25

Your question is a variant of the teabag problem. I don't believe an exact answer is known, but for the $1 \times 1$ square teabag, the maximum volume is about $0.2$:                     (Image from Wikipedia article.) The primary reference is Anthony Robin's 2004 article, "Paper Bag Problem". ...


22

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 reading of his paper). This also addresses the question of time-dependent Lagrangians originally raised by the OP. Have a look at Jesse Douglas' paper Solution ...


19

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 principle itself turns out not to hold in general. For example, every smooth (complex-valued) function $f$ on the unit circle $S^1\subset\mathbb{C}$ can be ...


16

There is a book on the subject: "Information Theory and The Central Limit Theorem" by Oliver Johnson. The article by Anshelevich mentioned by Yemon considers the operator $T$ acting on probability densities and corresponding to going from the law of a random variable $X$ to that of $(X+Y)/\sqrt{2}$ where $Y$ is an independent copy of $X$. The entropy is a ...


16

If you just calculate using the moving frame, you'll get the answer for the variation of the principal curvatures in a few lines: $$ \delta\kappa_i = \mathrm{Hess}(u)(e_i,e_i) + \kappa_i^2\,u . $$ Here, $\delta\kappa_i$ is the first $t$-derivative of $\kappa_i$ at $t=0$ (i.e., the 'first variation of $\kappa_i$'), $\mathrm{Hess}(u)$ is the quadratic form ...


16

There are some differences. For example Bishop-Phelps theorem, which holds only in real Banach spaces. In my opinion, this qualifies as a "major theorem". MR1749671 Lomonosov, Victor A counterexample to the Bishop-Phelps theorem in complex spaces. Israel J. Math. 115 (2000), 25–28. Remark. Your statement "natural to assume that the field is real if the ...


15

The answer seems to be $\frac{1}{2\pi}$, using a semi circle. See Moran, P. A. P. "On a problem of S. Ulam." Journal of the London Mathematical Society 1.3 (1946): 175-179.


15

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 $\pi'(v):T_v(TM)\to T_{\pi(v)}M$ is surjective, and the $\pi$-fiber through $v$ is equal to $T_{\pi(v)}M$, a vector space. It follows that the kernel of $\pi'(v)$ ...


15

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 identity (1) below as an identity on densities, so the orientability hypothesis is not essential, but I'll leave that detail for the interested.) If $g$ is a metric ...


15

I do not know if it is good form for MO to cite one's own papers when answering a question, but I will take the chance. This matter is addressed in quite a bit of detail in my joint paper with Romeo Brunetti and Klaus Fredenhagen, R. Brunetti, K. Fredenhagen, P. L. Ribeiro, Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics ...


14

There's a 1985 article by Derriennic called "Entropie, theoremes limite et marches aleatoires" (entropy, limit theorems and random walks). In it there is a section where the connection between your observation that the Gaussian maximizes entropy (which is attributed to Shannon) and the central limit theorem is discussed. He begins by discussing a proof (...


14

I just bumped into your post after doing some google search to find a bibtex entry for Almgren's notes. I have a copy of them: I could copy it and send it to you (or maybe scan it and share it via dropbox). However before getting to the combinatorial arguments you would have to go through quite a lot of material. I am writing a paper with a PhD student ...


14

You are asking whether a least area surface depends continuously on its boundary curve, or in mathematical terms: whether for every curve $\gamma$ there exists a least area surface $S$ with $\partial S=\gamma$ such that for any neighborhood of $S$ in the space of surfaces there exists a neighborhood of $\gamma$ in the space of curves such that for any curve $...


12

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 defined as $x^2+y^2 = 1$ and $z = \lambda$. Let $\lambda>0$ be fixed and orient $C_{-\lambda}$ counterclockwise and orient $C_\lambda$ clockwise. Then for $\...


12

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 $$ \alpha_n = \cos(2\pi n z)\,\mathrm{d} x - \sin(2\pi n z)\,\mathrm{d} y, $$ where $n$ is an integer, which are well-defined on $M$. One finds by calculation that $$...


11

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 $S^3\times S^2$ of any differentiable map $f:S^3\to S^2$ that is homotopic to $\pi$ is strictly greater than the area of the graph of a constant map, i.e., of $...


11

A discussion of Tonelli's contributions and their relation to Hilbert's work can be found in this AMS bulletin. The original work was published in Italian, Fondamenti di Calcolo delle Variazioni (Bologna, 1921 & 1923) --- I have not found an English translation, but an English summary by Tonelli is also in an AMS bulletin. It is noteworthy that Tonelli ...


11

Actually this is one of the oldest problems in the calculus of variations. It's named "the Newton problem" after Sir Isaac Newton, who studied it in 1685. It arises from the determination of the optimal profile for the motion of bodies (projectiles, ships, etc), that is, the profile giving the minimal aerodynamic or hydrodynamic resistance. Here you are ...


11

In the form (1), if you compute the variation $\delta S / \delta x(t) = E(t)$, you find that $E(t) = E(x(t),\dot{x}(t), \ddot{x}(t) ,t)$ is a local/differential expression (the value of $E(t)$ does not depend on $x(t')$ or its derivatives at other times $t'\ne t$). This is no longer true if you use $\exp(S)$ instead of $S$. There is no dispute that $S$ and $\...


11

Not necessarily- let $\Omega = B_1 \cap \{x_3 > 0\}.$ Then $u(x) := (1-|x|^2)\frac{x_3}{|x|}$ is in $H^1_0(\Omega) \cap C^{\infty}(\Omega),$ but $u$ is discontinuous at the origin.


10

The principle you mention is not always true ! V. Arnold proved that every continuous function in $N$ real variables is a composition of continuous functions of two variables only. More precisely, there exist $N(2N+1)$ universal functions $\phi_{ij}:[0,1]\rightarrow[0,1]$ such that the map $$(g_1,\ldots,g_{2N+1})\mapsto\sum_jg_j\left(\sum_i\phi_{ij}(x_i)\...


10

There is a trivial sense in which the answer is "yes": the solutions to the Maxwell equations are (formally, at least) the global minimisers to the functional $$ \int\int |\mathrm{div} E|^2 + |\mathrm{div} B|^2 + |\mathrm{rot} E + \frac{1}{c} \dot{B} |^2 + |\mathrm{rot} B - \frac{1}{c} \dot{E} |^2\ dx dt.$$ I would suppose you consider this as ...


9

Yes indeed, the Maxwell's equations are Euler-Lagrange equations. And this is quite interesting. Let me give here a presentation within Special Relativity, in which the light speed is set to $c=1$. The ambiant space is therefore a Minkowski space $\mathbb R^{1+3}$ with metric $dt^2-d{x_1}^2-d{x_2}^2-d{x_3}^2$. I restrict myself to the case of a vacuum. The ...


8

The book by Gelfand and Fomin is quite good (and its Dover ...). Another one I like a great deal are those of Giaquinta and Hildebrandt (specially volume 1), but those are not Dover: check them out from the library!


8

Yes, this is called the nonlinear variation of constants formula due to Alekseev: “An estimate for the perturbations of the solutions of ordinary differential equations”, in: Vestnik Moskov. Univ. Ser. I Mat. Meh. 2 (1961), pp. 28–36. I don't think that that article is available in English. It can also be found in the book by V. Lakshmikantham and S. Leela "...


8

Your question actually is quite well answered around page 10 of Giaquinta and Hildebrandt's Calculus of Variations I: the Lagrangian formalism. The upshot is that the correct phrase you are looking for is the (possibly nonlinear) Gateaux differential, and not the Frechet derivative, and that is for good reason (with my apologies to French people everywhere ...


8

Here is the story behind these notes, and a redirect to On the First Variation of a Varifold, W.K. Allard (1972). a quote from: Selected Works of Frederick J. Almgren


8

In optimization, you typically have a function $f \colon X \to \mathbb{R}$ which you are going to minimize. In order to apply first-order optimality conditions or first-order methods, you would like to compute the derivative of $f$. However, if $X$ is a complex Banach space, the derivative will never be $\mathbb{C}$-linear, since the range of the derivative ...


8

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