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

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3
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60 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 ...
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0answers
25 views

Minimize $J(F)$ = $\int_0^1F(x)dx$ subject to given conditions [on hold]

Minimize $J(F)$ = $\int_0^1F(x)dx$ subject to $F(0)$ = $0$, $F(1)$ = $c \in (0,1)$, $F$ continuously differentiable, and $F'(x) \le G(x)$ for all $x \in [0,1]$. $c$ and $G$ are stipulated. $F$ can ...
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0answers
13 views

Missing explanation in this paper of Masmoudi.

In this paper, on page 4, beginning in the line above 3.8, the authors begin a discussion of a given variational problem. I follow their argument until they begin the line of reasoning that begins ...
3
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1answer
123 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
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1answer
30 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
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0answers
47 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 ...
1
vote
1answer
63 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, ...
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0answers
46 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 ...
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0answers
28 views

Problem calculating the average power of a vector?

I am calculating the average power of a vector. I would like to compare the final expression with the simulation. However, they are not equal. Please help me to point out which steps are wrong. Thank ...
0
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0answers
56 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 ...
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0answers
35 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 ...
2
votes
1answer
49 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 ...
0
votes
1answer
53 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
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0answers
127 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
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0answers
35 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} ...
1
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0answers
35 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 * ...
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0answers
3 views

Perturbation of the boundary of a strictly stable minimal surface

Let $\Sigma \subseteq \mathbb{R}^3$ be a minimal surface with boundary $\Gamma$. Now let us assume that $\Sigma$ is strictly stable, that is, $\lambda_1(\Sigma,L) >0$, where $L$ is the stability ...
1
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0answers
27 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
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1answer
55 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 ...
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0answers
19 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 ...
0
votes
1answer
39 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
68 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 ...
1
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1answer
18 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: ...
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0answers
64 views

Partial Derivative of Integration

Suppose that I have some set of weight functions, $W = \{w_1(i,j), w_2(i,j),..., w_k(i,j)\}$, where each weight function is a Taylor polynomial in $\mathbb{R}^2$ with constants $c_{kn}$ where $n$ is ...
3
votes
0answers
41 views

Regularity of a Weak Solution

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 ...
4
votes
2answers
233 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, ...
0
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0answers
10 views

weak local minima sufficient conditions

I have the functional $F(u)=\int^b_a f(x,u(x),u'(x)) dx$ with $u \in C^1([a,b])$ such that $u(a)=\alpha,u(b)=\beta$ and I'm trying to prove the following fact if $\delta F(u_0,v)=0$ and $\exists ...
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0answers
20 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 ...
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0answers
26 views

Path followed by a light ray-calculus of variation

Find the path followed by a light ray if the index of refraction (in polar coordinates ) is proportional to $r^{-2}$. In the given answer in the book it says we want to make stationary $\int nds$ or ...
0
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0answers
27 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
28 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 ...
2
votes
1answer
43 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
59 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
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1answer
17 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 ...
1
vote
1answer
50 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 ...
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0answers
45 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 ...
0
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1answer
49 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
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1answer
61 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
52 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 $$ ...
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0answers
26 views

Minimization Problem for Winding number

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

Quadratic Minimization

Consider a functional $I\colon H \to R$ on $H$ Banach space, sufficiently regular. Is in generally true that $$ \inf_{\rho \in H}{I^2(\rho)}=\Big(\inf_{\rho \in H}{I(\rho)} \Big)^2 \quad ? $$ If ...
1
vote
1answer
44 views

Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
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0answers
25 views

Isoperimetric Euler Equation When Objective Does Not Depend on Derivitive

I'm solving the optimization problem (simplification of my actual problem) $$ \max \int_{0}^{1} b(t) \left(t - \frac{b(t)}{2}\right)dt \\ \text{st} \\ \int_0^1 \frac{b^2(t)}{2}dt = B $$ Without the ...
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0answers
39 views

Brachistochrone Problem to find out the path by which a bead travels in least time

The question is to find the shape of the curve down whcih a bead sliding from rest and accelerated by gravity will slip(without friction) from one point to another in the least time. So I proceeded ...
3
votes
1answer
32 views

how to use variational principle to find the best value for parameter $\lambda$?

I need to minimize the following integral by varying parameter $\lambda$: $$\int_0^\infty(f(x)-g(x,\lambda))^2dx$$ The functions $f(x)$ and $g(x,\lambda)$ are known and they satisfy ...
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0answers
18 views

Double standards on recognizing expression as functional

The Calculus of Variations starts with a definition of functional Such an expression, the argument of which is a function, is called a functional. Particularly, they say that $J = ...
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0answers
12 views

Minimization Problem and Winding number

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

Eikonal Equation [duplicate]

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $A \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=A$ and ...
4
votes
1answer
62 views

Shortest distance between two points calculus of variation

Ok this problem might be trivial but when solving it using calculus of variations it's not so stupid. Suppose we have a fixed boundary condition $f(a) = f(b) = 0$ and we want to find the shortest ...
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0answers
37 views

First variation of convolution of two nonlinear functions, how to reexpress $\left[x \delta x * x^2 \right]$?

A new variational principle is presented in this paper: Mixed Convolved Action When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach: Set ...