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

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25 views

Is every set a subset of a vector space?

I was taught that a functional is a map from a subset (not subspace) of a vector space into the reals, $F: D\subset V \to \mathbb{R}$. I know there are other definitions, but is there any reason to ...
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2answers
19 views

Euler-Lagrange equation: Differentiation with respect to x

I got stuck in my lecture notes after a supposed differentiation of the Euler-Lagrange equation: $$\dfrac{\partial f}{\partial y}-\dfrac{d}{dx} \left( \dfrac{\partial f}{\partial y'}\right) = 0$$ ...
0
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2answers
33 views

Does the square root function change the variations of a function?

If I have $$f(x) =\sqrt{g(x)}$$ Will the variations of $f(x)$ be the same than the variations of $g(x)$ ?
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0answers
17 views

Formulation of variational problem

I've been stuck on this problem for a long while. I'd be grateful for your help. The problem A variational problem is given by the functional: $$I[y]=\int_a^b{F(x,y,y',y'')dx}+[y'(a)]^2$$ Where $F$ ...
0
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1answer
12 views

Strong and weak extrema

I am confused about the "strength" of the two definitions. The definitions I use are the following: Let $y$ be a function defined on the set $M$. Neighborhood (0. order) of the function $y$ is the ...
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1answer
14 views

Finding the extremal curve satisfying a variable endpoint

Below is a question I am trying to solve, and my attempt. $\int_0^T \frac{\dot{x}^2}{t^3} \mathrm{d} t$, where $x(0)=1 $ and $x(T)$ lies on the curve Transversal condition: $$f-(\dot{c} ...
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0answers
11 views

Extremal for variable end point problem. Optimization

I feel like this is not correct, could anyone verify? $\int_0^T (\dot{x}^2 + 2x\dot{x} + x^2) \mathrm{d} t,\;\; x(0)=1 \text{ and for } T\gt 0, x(T) \text{ lies on a given curve } x = c(t) = 3$ ...
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1answer
12 views

Proving isoperimetric inequality using calculus of variations

I was trying to prove isoperimetric inequality, which states that for any simple closed curve of length $l$, the area that it encloses is $\leq \frac{l^2}{4\pi}$. I wanted to use calculus of ...
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0answers
21 views

Denominator of a function

I have a function $S(x,y)$ which satisfies the following PDE $$\frac{\partial S(x,y)}{\partial y}=-H\left(x,\frac{\partial S(x,y)}{\partial x}\right)$$ where the known function ...
2
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0answers
23 views

Given a point $A$, describe those points to which a catenary cannot be drawn from $A$.

Background An elementary problem in the calculus of variations shows that among all curves joining two points $A$, and $B$ in the first quadrant, the one which generates the surface of minimum area ...
4
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1answer
266 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 ...
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1answer
48 views

Why does the arc-length formula have form $\int_a^b\left|\left|\frac{d\vec{f}(t)}{dt}\right|\right|_2dt$ for C1 curves?

This discussion focuses on $\mathcal{C}^1$ curve on $\mathbb{R}^n$. But feel free to talk about the case where we only have a continuous curve or the scenario with a manifold with a metric in general. ...
1
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1answer
18 views

variation of functional

A little confused about finding the variation of the functional J = $\int_{t0}^{tf}(e^{x_1(t)+x_2(t)})dt$ When I perturb and find the increment, I get: $\Delta J = \int_{t0}^{tf} (e^{x_1(t) + ...
6
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2answers
670 views

Shortest path on hyperboloid

On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the ...
0
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1answer
23 views

Prove that solution of a variational problem exists

Let $X$ is a Hilbert Spaces We define two operators $$a:X\times X\rightarrow\mathbb R$$ and $$b:X\rightarrow\mathbb R$$ where $a$ is a symmetric, bounded, strongly positive operator, and $b$ is a ...
0
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1answer
42 views

Are all definite integrals considered functionals?

In my Optimization class, we are messing around with some Calculus of Variations in an effort to find functions which minimize functionals. In these cases, the spaces we're working with are spaces ...
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2answers
32 views

A derivation of Euler–Lagrange equations with a general metric

$g_{\mu \nu}(x)$ is a metric of a space and $X^{\mu}(\lambda)$ a curve with $\lambda$ a parameter that varies monotonically along the curve: $$0 = \delta \int d\lambda L = \int d\lambda \delta L = \\ ...
2
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1answer
33 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 ...
0
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1answer
32 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 ...
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0answers
32 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: ...
0
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1answer
19 views

Finding extremal of a fixed end point problem. Optimisation

I want to find the extremal of the fix-end point problem $\int_1^2 \frac{\dot{x}^2}{t^3}$ with $x(1)=2,x(2)=17$ First I check the euler-lagrange equation is equal to $0$. We have: ...
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0answers
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Local minima: Sufficient conditions. Comparison of Calculus verses Calculus of Variations

My lecturer has written: Let $y=x^*+\epsilon \eta$ where $x^*,\eta,y\in \mathbb{R}^2$ $0\leq f(y) - f(x^*) = \epsilon V_1 + \epsilon^2 V_2 + O(\epsilon^3)$ $V_1 = \nabla f(x^*)\eta$ $V_2 = ...
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0answers
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Constrained optimization minima and maxima and non-degeneracy answer check

Find the critical points of $$\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\f{f(\1,\2,\3)}\def\l{\lambda}$$ $$\f=\1\2+\2\3+\3\1$$ subject to constraint $\1+\2+\3=1$ First I will construct the Lagrangian: $$L ...
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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 ...
1
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1answer
23 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 ...
1
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1answer
21 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 ...
0
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0answers
6 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 ...
3
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3answers
384 views

Procedure for Gâteaux Derivative with functionals

Not after an answer, just the method/procedure as I'm stumped... We have the functionals: $$ T[y] = \int_2^3 \left( 3\left| \frac{dy}{dx}\right|^2 - 8y \right)dx $$ $$ S[y] = \cosh(T[y]) $$ Now, to ...
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0answers
28 views

how to find the optimal function with lagged cost? (calculus of variations)

I need to find the function $b( )$ that maximizes this guy ($c()$ and $\beta()$ are functions too, and $c()$ is convex): $$\int_{0}^{T} \! e^{-\delta v}\beta(v) \left[\int_{0}^{v} b(s) \; ds - ...
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0answers
48 views

Open problems in variational analysis/PDEs

I wasn't sure whether this question was more appropriate for StackExchange or Overflow, but in any case I would really appreciate it if any active researchers in the field responded. I'm a PhD ...
1
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0answers
18 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? ...
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0answers
15 views

Random variable variance

I have the model yi=β1+β2Xi+ui where ui∼iid N(0,σ2). I estimate β1 and β2 by drawing a straight line between the first (x1,y1) and last dot (xn,yn). So, β̂ 2 will be the slope of this straight line. ...
0
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1answer
24 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 $$\frac{\int_0^{2\pi}r^3\cos\theta\, d\theta}{\int_0^{2\pi}r^4\, d\theta}$$ ...
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0answers
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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 ...
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0answers
22 views

Minimum travel time of a fuel-less train: brachistochrone problem

It is suggested that a rail network should include a frictionless tunnel where fuel-less trains run under gravity. The trains are released from rest at the point of departure and run freely until ...
0
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1answer
39 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
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1answer
63 views

Pontryagin principle: does the abnormal multiplier define a minimum

The Pontryagin principle PM provides the necessary condition for a local minimum of the functional $ J(u)=\int L(x(t),u(t))dt \\$ subject to: $\dot x = f(x(t),u(t)) \ \ \ \ x(t0)=x0, \ \ ...
1
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1answer
30 views

On the controllability function (minimising a functional)

Consider a system of ODEs $$\dot{x}(t)=f(x(t))+g(x(t))u(t),$$ where $f:\mathbb{R}^n\to\mathbb{R}^n$ and $g:\mathbb{R}^n\to\mathbb{R}^{n\times m}$ are smooth. Let $L:\mathbb{R}^n\times ...
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1answer
70 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 ...
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1answer
40 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) ...
5
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1answer
396 views

Euler-Lagrange Equation example

I have been working on solving Euler-Lagrange Equation problems in differential equations, specifically in Calculus of Variations, but this one example has me stuck. I am probably making mistakes in ...
1
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1answer
113 views

Minimum calculus of variation

Hi I am looking for a criterion that is sufficient to prove that a solution to a functional depending on two functions y(t) and x(t) is an extremum. it is about the following functional$$ \int_0^b ...
0
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1answer
16 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 ...
0
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1answer
51 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: ...
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0answers
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Minimum surface attaching two parallel (non-planar) circles.

While studying for a comprehensive exam, I found this old problem: Consider two parallel coaxial wire circles, not in the same plane, to be connected by a surface of minimum area that is ...
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1answer
33 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
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1answer
34 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?
8
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1answer
68 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 ...
5
votes
2answers
89 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 ...
3
votes
1answer
64 views

Motivating the Legendre Transform Mathematically

If I begin with a functional of the form $$J[y] = \smallint_a^b f(x,y,y')dx$$ & find it's Euler-Lagrange equations $$\tfrac{\partial f}{\partial y} - \tfrac{d}{dx}\tfrac{\partial f}{\partial ...