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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)
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)
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)
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
is very confusing. Can anyone expand it to make it clearer?-- 132.239.27.145 19:14, 21 September 2007 (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)
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 from partial derivative of F with respect to to partial derivatives of F with respect to and ? I don't get it.
The proof is very hard to follow- I don't understand what everything means- for instance, why do you introduce Lx, rather than stating it explicitly? Is it dL/dq(t)?
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)
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)
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)
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)
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)
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)
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,
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:
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)
Sources such as this pdf indicate that the euler-lagrange equations for several functions of one variable are:
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.
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)
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 and 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 . Then due to the theorem of Schwarz there would be
So I think summation should be taken over all combinations with repetition, so that . If this is the case, the sum should be noted with
— Preceding unsigned comment added by 188.103.118.233 ( contribs)
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 ∧Ŝ c2ħε Иτlk 17:29, 11 January 2016 (UTC)
Amended the post above after deleting the section to clarify and add links. M ∧Ŝ c2ħε Иτlk 19:06, 11 January 2016 (UTC)
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)
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)
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"... and denote the partial derivatives of 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)
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)
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. — Preceding unsigned comment added by 173.206.33.141 ( talk) 22:31, 29 August 2020 (UTC)
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 symbol 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)
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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)
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)
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)
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
is very confusing. Can anyone expand it to make it clearer?-- 132.239.27.145 19:14, 21 September 2007 (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)
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 from partial derivative of F with respect to to partial derivatives of F with respect to and ? I don't get it.
The proof is very hard to follow- I don't understand what everything means- for instance, why do you introduce Lx, rather than stating it explicitly? Is it dL/dq(t)?
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)
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)
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)
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)
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)
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)
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,
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:
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)
Sources such as this pdf indicate that the euler-lagrange equations for several functions of one variable are:
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.
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)
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 and 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 . Then due to the theorem of Schwarz there would be
So I think summation should be taken over all combinations with repetition, so that . If this is the case, the sum should be noted with
— Preceding unsigned comment added by 188.103.118.233 ( contribs)
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 ∧Ŝ c2ħε Иτlk 17:29, 11 January 2016 (UTC)
Amended the post above after deleting the section to clarify and add links. M ∧Ŝ c2ħε Иτlk 19:06, 11 January 2016 (UTC)
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)
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)
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"... and denote the partial derivatives of 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)
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)
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. — Preceding unsigned comment added by 173.206.33.141 ( talk) 22:31, 29 August 2020 (UTC)
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 symbol 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)