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Results tagged with calculus-of-variations
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user 13972
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
12
votes
Accepted
Is there a geometric interpretation for this quantity?
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
$$
\alph …
- 106k
13
votes
Accepted
Who came up with the Euler-Lagrange equation?
According to Giaquinta and Hildebrandt (Calculus of Variations I, p. 70): "Euler's differential equation was first stated by Euler in his Methodus inveniendi [2], Chapter 2, no. 21. Quite often, one s …
- 106k
13
votes
Accepted
Characterizing maximal powers of closed 2-forms in odd-dimensional manifolds
Thanks for explaining your motivation, because I think that the general problem as you stated it is impossibly hard, but that, fortunately, for the problem that you are really trying to tackle (the in …
- 106k
4
votes
Accepted
Calculus of variations when functional involves inverse of the function
Probably, the best thing to do would be to write $x = f(u)$ and then use
$$
\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), …
- 106k
10
votes
Accepted
Convex curves with many inscribed triangles maximizing perimeter
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 …
- 106k
16
votes
Accepted
Invariance of the l.h.s. of Euler-Lagrange equation
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' …
- 106k
5
votes
Accepted
Least-squares regression and differential geometry
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 …
- 106k
20
votes
Accepted
For what metrics are circles solutions of the isoperimetric problem?
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 …
- 106k
22
votes
Accepted
Example of ODE not equivalent to Euler-Lagrange equation
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 …
- 106k
4
votes
Accepted
Numerical or exact solution for a system of differential algebraic equations
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 = …
- 106k
3
votes
A Lagrangian problem with a countable family of local extrema ?
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 …
- 106k
11
votes
Accepted
Formulating the calculus of varations with exterior calculus
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 …
- 106k
6
votes
Accepted
Are all null curves of a Lorentzian metric extrema?
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 …
- 106k
12
votes
Accepted
Stability of minimal surfaces
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 …
- 106k
17
votes
Accepted
Variation of curvature with respect to immersion?
If you just calculate using the moving frame, you'll get the answer for the variation of the principal curvatures in a few lines:
$$
\delta\kappa_i = \mathrm{Hess}(u)(e_i,e_i) + \kappa_i^2\,u .
$$
Her …
- 106k
5
votes
Prove/disprove $(\int_{0}^{2 \pi} \!\!\cos f(x) d x)^{2}+(\int_{0}^{2 \pi}\!\!\! \sqrt{(f'(x...
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 …
- 106k
17
votes
Accepted
Tweetable way to see Riemannian isometries are harmonic?
Not exactly 'tweetable', but perhaps the identity (1) may help, if all you want to do is avoid the Euler-Lagrange equations. For simplicity, assume that $M^n$ is oriented. (One can write the identit …
- 106k
19
votes
Accepted
Rigorous justification that overdetermined systems do not have a solution
There is probably no single proof that would provide a rigorous justification of the OP's principle in all cases. Moreover, without specifying more clearly what is meant by a 'natural map', the princ …
- 106k
4
votes
Accepted
Calculating the geodesic equation for a particular set of phase-space coordinates
The answer that you want, namely div $U_g$, is not going to expressible in terms of a geometrically invariant quantity (such as, say, the scalar curvature of $g$) because it depends on the underlying …
- 106k
6
votes
Accepted
Are all the mappings which satisfy this equation scaled isometries?
Here's a simple counterexample: Let $M=N=T^2$ (the standard torus, thought of as $\mathbb{R}^2/\mathbb{Z}^2$). Let $f:M\to N$ be the identity, and let the metrics on $M$ and $N$ be any two translati …
- 106k
11
votes
"Small" maps from sphere to sphere
Here's an example to show that the infimum is not always attained:
Consider the standard Hopf map $\pi:S^3\to S^2$, which is not null-homotopic, of course, so it follows that the area of the graph in …
- 106k
2
votes
Accepted
Are all symmetries of the Dirichlet functional isometries?
The answer to your question is "yes, every symmetry (in the sense you have specified) is an isometric immersion".
To see why, first note that, if $(M,g)$ is a compact Riemannian $m$-manifold and $h$ …
- 106k