by 70.53.208.149 (talk) 17:23, 30 January 2016 (UTC)

Nationality of Euler

Was Euler truly "Swiss-Russian"? Yes, so he spent much of his adult life in St. Petersburg, but I'm not sure (and I've never seen anything written down anywhere) that he actually took on dual nationality. (Admittedly the rules on what nationality you are may have changed since those days, and it may have been that all you needed to do to be declared "Russian" was to live there for a period of time -- I haven't a clue). But, as I say, I have never anywhere else seen a suggestion that he actually was "Swiss-Russian" -- only that he was Swiss. I'm asking the question because I don't actually know one way or the other, I'm asking in case anyone does know. --Matt Westwood 06:51, 6 May 2016 (UTC)

I don't know much about him (aside from this long List of things named after Leonhard Euler). This is probably a question for talk:Leonhard Euler anyway. M∧Ŝc²ħεИ_τlk 08:56, 6 May 2016 (UTC)

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Unclear what $L_{x}$ means

"... $L_{x}$ and $L_{v}$ denote the partial derivatives of $L$ with respect to the second and third arguments, respectively."

To get the partial derivative with respect to the second argument, we need to vary the second argument while keeping the third argument the same. But the third argument belongs to the tangent space of the second argument, so it doesn’t make sense to keep the third argument the same while varying the second argument, because we have no canonical way of identifying nearby tangent spaces. — Preceding unsigned comment added by Tprice7 (talk • contribs) 23:17, 1 May 2018 (UTC)

$L$ is a function of three arguments and its partial derivatives with respect to the three arguments only depend on the definition of $L$ not on the way you are going to use this function. In the definition of the Euler-Lagrange equation this $L$ is used to define a functional by composition of $L$ with functions in the second and third argument, ${\boldsymbol {q}}$ and ${\boldsymbol {\dot {q}}}$ (which are related as you state), and by integrating this composition over the common domain of ${\boldsymbol {q}}$ and ${\boldsymbol {\dot {q}}}$ (a "time" interval). In this way, the functional maps a function ${\boldsymbol {q}}$ to a scalar value, but in this construction $L$ is an ordinary function. Bas Michielsen (talk) 09:17, 2 May 2018 (UTC)

I understand that partial derivatives do not depend on how you are going to use the function. Maybe I wasn't clear about what the issue is. The notation $L(t,q(t),{\dot {q}}(t))$ suggests that L is a three-argument function, but this is actually misleading, and it’s not truly a three-argument function. This is because the third argument must be in the tangent space of the second argument. I don't just mean that it ends up that way because of the arguments we happen to use in this particular application of L; I mean that it has to be that way in order for L to even be defined for that combination of arguments.

L is originally defined as a *two* argument function, taking one real number from the interval [a, b] and one element from the tangent bundle of X. In order for L to be a true 3-argument function, where any input from some product of a time space, a position space, and a velocity space is valid, we need to factor the tangent bundle of X into a product of a position space and a velocity space. There is no canonical way of doing this in general.

My recommendation would be to introduce a coordinate system at the beginning, and make no attempt at stating a coordinate-free version of the Euler-Lagrange formula. With a coordinate system, it is possible to make sense of a position partial derivative (the underlying reason for this is that, with a coordinate system, a tangent vector at a point can be canonically extended to a vector field in a neighboorhood of that point, and there is an induced flow on the tangent bundle, whose corresponding Lie derivative is the position partial derivative). Giving a coordinate-free version of the Euler-Lagrange formula is a tricky problem, perhaps harder than it appears at first sight. --Tprice7 (talk) 18:31, 2 May 2018 (UTC)

OK, I apologise for the misinterpretation. Perhaps, we could make the text more clear by replacing $X$ by $\mathbb {R} ^{d}$ for some d, which had already been suggested for other reasons. I did not trace back the history of the article, but I think that, originally, the idea was not to make a coordinate free version of the Euler-Lagrange equation. The tangent bundle over $\mathbb {R} ^{d}$ is well-defined and remains useful in my opinion. In this way, we come close to what you want, without mentioning possible generalisation to differentiable manifolds. Bas Michielsen (talk) 22:02, 2 May 2018 (UTC)

Differentiability of q(t)

Hi all,

The article says function q is differentiable, but the domain of q is not open, hence q is not differentiable. Could anyone fix that please.

Thanks, Konstantin Pavlovskii (talk) 12:27, 12 May 2018 (UTC)

the proof is simple but not shown in article

I am not complaining, but others seem to say "the proof isn't right". the proof is somewhat long (it varies depending on who's is using what method and the "realm of influence" proven).

for me the most rapid and SIMPLE proof is as by farlow's PDE book and my solving (farlow asks the student to complete steps):

   let Gamma[x] represent minimized y[x] of J[y[x]] == Integral[b,a] F dx
   (y is any y substituted, y is not a specific function)

   y[a]==A, y[b]==B (Gamma[x] found must comply)

   where F == F[x,y[x],y'[x]]

   let Xi be a value which is small and approaches zero

   let Eta[x] be a smooth curve (any function)

For minimization we must have:

   J[Gamma[x]] <= J[Gamma[x] + Xi Eta[x]]

The above is a calculus of variations premise.

   let Psi[Xi] == J[Gamma[x] + Xi Eta[x]]

You can visualize a graph of J[g+xe] now as a parabola, Psi' it's slope, which has a minimal value where Psi'[Xi]==0, when Psi'[Xi] is not 0 the we have the J[g+xe] that isn't minimized. Furthermore note Psi is critical when Xi==0 (critical points are part of calculus derivative function minimization). This gamma, if we find it, is the smallest of all y[x] in interval [a,b].

   (Psi'[Xi])[0] == 0 == [Eta][x] D[J[Gamma[x]],Xi]

   (the right we discard, so ignore; it is D[J[g+xe],xi] then xi->0)

This is the step that is more rapid than texts I've seen. Expressing F in it's full form is key. (this is just F with our variation substituted in)

   Integral[b,a]  F[x, Gamma[x] + Xi Eta[x], Gamma'[x] + Xi Eta'[x]]  dx

   Integral[b,a]  (@/Xi) F[x, Gamma[x] + Xi Eta[x], Gamma'[x] + Xi Eta'[x]]  dx  Xi->0

Just work in parallel: do to F what we did with Psi (to set them equal, the integral must undergo the same steps). Take one derivative of F with respect to Xi (perform @/Xi). Then substitute 0 for Xi after that. The result is the same as seen at the end of proofs. DONE, our F now matches the theorem's very simply. (use of chain rule for partial derivative of a function with arguments is used, simple but is a pre-requisite). Mathematica can do the step easily. Many say the next step is "to use integration by parts to move to a simpler form". I would say i.p. would be used in solving the integral - which is a solution we do not want. We wish only an expression equating (F,Gamma,x).

   Integral[b,a] Eta'[x] F^{0,0,1} + Eta[x] F^{0,1,0} dx == 0 == @J[Gamma[x] + Xi Eta[x]]

   Eta'[x] F^{0,0,1} + Eta[x] F^{0,1,0}

I next would claim by ODE, "N[x] F dx + M[x] F dy == 0", we can move around coefficients by simple algebra principles and that n[x] qualifies.

   ((d/dx) Eta[x]) F^{0,0,1} + Eta[x] F^{0,1,0} ==
    Eta[x] (d/dx) F^{0,0,1} + Eta[x] F^{0,1,0} ==
   ( (d/dx) F^{0,0,1} + F^{0,1,0} ) Eta[x]

We assumed Eta[x] is a smooth differentiable curve, it is not zero. For the integral to be zero:

   (d/dx) F^{0,0,1} == F^{0,1,0}
   -(d/dx) F^{0,0,1} + F^{0,1,0} == 0

In solving J[y] using the above it will become clear Gamma[x]==y[x], that is: while solving problem use J[y] and F[x,y,y'] not J[Gamma[x] + Xi Eta[x]].

We could now write out Psi'[x] == 0 == Integral[ result ] but that is discarded too; we only sought an expression relating (F,Gamma[x],x). (most) proofs use i.p. to do the last step (not solved as ODE) and it is "tricky" to find the (F,Gamma,x) form that way but well documented.

— Preceding unsigned comment added by 2601:143:480:A4C0:88F:739C:2A43:F405 (talk) 23:14, 26 January 2020 (UTC)

Proof box does not appear to be expandable

Proof box does not appear to be expandable. — Preceding unsigned comment added by 173.206.33.141 (talk) 22:31, 29 August 2020 (UTC)

Characteristic of the set X in section "Statement"

Since there is a tangent space on X defined I assume that X must be a differential manifold. If so I think that should be mentioned.

Jyyb (talk) 17:31, 18 March 2021 (UTC)

Done. StrokeOfMidnight (talk) 15:04, 2 October 2021 (UTC)

The notation ${\mathcal {L}}$ needs clarification

The symbol ${\mathcal {L}}$ is introduced in the section Euler–Lagrange_equation#Generalizations without definition -though one can deduce it is a differentiable enough function of several variables-, and then used also in Euler–Lagrange equation#Generalization to manifolds with a different definition which is here clearly, it is here the Lie derivative operator on differential forms. It would be good to define this symbol at its first use, and warn in its second appearance that it is redefined, or overloaded. Plm203 (talk) 10:33, 12 November 2023 (UTC)