Questions on the calculus of variations, a subfield of calculus that deals with the optimization of functionals.

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Maximium of area/perimeter^2 of a function

Let $a,b$ be two reals. Does there exists a constant $C$ such that for all functions $f:[a,b]\to\mathbb{R}$, continuous on $[a,b]$ and differentiable on $(a,b)$ with $f(a)=f(b)=0$, \begin{equation} ...
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1answer
365 views

Arnol'd's trivium problem #68

I came across this blog that says that its French version has answers to most of Arnol'd's trivium problems, and I figured I'd try my hand at some of the ones they don't have. Number 68 raised my ...
2
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0answers
539 views

Find the extremal to the functional and discuss whether they provide a max/min

I am having a hard time getting my head around Functionals and Calculus of Variations, My question is: Given a functional and using the Euler-Lagrange equation to find an extremal, how do we show ...
2
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1answer
320 views

Separation of variables and substituion; first integral from the Euler-Differential Equation for the minimal surface problem

Let $P_1=(a,y_a),P_2=(b,y_b), y\in C^1 (a,b), y_a>0,y_b>0$ And the area integral: $\int^b_a y(x) \sqrt{1+y'(x)}dx$ From the Euler differential-equation we obtain: $$y'=1/\alpha ...
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1answer
180 views

Noether's theorem

I am now reading the book Calculus of Variations written by Jost and I have a problem in the proof of Noether's theorem: Theorem 1.5.1. Let $F\in C^2([a, b]\times \mathbb R^d \times \mathbb R^d, ...
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1answer
146 views

Smoothness of a non-local functional

While studying a nonlinear PDE arising from quantum mechanics, I met a statement that I cannot prove easily. Let us write $E=W^{1,2}(\mathbb{R}^3)$ for the usual Sobolev space, and define the ...
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3answers
380 views

calculus of variations question

I would like to find a continuous function $y : [0,4] \to \mathbb{R}$ that minimizes the following functional $$I (y) := \displaystyle\int_{0}^4\sqrt{y\left(1+(y^{\prime})^2\right)} dx$$ subject to ...
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421 views

Finding a proper solution of a given functional

It's my first post here, but I worked very hard to find solution and I failed. Hereinafter, I skip physical background and directly proceed to my mathematical problem. No matter how, you know the ...
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1answer
130 views

how to obtain Euler equation for smoothing spline minimization problem?

The question might be trivial, but I don't understand why this minimization problem in Sobolev space $$ \min_{g}\int_{0}^{1}\left\{ f(x)-g(x)\right\}^{2} dx+\lambda\int_{0}^{1}\left\{ ...
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83 views

Is this scenario possible and if so, find the equation of the hill

Assume a ball is placed on a hill and the time is measured for the ball to reach the ground (y=0). If the time taken for the ball to reach the ground is independent on the position the ball is placed ...
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1answer
557 views

Taking derivatives in index notation

So I'm working out some calculus of variations problems however one of them involves a fair bit of index notation. I'm familiar with the algebra of these but not exactly sure how to perform ...
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1answer
68 views

Pf. of weak lower semicontinuity for convex Lagrangians

This question is about the proof of Theorem 1 in Chapter 8 of Evans' PDE book (p. 468 in the 2nd edition). Let $u,u_k\in\mathrm{W}^{1,q}(U)$ for all $k\in\mathbb{N}$, $U\subset\mathbb{R}^n$ be open, ...
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3answers
295 views

Euler-Lagrange equation has no solution

I need to find a function $f(x)$ that maximizes a functional: $$ J(f)= \int\limits_{-\infty}^{+\infty} e^{-x^2/2}f(x) \,dx$$ Where $$f(x)>0 \ \text{ and} \int\limits_{-\infty}^{+\infty} f(x) ...
3
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1answer
99 views

Limit with functionals

I need to evaluate the following limit which intuitively I know its equal to 0 but I can't really prove it, so I need some help: $$\lim_{\epsilon \to 0}{\frac{F[\rho + \epsilon\rho' + ...
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2answers
60 views

Another partial derivatives question

I think this is a common question in applied math but I find no occurrence of it in MSE. $$u=u(x), v=v(x)$$ $$f=f(u(x),v(x))$$ 1) $u$ and $v$ are both functions of the variable $x$. If $u$ varies, ...
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134 views

Is this calculus of variations intuition justifiable?

I'll preface this by saying that I haven't taken an in depth study of the calculus of variations and have only come across it recently in applications; in depth study is on my to do list. I'm ...
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43 views

Gradient/sub-differential of a minimum of a function

Suppose I have a function $F(x,D) = ||y-Dx||_2^2$, such that $x^{*}(D)= \displaystyle arg \min_{x} F(x,D)$ (that is given $y$ and for a fixed $D$) and subject to some constraint $h(x) <\epsilon$, ...
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1answer
185 views

Frechet Differentiabilty of a Functional defined on some Sobolev Space

How can I prove that the following Functional is Frechet Differentiable and that the Frechet derivative is continuous? $$ I(u)=\int_\Omega |u|^{p+1} dx , \quad 1<p<\frac{n+2}{n-2} $$ ...
4
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1answer
215 views

Divergence Theorem/ partial integration of higher order

I'm in the middle of a proof and i'm trying to understand a step of the proof which does give me a hard time. The proof is about minimal surfaces and at the moment I'm trying to understand why the ...
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1answer
275 views

What is a differentiable functional?

I saw in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. That the functional ...
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124 views

Prove that a flat shape minimizes a functional

The following functional arises in an information theoretic problem that I work on currently. $$I(G(\omega)) = \int_{-\kappa\pi}^{\kappa\pi} \frac{A}{G(\omega)+A}d\omega-\frac{\left| ...
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1answer
170 views

Who came up with the Euler-Lagrange equation first?

Could someone explain who came up with the specific equation first? http://en.wikipedia.org/wiki/Euler-Lagrange makes it sound like Lagrange got it first, in 1755, then sent it to Euler. but: ...
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188 views

variation problem of constrained area and minimized distance

$$c=\int_{x_1}^{x_2}f_{gr}(x)\;dx$$ The integral is a time-like curve between $x_1$ and $x_2$ and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and ...
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3k views

Simple simple Euler Lagrange Equation

Just starting a course on Lagrangian Mechanics and I'm just wondering what about the Euler-Lagrange equation, and more specifically what I'm meant to be trying to do .. One of the questions from my ...
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95 views

Am I allowed to move around an operator like this?

Can I take this product: $$\frac{dL}{dt}\frac{d L}{d \dot{x}}$$ And factor out one of the $L$'s to get: $$L\frac{d}{dt} \left( \frac{d L}{d \dot{x}}\right)$$ Where the operator $\frac{d}{dt}$ now ...
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1answer
216 views

Why can't we construct a counter-example to the Fundamental Lemma of the Calculus of Variations?

"The fundamental lemma of the calculus of variations states that if the definite integral of the product of a continuous function $f(x)$ and $h(x)$ is zero, for all continuous functions $h(x)$ that ...
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1answer
224 views

Find function to make maximum value

Let ${f : [0, 1] \rightarrow [-1, 1] }$ is a continuous function such that ${ \int_{0}^{1} x f \left(x\right) dx =0}$ Find $f(x)$ such that ${ \int_{0}^{1} \left(x ^{2 } + \frac{1}{4} \right) f ...
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204 views

Closed Geodesics as minimisers of action functional

Suppose I have a Riemannian surface $(M,g)$. It's clear that closed geodesics are critical points of the length functional $l(\gamma)=\int\left|\gamma(t)^{\prime}\right|_{g(\gamma(t))}dt$ over the ...
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2answers
558 views

minimal surface of revolution when endpoints on x-axis?

What is the formula for the planar curve through $(\pm a,0)$ of fixed length $l$ which has minimal-area surface of revolution when rotated about the x-axis? I get the area of the surface to be ...
3
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1answer
208 views

Finding the Euler-Lagrange operator $\mathcal M$ of a functional $\mathcal F$

I'd appreciate some help with the following problem: Let $F = F(x, \{p_\alpha\}_{|\alpha|\le m})$ be a smooth function of the variables $x\in \overline \Omega$, and $p_\alpha \in \mathbb R, ...
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91 views

First Weighted Eigenvalue of the Laplacian

Let $\Omega$ be a ball centered in the origin and let $\lambda_1(\Omega)$ be the first (or lowest) eigenvalue of the Dirichlet Laplacian in $\Omega$: $$\lambda_1 (\Omega) =\min_{u\in H_0^1 (\Omega),\ ...
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1answer
236 views

An elementary (?) minimization problem

This morning, in Italy, there was the national exam of mathematics for students of high schools. One of the exercises asked to solve Heron's problem: given a straight line and two points lying on the ...
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217 views

Problem of finding strong maxima or minima of a functional

I have got this problem in exam where I have to to check for strong maxima (or minima) or weak maxima(or minima) of the functional given by $\int_{0}^{1} (1+x)(y^')^2 dx ~~~~~ y(0) = 0, ~~ y(1) = ...
3
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1answer
97 views

how to solve differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$?

What's the solution of the differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$, where $y$ is a function of $x$ and $\alpha$ is a constant? Polynomial solutions don't seem to work, because the ...
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2answers
203 views

Intuition of 'Transversality conditions' needed

I have read different matrials on Calculus of Variations, but I still do not grasp the intuition of transversality condition. From textbooks, I can only roughly get an idea that with a transversality ...
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90 views

$u''+\frac{4}{x+1}u'+\frac{2}{\left(x+1\right)^{2}}u=0$ variational solution

This is a concept solution scheme derived from a particular example that I have not been able to generalise sufficiently. The objective is to find a particular solution to a certain second-order ...
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1answer
89 views

Lagrange multipliers — What have I done wrong?

I am trying to find the stationary points of the potential $U(x,y)=x^2+y^2$ with constraint $x^2-2y^2=1$ So I set the Augmented potential $U^*=x^2+y^2+m(x^2-2y^2)$ where $m$ is the Lagrange ...
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1answer
123 views

Volume integral and Variations

Suppose I wish to find the Euler-Lagrange equation for an integral $\int_V f(u,\mathop{\mathrm{grad}} u)\,dV$ where $V$ is a volume given by some equation, for example say $x^2+y^2+z^2\le 1$, and ...
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1answer
114 views

Could someone please explain what this question is asking?

I have some trouble understanding the following question: Suppose we have 1st fundamental form $E \, dx^2+2F \, dx \, dy+G \, dy^2$ and we are given that for any $u,v$, the curve given by $x=u, y=v$ ...
6
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1answer
178 views

Is there a fundamental misunderstanding here or have I made an algebraic slip?

Is there a fundamental misunderstanding here or have I made an algebraic slip? I have a Riemannian metric of the form $ds^2={du^2+dv^2\over 1-u^2-v^2}$ on an open disc and I want to prove that radial ...
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453 views

Simple form of Euler-Lagrange equations in cylindrical polars for path of light?

What are the Euler-Lagrange equations in cylindrical coordinates $(r,\theta,z)$ for light moving at speed $v(r,\theta)$, where $r$ and $\theta$ depend on $z$? I.e. for the problem of minimising the ...
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348 views

Constraint of a Lagrange multiplier

My question concerns Lagrange multipliers and the possibility to impose constraints on the multipliers themselves. I have a Stokes flow which is solved using the Finite Element Method on a domain ...
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1answer
137 views

Find an upper bound for lowest eigenvalue using calculus of variations.

So I'm doing a little calculus of variations on an eigenvalue problem. The goal of this is to find an upper bound for the $\omega_0$ as follows: $\omega_0^2 \leq ...
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121 views

Nonlinear BVP pde and variational inequality

Suppose $f \in L^2(\Omega)$ where $\Omega$ is bounded. The problem: for $a \in \mathbb{R}$ find $u_a \in H^1_0(\Omega)$ s.t $$-\Delta u_a + \frac{m(u_a)}{a} = f$$ where $m(r) = \begin{cases} r ...
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0answers
181 views

A “bounded” constraint in a variational problem

Is there any standard approach to solve the following kind of variational problem? Maximize $F=\int_0 ^1 L(x,y,y')dx $ subject to the constraint $|\int_0 ^1 M(x,y,y')dx| \lt k$ where $y$ ...
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161 views

Finding a force function from bodies in equilibrium

(This is an edited version of the original question, since I'm starting a bounty) I'm trying to find a function $y$ from given data. Reverse optimization, so to speak. Say we have two ...
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2answers
266 views

How do I obtain an appropriate energy functional from the weak formulation of a partial differential equation?

I'm reading a textbook example on the finite element method: $\nabla^T[D(x,y,z)\nabla u] - a(x,y,z)u + f = 0 $ in R $\partial R= \partial R_1 \bigcup \partial R_2$, $\partial R_1 \bigcap ...
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1answer
135 views

Equation for stationary values of $px^2+qy^2+rz^2$ given sphere and plane constraints

Consider stationary points of the function $V=px^2+qy^2+rz^2$ subject to the constraints $x^2+y^2+z^2=1$ and $lx+my+nz=0$, where $l,m,n$ not all zero and $p,q,r$ not all equal. How can we show that ...
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87 views

Extremizing an Integral under a cyclic condition

Let $h$ be a nonnegative, smooth and convex function on $[0,1]$ and let $f(x,y):[0,1]\times[0,1]\rightarrow[0,1]$ with $f(x,y)=f(y,x)$ and $f$ continuous. Suppose I fix $r>0$ and demand that $$ ...
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1answer
77 views

Simplifying the search for a geodesic

How can calculations for the geodesics on the surface $U=\{(x,y,z): c(x^2+y^2)-z^2=0, z>0\}$ be simplified by noting that is locally Euclidean? I can see that the property means that when we open ...