Questions tagged [calculus-of-variations]

Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.

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48 votes
4 answers
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Central limit theorem via maximal entropy

Let $\rho(x)$ be a probability density function on $\mathbb{R}$ with prescribed variance $\sigma^2$, so that: $$\int_\mathbb{R} \rho(x)\, dx = 1$$ and $$\int_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$ ...
Paul Siegel's user avatar
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26 votes
3 answers
1k views

Largest possible volume of the convex hull of a curve of unit length

What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?
Vladimir Reshetnikov's user avatar
23 votes
2 answers
3k views

How can you compute the maximum volume of an envelope(used to enclose a letter)?

It's obvious that the volume of a envelope is 0 when flat and non-0 when you open it up. However, if you were to fill it with liquid, there must be some shape where it has a maximum volume. Is there a ...
Victor Stone's user avatar
22 votes
2 answers
2k views

Functional approach vs jet approach to Lagrangian field theory

Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...
Bence Racskó's user avatar
21 votes
2 answers
5k views

Surface equivalent of catenary curve

A catenary curve is the shape taken by an idealized hanging chain or rope under the influence of gravity. It has the equation $y= a \cosh (x/a)$. My question is: What is the shape taken by an ...
Joseph O'Rourke's user avatar
21 votes
0 answers
2k views

Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$

$\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\...
Asaf Shachar's user avatar
  • 6,611
18 votes
4 answers
4k views

What was Weierstrass's counterexample to the Dirichlet Principle?

Shortly after his work on the foundations of geometry David Hilbert turned his attention to finding a suitable statement of the Dirichlet principle, from which to prove the Riemann mapping theorem and ...
Jeremy Shipley's user avatar
17 votes
4 answers
2k views

What are the major differences between real and complex Banach space?

Most theorems under real Banach space settings have their twin brothers for complex ones, say, the Hahn-Banach theorem. However, some theorems are not valid in complex Banach spaces, and vice versa. ...
Ice sea's user avatar
  • 407
17 votes
2 answers
1k views

Example of ODE not equivalent to Euler-Lagrange equation

I am looking for an explicit (preferably simple) example of an ODE with time-independent coefficients in $\mathbb{R}^3$ such that there does not exist an Euler-Lagrange equation $$\frac{\partial L}{\...
asv's user avatar
  • 21.1k
17 votes
1 answer
1k views

What variational problem does the parabolic suspension bridge solve?

(Posted to MSE here, no answers) The catenary curve $y(x)$ minimizes the gravitational potential energy $$\int \rho g y\,ds=\int \rho g y \sqrt{1+y'^2}dx,$$ subject to a fixed length, $L=\int \sqrt{1+...
thedude's user avatar
  • 1,417
17 votes
1 answer
494 views

Is a smooth action of a semi-simple Lie group linearizable near a stationary point?

Suppose that $G$ is a semi-simple Lie group that acts smoothly (i.e., $C^\infty$) on a smooth, finite dimensional manifold $M$. Does it follow that the action of $G$ is linearizable near any ...
Dick Palais's user avatar
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16 votes
6 answers
3k views

Smallest area shape that covers all unit length curve

On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve. What are the bounds of the shape's area if this ...
Chao Xu's user avatar
  • 583
16 votes
1 answer
1k views

What braking strategy is most fuel-efficient?

You notice a stop-light ahead of you and it is currently red. You can't run the red light, so you will have to brake, but braking wastes energy and you want to be as fuel efficient as possible. What ...
David Spivak's user avatar
  • 8,549
15 votes
6 answers
3k views

Maxwell equations as Euler-Lagrange equation without electromagnetic potential

In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) ...
asv's user avatar
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15 votes
1 answer
1k views

Who first resolved Hilbert's 20th problem?

Hilbert's 20th problem concerns the existence of solutions to the fundamental problem in the calculus of variations. I understand that Hilbert, Lebesgue and Tonelli were pioneers in this area. In ...
Nirav's user avatar
  • 347
15 votes
3 answers
2k views

Good reference for globally formulated calculus of variations on Riemannian manifolds?

I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant ...
Victor Dods's user avatar
15 votes
1 answer
1k views

Is $\delta(df \wedge df)=0$ an Euler-Lagrange equation?

$\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\tr}{\...
Asaf Shachar's user avatar
  • 6,611
14 votes
2 answers
881 views

Tweetable way to see Riemannian isometries are harmonic?

$\newcommand{\al}{\alpha}$ $\newcommand{\euc}{\mathcal{e}}$ $\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\Det}{\operatorname{Det}}$ Smooth Riemannian isometries are harmonic. Can one conclude ...
Asaf Shachar's user avatar
  • 6,611
14 votes
2 answers
926 views

A riemannian manifold with finitely many closed contractible geodesics

By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction. This means that any two closed geodesics are equivalent if ...
Malte's user avatar
  • 817
14 votes
2 answers
513 views

Variation of the Einstein Hilbert action in a coordinate-free way

I am currently writing an expository paper on gauge theory and gravity and throughout the gauge theory part I have been able to mostly stay away from coordinates unless I wished to provide specific ...
Chris's user avatar
  • 255
14 votes
1 answer
2k views

Invariance of the l.h.s. of Euler-Lagrange equation

Let $M^n$ be a smooth manifold equipped with a nondegenerate Lagrangian $L:TM\to\mathbb R$, $L=L(x,y)$, $x\in M$, $y\in T_xM$. The stationary points of the corresponding integral functional on curves ...
Sergei Ivanov's user avatar
14 votes
1 answer
464 views

"Small" maps from sphere to sphere

Start with a continuous map $f:S^{n+k} \rightarrow S^n$ (round unit spheres). The graph of $f$ lives in $S^{n+k}\times S^n$ and suppose it has a surface area (as a subspace of co-dimension $n$). Now ...
David Feldman's user avatar
14 votes
1 answer
954 views

Area of the minimal surface of a non-planar quadrilateral in 3d

Consider a non-planar quadrilateral in three dimensions, i.e. four points $x_1,\dots,x_4$ in $\mathbb{R}^3$ that do not lie on a plane and connected by straight lines. Then, by general theory of ...
Dirk's user avatar
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13 votes
5 answers
12k views

Beginners text on calculus of variations

I want to begin learning Calculus of Variations. What texts would MathOverflow recommend? Amazon shows up quite a few options. I work on Machine Learning, and that where I intend to apply this. ...
13 votes
3 answers
2k views

How to get to the earliest time zone?

You are in a plane at some point on Earth. You want to be at the earliest time zone possible at the end of your flight. What is the optimal path to take? Formally, fix spherical coordinates $(\theta, \...
Nate River's user avatar
  • 4,822
13 votes
1 answer
430 views

Choquet theory and Hilbert's fourth problem

The following text is an attempt to see Hilbert's fourth problem in a new light. Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line ...
alvarezpaiva's user avatar
  • 13.2k
13 votes
1 answer
710 views

Is this expression for the Laplacian of conformal maps between Riemannian manifolds known?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\...
Asaf Shachar's user avatar
  • 6,611
13 votes
0 answers
257 views

dual to Hodge theory

Let $(M,g)$ be a closed Riemannian manifold. In my understanding Hodge theory shows that any de Rham cohomology class can be represented uniquely by a harmonic form. Moreover the harmonic form ...
Thomas Rot's user avatar
  • 7,363
12 votes
2 answers
836 views

For what metrics are circles solutions of the isoperimetric problem?

A classical result is that solutions of the isoperimetric problem on the plane, the sphere, and the hyperbolic plane are circles. Are there any other Riemannian metrics on these spaces that share this ...
alvarezpaiva's user avatar
  • 13.2k
12 votes
1 answer
898 views

Generalizing "variation of parameters"

I'm stuck on generalizing an ODE formula and could use your help! One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z_0$ of $z'=Az$ (here $z(t)\in\mathbb{R}^n$...
Justin's user avatar
  • 695
12 votes
1 answer
3k views

Results about existence/uniqueness of solution to Euler-Lagrange equations?

While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading: What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange ...
José Navarro's user avatar
12 votes
1 answer
506 views

Different smooth structures on the infinite jet bundle (for the purposes of calculus of variations)

Let $\pi:Y\rightarrow X$ be a (smooth, finite dimensional) fibred manifold. Since no other fibrations will be considered on $Y$, I will identify $(Y,\pi,X)$ with $Y$. The finite order jet bundles are ...
Bence Racskó's user avatar
12 votes
0 answers
388 views

A model of pillows

(The same system with slightly different questions has been asked in MSE.) Let $\Omega\subset \mathbb{R}^2$ be some simply connected planar domain. We seek for a mapping $\mathbf{r}:\Omega\rightarrow \...
Daniel Castro's user avatar
11 votes
2 answers
2k views

Textbooks or notes on gradient flows in metric spaces

What is a good introduction in gradient flows in metric spaces? I know the book Gradient flows: in metric spaces and in the space of probability measures by Luigi Ambrosio, Nicola Gigli and Giuseppe ...
11 votes
1 answer
587 views

Countable (?) dependent choice

In some circumstances I've been using a form of choice over the first uncountable ordinal knowing a priori that only a countable number of choices were going to be made (without any a priori upper ...
Nicola Gigli's user avatar
11 votes
1 answer
839 views

Vector fields on path spaces

I've been reading Chen's original works on iterated integrals and in order to consider differential forms on the path space $PM$ of a smooth manifold $M$ he gives $PM$ the following "differentiable ...
Manuel Rivera's user avatar
11 votes
1 answer
636 views

What is the first eigenvalue of $p$-Laplacian on unit sphere $S^n$?

We know that the first eigenvalue of Laplacian on the Riemannian unit sphere $S^n$ is $n$, then what is the explicit expression for the first eigenvalue of $p$-Laplacian on $S^n$? The $p$-Laplacian ...
mathsnail's user avatar
  • 189
11 votes
0 answers
424 views

Maximizing the volume in a family of subsets of a cube

Starting from a question in probability, I arrived to the following optimization problem. Let $I:=[0, 1],$ and let $A$ be a Lebesgue measurable subset of the $n$-dimensional cube, $A\subset I^n.$ ...
Pietro Majer's user avatar
  • 56.5k
11 votes
2 answers
733 views

Prove/disprove $(\int_{0}^{2 \pi} \!\!\cos f(x) d x)^{2}+(\int_{0}^{2 \pi}\!\!\! \sqrt{(f'(x))^{2}+\sin ^{2} f(x)}dx)^{2}\ge 4\pi^{2}$

This problem has been posted on Math.SE but didn't receive any correct answer after a long time. Let $f(x)$ be a differentiable function on $[0,2\pi]$ s.t. $0\leq f(x)\leq 2\pi$ and $f(0)=f(2\pi)$. ...
FFjet's user avatar
  • 282
10 votes
3 answers
819 views

Rigorous justification that overdetermined systems do not have a solution

There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions ...
asv's user avatar
  • 21.1k
10 votes
4 answers
12k views

Good book on Calculus of Variations

What is a good book on the Calculus of Variations, for a second year PhD student?
Edger's user avatar
  • 101
10 votes
5 answers
1k views

Convex curves with many inscribed triangles maximizing perimeter

A classical nice result of Euclidean geometry states that the triangles maximizing the perimeter among all inscribed triangles of a given ellipse constitue a one-parameter family. Precisely, for each ...
Pietro Majer's user avatar
  • 56.5k
10 votes
1 answer
2k views

Coordinate-free derivation of the Einstein's field equation from the Hilbert action.

It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional (without matter and cosmological constant, which is irrelevant here): $$S = \int_M R \mu_g,$$ is given by ...
Lizao Li's user avatar
  • 473
10 votes
1 answer
689 views

How to shrink a square with minimal distortion?

$\newcommand{\CO}{\text{CO}_2}$ $\newcommand{\euc}{\mathfrak{e}}$ $\newcommand{\SO}{\text{SO}_2}$ $\newcommand{\al}{\alpha}$ $\newcommand{\dist}{\text{dist}}$ $\newcommand{\Lip}{\text{Lip}_{\text{inj}}...
Asaf Shachar's user avatar
  • 6,611
9 votes
1 answer
363 views

Largest convex hull of a unit length path

What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?
ARi's user avatar
  • 841
9 votes
1 answer
779 views

Why the least action principle is always (?) used in this particular form?

The least action principle in (mathematical) physics says the following. Given a system, e.g. collection of particles, whose motion satisfies a known system of differential equations (of second order)...
asv's user avatar
  • 21.1k
9 votes
4 answers
1k views

Boundedness of nonlinear continuous functionals

Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$. Is it true that there exists an infinite dimensional reflexive subspace $E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ? ...
Ady's user avatar
  • 4,030
9 votes
1 answer
510 views

Do these surfaces intersect?

For any real numbers $a_{1},a_{2},\cdots a_{6}$ and $b_{1},b_{2},\cdots b_{6}$ with $\sum_{i=1}^{6}a_{i}^{2}=1$ and $\sum_{i=1}^{6}b_{i}^{2}=1$, does the equation $$ x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{...
mathers1's user avatar
9 votes
1 answer
282 views

Which domain maximizes the energy of the Lebesgue measure?

This could be asked in more generality, but let me stick to a concrete case. Usually one considers a fixed domain $E \subset \mathbb{C}$ and attaches to it the equilibrium probability measure $\nu_E$...
Vesselin Dimitrov's user avatar
9 votes
0 answers
287 views

Is there a variational interpretation for the equation $\operatorname{div}(\star \circ \bigwedge^k df\circ \star )=0$?

$\newcommand{\id}{\operatorname{Id}} \newcommand{\R}{\mathbb{R}} \newcommand{\TM}{\operatorname{TM}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Cof}{\operatorname{Cof}} \newcommand{\Det}{\...
Asaf Shachar's user avatar
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