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

learn more… | top users | synonyms

1
vote
1answer
50 views

Confusion with Euler-Lagrange Derivation

This is mostly a re-hash of this thread, but it did not receive an adequate answer. In the derivation that I am reading, there is one step that is not justified. Perhaps obvious, but it is not clear ...
0
votes
1answer
98 views

isoperimetric problem:how to solve the given question

Determine $y(x)$ for which $\int_{0}^{1} x^{2} + y^{'2}dx$ is stationary, subject to $\int_{0}^{1}y^2=2$, $y(0) = 0$, $ y(1) = 0$. how to solve it? I tried it: $f=x^{2} + y^{'2}$ and $g=y^2$ then $...
0
votes
1answer
361 views

extremal problem-how to check istrong minima,maxima condition

The functional $I[y(x)]=\int_{0}^{2}(xy^{'}+y^{'2})dx$,y(0)=1,y(2)=0 possess a.strong minima b.strong maxima c.strong maxima but not weak minima d.weak maxima but not strong minima How do we show ...
0
votes
0answers
1k views

Rayleigh-Ritz-method-how to solve the given problem

how to solve this: An approximate solution of the problem $y^"-y^{'}+4x\epsilon^x =0$, $y^{'}(0)-y(0)=1$,$y^{'}(1)+y(1)=-\epsilon$ is: here we have to calculate the value of y(x)? what i did is: ...
2
votes
0answers
94 views

green function, functional derivative

I am trying to find ${\delta F}/{\delta u}$ for the functional: $F[u]=\int u(x)\int G(x,y)u(y)dy dx $ G is green function for laplace operator. is there Euler-Lagrange version for double intrgral? ...
0
votes
1answer
38 views

Maximization of ratio of two functionals

I am trying to find a function prescribed in polar coordinates $r = f(\theta)$ that maximizes the following quantity $$\frac{\int_0^{2\pi}r^3\cos\theta\, d\theta}{\int_0^{2\pi}r^4\, d\theta}$$ ...
1
vote
1answer
101 views

Least surface of volume with constraints

We know that in 2D/3D the shape with the least surface of a certain volume is a circle/sphere (e.g. soap bubbles). Now Imagine we have a flat surface (tabletop) that can be used as part of the surface ...
2
votes
1answer
58 views

Maximizing the uniformity of density function subject to moment constraints

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
0
votes
1answer
1k 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 ...
1
vote
1answer
46 views

Local minimum of the function:

Find the local minimum of the function: $$\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}$$ $$\f=\1^2-2\1\2+2\2^2+\3^2 \text{ in } \mathbb{R}^3$$ $\n\f=(2\1-2\2,-2\1+4\2,2\3) =...
0
votes
1answer
64 views

Energy functional in Geodesic Active Contours

I have read some papers about Geometric active contours of the author C.Gout and Le Guyader [1] Segmentation under geometrical conditions using geodesic active contours and interpolation using level ...
1
vote
1answer
502 views

Minimize Energy in Image processing - Geodesic active contours

I've read some papers in Geodesic active contours (Image processing), which use the minimization of an Energy, consist of Internal Energy and External energy, for example, in the paper of Kass (Snake: ...
1
vote
1answer
48 views

Prove that the maximizing point configuration on the unit circle for a Vandermonde like functional is a picket fence

For $\lambda_i \in S^1 \subset \mathbb{C}$, consider the functional $H(\{\lambda_1, \ldots, \lambda_n\}):= \sum_{j < k} | \lambda_j - \lambda_k | $. I want to show that $H$ is globally maximized by ...
0
votes
1answer
102 views

Calculus of variations: big-O notation?

I have a formula in my text-book $$y(x+C) = y(x) + \frac{dy}{dx}C + O(C^2)$$ Can someone explain this formula?
10
votes
1answer
172 views

Shallow tent like soap film

A soap film circle in $x-y$ plane with center at origin can be carefully pricked with a blunt soapy pin at center and drawn out a little bit on $z$-axis forming a surface of revolution somewhat like a ...
7
votes
2answers
224 views

Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$ Called the "energy" of $f$? I am ...
2
votes
0answers
82 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 $C^1$ ...
6
votes
2answers
692 views

The relationship between two definitions of star-shaped domain

There are two definitions of star-shaped domain. One is given in wikipedia as follows. Def1: A set $S$ in the Euclidean space $\mathbb{R}^n$ is called a star domain (or star-convex set, star-shaped ...
4
votes
1answer
247 views

Erroneous calculus of variations reference in V. I. Arnold's Mathematical Methods of Classical Mechanics?

The beginning of section 12, Calculus of variations (chapter 3, Variational principles) in V. I. Arnold's Mathematical Methods of Classical Mechanics (2nd edition, p. 55) reads: For what follows, ...
2
votes
1answer
63 views

Continuity of a functional with respect to two different norms

Let $J$ be a functional defined on $E = C^1[a,b]$ by $$J(y) = \int_a^b \sqrt{1 + (y^{\prime}(x))^2} \, dx.$$ Define the following two norms on $E$: $$\|y\|_{\infty} = \max_{a\leq x\leq b} |y(x)|$$ $$...
1
vote
1answer
146 views

Maximizing a particular integral / functional

I have a (probably simple) question whose answer seems obvious but I cannot prove it. It relates to the calculus of variations. Let scalar $A = \Re[\int_a^bB(x)C(x)dx$], where $B$ and $C$ map $\...
2
votes
1answer
92 views

Gradient of norm of embedding

Let $\varphi:(M,g,\nabla)\to\mathbb{R}^n$ be a smooth embedding of a convex hypersurface. I want to explicitly calculate $$\langle \varphi,\varphi_{\ast}(\nabla\|\varphi\|^2)\rangle.$$ In particular, ...
1
vote
0answers
69 views

Application of a general “Weierstrass theorem”

http://books.google.at/books?id=9OSrV73a40gC&pg=PA45&lpg=PA45 gives a general Weierstrass theorem. Are there notable applications of this theorem, say in the calculus of variations? (I could ...
1
vote
0answers
65 views

How to prove $\gamma$ is continuous?

In the paper A remark on least energy solutions in $\mathbb{R}^N$, page 2407, it said, if $u_0\in H^1(\mathbb{R}^2)$, set $\gamma(t)=t^{-1/4}u_0(x/t)$. Then $\gamma(t)$ is a continuous path in $L^2(\...
0
votes
0answers
67 views

Newton's method for the brachistochrone

Consider the potential $V(x,y)=-y$ and a particle at rest in the beginning of the coordinate system. We are going to examine the brachistochrone - the smooth curve of fastest descent. Assume we are ...
3
votes
1answer
200 views

Variational Principles: Lagrange Multipliers

I am trying to minimize the functional $$I[\textbf{x}] = \int ||\dot{\textbf{x}}||^2 dt$$ subject to the constraint $\textbf{x}(t) \in \{\textbf{s} \in \mathbb{R}^3 : ||\textbf{s}|| = 1 \}$. The ...
1
vote
1answer
166 views

Does calculus of variations have a close connection to Feynman's ''differentiation under the integral sign''?

Most of the calculus I've studied seems separate math problems in to "derivative" or differential applications and integral applications. The one exception seems to be "calculus of variations," which ...
3
votes
0answers
281 views

Question on Moment of inertia about center of mass of a smooth plane curve.

This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question. Consider two smooth plane curves $C \equiv ...
3
votes
0answers
148 views

Functional Extremum

Let a functional $H[\phi]$ of a map $\phi\in\mathbb{R}^{\mathbb{R}^4}$ be given by: $$ H(x^0) = \int_{\mathbb{R}^3} \left[\frac{1}{2}\sum_{j=0}^{3}\left(\partial_{x^j}\phi\right)^2-\frac{1}{2}\mu^2\...
0
votes
0answers
162 views

Finding maximum rate of change of total derivatives

consider $PV =nRT , P,V,T =$ pressure , volume , temperature respectively. $nR =$ constant let $n=R=1$ differentiate with respect to $t$ (time) $dP/dt = ∂P/∂T * dT/dt + ∂P/∂V * dV/dt.................
1
vote
0answers
30 views

Why $\widetilde m = argmin_{m \in \mathcal P(X)} E[m]$ implies $\widetilde m (\arg \min \frac{\delta E}{\delta m}[\widetilde m]) =1$?

Consider $E: \mathcal P (X) \rightarrow \mathbb R \cup \{ \infty \}$ a functional (with a convex and dense domain, $E< +\infty$) over $\mathcal P(X)$ the set of probability measures of a metric ...
1
vote
1answer
116 views

Variation of the determinant of a Jacobian

I am following a derivation in a Calculus of Variation problem. After introducing a one-parameter family of one-to-one mappings from $R^{2}$ to itself, $$z({x},\epsilon)$$, $x = (x_1,x_2)$, such that $...
1
vote
0answers
70 views

Integral invariant under parametrization

Consider a continuous function $F(z,p)\colon \Omega\subset\mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ and the functional $$ \mathcal{F}(u)=\int_{a}^{b}{F(u(t),u'(t))\,dt}. $$ Prove that $\mathcal{...
0
votes
1answer
59 views

The curve of shortest length bounding given area

Is formula #7 in this MIT OCW course incorrect? I think it should be $$f(x)=(\sqrt{1-(mx-c)^2}+d)/m$$ Also, presumably this answer to a very similar problem is also wrong. Because this is not the ...
4
votes
1answer
130 views

Division of plane into equal area regions

We divide a plane ($\mathbb{R}^2$) into infinite number of regions each of area equal $1$. We can use only (one-dimensional) curves which may meet at points. Fix a point $p$ on a plane and consider ...
2
votes
1answer
105 views

Spherical rearrangement

Let $u\colon\Omega\subset\mathbb{R}^N\to\mathbb{R}$ be a non negative measurable function, and $\Omega$ open and bounded. Consider $u^*$ the spherical rearrangement $$ u^*(x)=\sup\{t\geq0 : \mu\{x: ...
3
votes
0answers
150 views

Regularity of a Weak Solution to Fokker-Planck Equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t =...
6
votes
2answers
445 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 refernece to this paper [Olivier Chapelle,...
2
votes
0answers
49 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
0
votes
0answers
1k views

Geodesics of a cone

To find Geodesics on a cone I used the cylindial coordinates $x=rcos\theta$ $y=rsin\theta$ $z=z$ Is this parameterization correct.How can I know how to parameterize? Then arc length $ds^2=r^2 ...
1
vote
1answer
140 views

minimal surface of revolution integrating Euler Lagrange result

In finding minimal surface of revolution after applying to euler lagrange equation $d u \over dx \sqrt{1+(u')^2}$$=0$. Then $ u \over \sqrt{1+(u')^2}$$=constant$. Then solving $\int{ c \over \sqrt{u^...
2
votes
1answer
106 views

Proof of Euler Lagrange equation in calculus of variation

I am learning calculus of variations from the article page 9 There it says $J[u]=\int_a^bL(x,u,u')\, dx$ and $u'$ is represented by $p$. $$\langle\nabla J[u],v\rangle={dJ[u+tv] \over dt} \text{ ...
2
votes
2answers
104 views

Doubt on calculus of variation

In wiki http://en.wikipedia.org/wiki/Calculus_of_variations#Example, the first example of calculus of variation is the minimize distance between 2 points. In my understanding, value of functional $J$ ...
0
votes
1answer
24 views

Lebesgue integral question using du Boise-Reymond lemma

This question was inspired a previous question of mine. If we are given that $\Omega \subset \mathbb{R}^{n}$ is open and bounded and $$\int_{\Omega}fv dx = 0$$ where $f \in C(\Omega)$ and $v \in C^{\...
1
vote
1answer
120 views

Variational Methods, why KL divergence is the difference between true distribution and approximating distribution.

Likelihood = $L(\textbf{w}) = P(V\mid \textbf{w})$. $$\ln P(V\mid \textbf{w}) = \ln \sum_H P(H,V\mid \textbf{w})$$ $$= \ln \sum_H Q(H\mid V)\frac{P(H,V\mid \textbf{w})}{Q(H\mid V)}$$ $$\geq \ell(Q,\...
0
votes
0answers
398 views

Integration of the Euler-Lagrange equation with explicit dependence on x

In the variational calculus, if the Lagrangian $L[y(x),y'(x)]$ is not an explicit function of $x$, the Euler-Lagrange equation takes on the following form: $$\frac{d}{dx}\left(L-y'\frac{\partial L}{\...
0
votes
1answer
153 views

Integrating the Euler-Lagrange equation

Let us have a Lagrangian $L(y,y') = f(y)\sqrt{1+y'^2}$, where $y=y(x)$. The corresponding Euler-Lagrange equation is $$\frac{f'}{\sqrt{1+y'^2}} - f\frac{y''}{(1+y'^2)^{3/2}}=0$$ This expression should ...
0
votes
1answer
113 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 $\...
2
votes
2answers
144 views

Equivalent norm in Sobolev space

Let $\rho\in H^{1}(0,\pi)$ be a function, and consider the functional $$ I(\rho)=\bigg(\int_{0}^{\pi}{\sqrt{\rho^2(t)+\dot\rho^2(t)}\,dt}\bigg)^2. $$ I'm asking if it is equivalent to the norm $$ \...
1
vote
0answers
32 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...