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

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12
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8answers
5k views

Why Circle encloses largest Area?

In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all i would ...
5
votes
0answers
500 views

Maximum and minimum of an integral under integral constraints.

Find the maximum and minimum of the following integral in terms of $f(x),a,C$: \begin{align}I=\int_{0}^{a} \frac{x}{f(x)}p(x)dx \end{align} s.t.: 1) $\int_{0}^{a} p(x)dx=1$ 2) $\int_{0}^{a} ...
4
votes
1answer
243 views

Assumption that $\delta q'$ is small in the derivation of Euler-Lagrange equations.

I have never completely understood the justification of this step in the derivation of the E-L equation: $\delta L = L(q + \delta q, q' + \delta q', x) - L(q, q', x) = \partial_q L \delta q + ...
7
votes
5answers
2k views

Introductory text for calculus of variations

I am currently working on problems that require familiarity with calculus of variations. I am fairly new to this field. Please suggest a good introductory book for the same that could help me pick up ...
11
votes
4answers
425 views

When is the moment of inertia of a smooth plane curve is maximum?

Given a smooth plane curve $(x(s),y(s))$, parameterized in arc length $s$, of fixed finite length $L$, its moment of inertia about its center of mass (axis perpendicular to the plane) is given as $$MI ...
7
votes
2answers
1k views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ...
4
votes
2answers
248 views

Optimization problem using Reproducing Kernel Hilbert Spaces

I am encountering a problem concerning Reproducing Kernel Hilbert Spaces (RKHS) in the context of machine learning using Support Vector Machines (SVMs). With refernce to this paper [Olivier Chapelle, ...
2
votes
2answers
519 views

satisfy the Euler-Lagrange equation

Two circles of unit radius, each normal to the line through their centers are a distance d apart. A soap film is formed between themas shown below; energetic considerations require the filem to ...
4
votes
2answers
806 views

Geodesics of a Sphere in Cartesian Coordinates

I want to minimize $I = \int |\dot{x}|^2 dt$ subject to the constraint $|x|^2=1$ (sphere) which gives an Euler equation of $\lambda x - \ddot{x} = 0$. I have to show that the Euler equation is ...
2
votes
2answers
53 views

A variational problem with a lagrangian , what is the lagrangian?

given the 2 PDE $$ \Delta u-au_{tt}+u_{t}=0$$ and $$ \Delta u + Du*Df=0 $$ here $ \Delta u $ is the Laplacian $ Du= grad(u) $ is the gradient and * means scalar product $u_{t} = \frac{\partial ...
1
vote
1answer
264 views

Maximizing a function by finding derivative

I want to find the value of $\vec{p}$, $p_s$, $p_t$ each of which is a function of the form $f:\mathbb{R}^2 \to \mathbb{R}$ that maximize the following function : $$\begin{align} \int_\mathbb{R^2} ...
0
votes
0answers
30 views

Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
-2
votes
2answers
182 views

Calculus of variations question from Darcogona

I asked the question in the next forum, the 4th post, hopefully someone can help me with this here or there: https://nrich.maths.org/discus/messages/7601/151442.html?1310911861 Thanks in advance.
56
votes
1answer
2k views

What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even, positive function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. View the ...
12
votes
1answer
349 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 ...
4
votes
1answer
267 views

Derivation of Euler-Lagrange equation

Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation. If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is $\dfrac{\partial ...
2
votes
0answers
58 views

A variational strategy for finding a family of curves?

In a recent question, I asked for examples of families of distinct smooth curves with fixed area and perimeter (which for this question I will dub as doubly-isometric). That wording allows $C^1$ ...
14
votes
1answer
253 views

Fastest curve from $p_0$ to $p_1$

I'm trying to solve a problem in path planning: Given points $p_0$ and $p_1$ and vectors $v_0$ and $v_1$, find a function $p(t)$ st. $p(0) = p_0$, $p(T) = p_1$, $p'(0) = v_0$ and $p'(T) = ...
5
votes
1answer
720 views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

I have a question about Euler-lagrange equation which you can check this file. http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf, specifically in page 6,equation 8 , not equation 9... There ...
1
vote
1answer
50 views

Calculation mistake in variation of length functional?

This should be pretty simple to check if you know the basics of variational calculus. I feel like I am making an obvious mistake somewhere like not using chain rule somewhere. Let $g : \mathbb{R}^n ...
1
vote
1answer
1k views

Derivative of $f(x,y)$ with respect to another function of two variables $k(x,y)$

Suppose that we have a function $f(x,y)$ of two variables: $$f(x,y) = g(x) + h(y) + 5(x-y) = x^2 + y^2 + 5(x-y)$$ where $g(x) = x^2$ and $h(y) = y^2$ are also functions of $x$ and $y$, respectively. ...
0
votes
0answers
25 views

zero mean curvature and a variational problem

given the lagrangian $$ J(u)= \int_{V} \sqrt{1+|gra(u)|^{2})} $$ with the constraint $ \int_{V}udx =1 $ why is the volumen constraint there ? for teh case of R^{3} i know this must satisfy the ...
0
votes
2answers
350 views

Moment of inertia about center of mass of a curve that is the arc of a circle.

Let $(x(s),y(s))$ be a smooth 2-d plane curve which is an arc of a circle of a certain radius $r$. Assume it is represented by an inelastic string $S$ of finite length, lying in a 2-d plane. Let there ...
4
votes
1answer
321 views

Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
4
votes
1answer
235 views

Step in derivation of Euler-Lagrange equations of motion

From http://www.mathpages.com/home/kmath523/kmath523.htm Variations in $x,y,z$ and $X$ at constant $t$ are independent of $t$ (since each of these variables is strictly a function of $t$), so we ...
4
votes
2answers
91 views

Laplace Boundary Problem

Consider a boundary given by vertices $(0,a)$, $(0,0)$ and $(1,0)$ (an 'L' shaped boundary). The problem is to find the equation that passes between the endpoints $(0,a)$ $(1,0)$ of minimum length ...
4
votes
3answers
341 views

Treacherous Euler-Lagrange equation

If I have an Euler-Lagrange equation: $(y')^2 = 2 (1-\cos(y))$ where $y$ is a function of $x$ subjected to boundary conditions $y(x) \to 0$ as $x \to -\infty$ and $y(x) \to 2\pi$ as $x \to ...
3
votes
1answer
96 views

Direct Method of Variationcalculation

Consider the Bolza problem $$ \inf\left\{F(u)=\int\limits_0^1 ((1-u'^2)^2+u^2)\, dx, u\in W^{1,4}(0,1), u(0)=0=u(1)\right\}. $$ Show that $\inf F(u)=0$, but that it does not exist an $u_0$ with ...
3
votes
0answers
217 views

Calculus of Variations: Contains an integral of my goal function

OK, using the calculus of variations, I want to find a function $f$ that maximizes: $$J = \int_0^n L(x,f(x)) \text{d}x$$. But $L$ has multiple integrals in it (for example, $\displaystyle \int_0^n y ...
2
votes
1answer
35 views

How to calculate this functional derivative?

How can I calculate the functional derivative of this functional? $$F[x](t) = \int_{0}^{t}x(t_1)a(t_1)\left \{ \int_{0}^{t_1}x(t_2)b(t_2) \,dt_2\right \} dt_1 .$$ Where $a(t)$ and $b(t)$ are real ...
2
votes
2answers
101 views

Function extremal - calculus of variations

Find a curve passing through (1,2) and (2,4) that is an extremal of the function: $J(x,y')=\int_1^2 xy'(x)+(y'(x))^2dx$ I don't know what methods to use at all.
2
votes
1answer
33 views

Minimising line intergral over a scalar field part 1

I'm self teaching myself calculus of variations, and decided to solve a problem to practice what I learned. Say we want to find a path $y=y(x)$ in the scalar field $S(x,y)$. Therefor we wish to ...
2
votes
2answers
111 views

Natural growth conditions and weak solutions for inhomogenous systems.

Let $\Omega\subset \mathbb{R}^n, \ n\geq 2$ be a bounded Lipschitz domain. Consider the following inhomogenous system (in divergence form) subject to zero Dirichlet boundary conditions: ...
2
votes
1answer
98 views

Calculus of Variations: Mul-variable-mul-function

My question is: How to find the necessary condition for minimizing/maximizing the functional $$J(f,g)=\int\int_{R}F(x,y,f(x),g(y))dxdy,~~~~~~~~~~(1)$$ where we have two functions $f(x)$ which only ...
2
votes
1answer
72 views

Interpretation of the variational principle for the Ritz approximation, solid Mechanics

Below $U$ and $V$ are recpectively the internal and external energy components of a given structural element: $$U+V=W$$ Expressing $U$ in terms of the strains $\varepsilon$ and the material ...
2
votes
1answer
86 views

Infimum length of curves

Let the unit disc $\{(x,y): r^2=x^2+y^2<1\}\subset\mathbb R^2$ be equipped with the Riemannian metric $dx^2 +dy^2\over 1-(x^2+y^2)$. Why does it follow that the shortest/infimum length of curves ...
1
vote
0answers
32 views

Minimising line integral over a scalar field part 2

This is a continuation of this question whose general point is summarised below Say we want to find a path $y=y(x)$ in the scalar field $S(x,y)$ that finds the extrema of of its line integral. ...
1
vote
0answers
26 views

variational problem: obtain the lagrangian from the PDE equations of motion [duplicate]

given the 2 PDE $$ \Delta u-au_{tt}+u_{t}=0$$ and $$ \Delta u + Du*Df=0 $$ here $ \Delta u $ is the Laplacian $ Du= grau$ and * means scalar product $u_{t} = \frac{\partial u}{\partial t}$ my ...
1
vote
1answer
156 views

How to check if a function is minimum to functional?

Given $\int_0^1(y')^3dx$ functional and $y(0) = 0 ,y(1)=1$ conditions. Using Euler–Lagrange equation I have got $y(x)=x$. So $y$ is a stationary point of the functional. How to check if it is the ...
1
vote
1answer
234 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 ...
1
vote
0answers
161 views

Legendre test re First Variation

The Legendre test (as mentioned in An Introduction to the Calculus of Variations by Charles Fox, requires that the sign of $\partial^2 F \over\partial y'^2$ is constant throughout the range of ...
0
votes
1answer
40 views

Using bordered Hessian matrix to determine non-degeneracy and type of constrained extremum

I have the following problem: $\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}\def\g{g(x_1,x_2,x_3)}\def\l{\lambda}\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}$ Find the ...
0
votes
1answer
69 views

Minimizing cost function (Eikonal)

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $S \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=S$ and ...
0
votes
0answers
55 views

minimal surface from a variational problem

Given the Lagrangian $$ J(u)= \int_{V} \sqrt{1+|\operatorname{grad}(u)|^{2})} $$ with the constraint $ \int_{V}udx =1 $ (1) Why is the volume constraint there ? (2) For the case of ...
0
votes
1answer
95 views

Fundamental lemma of calculus of variations, gradients

Let $D \subset \mathbb{R}^d$ be a smooth bounded domain. Let $C_c^\infty(D)$ denote smooth and compactly supported functions on $D$. Let $f \in [C_c^\infty(D)]^d$ be a smooth, compactly supported ...
0
votes
2answers
126 views

Calculus of variations, what is a functional

I'm reading a bit about the calculus of variations, and I've encountered this bit: Suppose the given function $F(.,.,.)$ is twice continuously differentiable with respect to all of its arguments. ...
0
votes
1answer
104 views

How to check if stationary point is extremum?

In this question the solution of Euler–Lagrange equation is $y=x$ function. $L = (y')^3$ so $L''_{y'} = 6y'$ and is positive when $y=x$. But from the answer of Emanuele Paolini follows that it is ...
0
votes
1answer
400 views

When the Euler Lagrange equation simplifies to zero

My question is rather simple, and I'm sure I'm missing something simple, and yet... I'm trying to calculate the Euler Lagrange Equations for the example function here: ...
-1
votes
1answer
231 views

Show that minimum exists (direct method)

Consider $$ F(v):=\int\limits_0^1\lvert v'(x)\rvert^2\, dx, $$ with $$ \left\{v\in H^{1,2}(0,1), v(0)=0=v(1)\right\}. $$ Show that a minimum exists and use the direct method. When I am right ...