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Probably, the best thing to do would be to write $x = f(u)$ and then use $$ \int_{u^{-1}(a)}^{u^{-1}(b)} L(x,u,u') dx = \int_a^b L\left(f(u),u,\frac{1}{f'(u)}\right)f'(u)\ du = \int_a^b M\left(u,f(u), …

According to Giaquinta and Hildebrandt (Calculus of Variations I, p. 70): "Euler's differential equation was first stated by Euler in his Methodus inveniendi [2], Chapter 2, no. 21. Quite often, one s …

Thanks for explaining your motivation, because I think that the general problem as you stated it is impossibly hard, but that, fortunately, for the problem that you are really trying to tackle (the in …

N.B. This is an edit of my original post, confirming the guess that I made originally. The answer is no, i.e., such curves are not forced to be ellipses. Here is a sketch of the argument. (The det …

There is no reason to believe that there is a supremum of this functional. For example, consider the $3$-torus $M = \mathbb{R}^3/\mathbb{Z}^3$ with the quotient metric and the unit $1$-forms $$ \alph …

There is a coordinate-free description using only natural objects on $TM$. Here is one way to do it. First, consider the basepoint submersion $\pi:TM\to M$. For each $v\in TM$, the linear map $\pi' …

If you assume that $f>0$ and $g < 0$ on $[0,1]$, then one can integrate the equations explicitly. Assume $0<t<1$, so that $F$ and $G$ are positive in $(0,1)$. Let $p = f/F = (\log F)' >0$ and $q = …

By calculus, the line $l_C$ is 'the' major axis of the ellipse of inertia of the finite point set $C$. (The reason for the quotes around 'the' in the previous sentence is that, if the ellipse of iner …

Your problem does not have a maximizing solution. Here is why: Assuming that $f$ is piecewise smooth and not identically constant, we can solve the Euler-Lagrange equations in an interval where $f …

Note: I'm updating my answer to give a better (i.e., simpler) example plus a little more information about how to derive the example from Douglas' results (which may not be entirely clear upon first …

It is known (and follows from an easy calculation) that solutions of the isoperimetric problem on a surface have constant geodesic curvature. In his 1887 classic Leçons Sur La Théorie Générale Des Su …

Now that your comment has clarified your question, we can answer it: The answer is 'no'. There is the following well-known example: Consider the following family of circles: $C_\lambda$ is defin …

There is a large literature on this, and the roots go back more than one hundred years. Some of the modern work along these lines can be found by looking for papers containing the term 'variational b …

Actually, your notation is causing some confusion. In one very real sense (probably not your intended one) the answer to your question is yes, not no which is probably the answer to the question that …

An approach that should work is to derive the differential equation that any minimizer would have to satisfy and check that its solutions are the known ones for which equality holds. To fill in the d …