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

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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 ...
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58 views

Extremal of functional $ I\left[ y(x) \right] = \int_{0}^{\frac{\pi}{2}} {\left((y')^2 - y^2 + 2xy\right)dy} $

I have the following functional: $$ I\left[ y(x) \right] = \int_{0}^{\frac{\pi}{2}} {\left((y')^2 - y^2 + 2xy\right)dy} $$ subject to boundary conditions: $$ \begin{align} y(0) &= 0 \\ ...
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0answers
20 views

Variational Calculus on unbounded domains reference

I've been studying Variational Calculs from Differential Equations and Variational Calculus by L. Elsgoltz. Everything is ok, but the whole theory is developed bounded intervals of the real numbers, ...
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2answers
162 views

Maximum area under a curve by calculus of variations

I am asked to find the function that has the maximal area for a given length L when x runs from -a to a. I calculated the integral to be varied as follows: $$ \int_{-a}^{a}\ y + \lambda \sqrt{1 + ...
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1answer
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Calculus of variations, minimizing $\int_0^\pi y' ^2 - ky^2 dx$. Please check my work.

I have to minimize the functional $$J[y] =\int_0^\pi y' ^2 - ky^2 dx$$ subject to $y(0)=y(\pi)=0$. The parameter $k$ is positive. Writing down the Euler-Lagrange equation, I have: $$y'' +ky =0,$$ ...
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0answers
81 views

Lagrange multipliers in the context of the calculus of variations

Suppose we wanted to extremise the function (of a finite number of variables) $f$ subject to the constraint $g = 0$. The Lagrange multiplier approach is to extremise without constraint the function ...
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52 views

stationary function of an integral

Find the stationary function $y=y(x)$ of the integral $\int_o^4[xy'-(y')^2]dx$ satisfying the conditions $y(0)=0$ and $y(4)=3$. I don't know what a stationary function is. Can you anyone suggest me ...
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1answer
53 views

Palais-Smale conditions of functional involving noncoercive differential operator

I am working on mountain-pass like theorems for the problems $$ - u_{xx} - a u = \pm |u|u+|u|^2u , \ x \in (a,b), \quad u(a)=u(b) = 0$$ where $a \in L^\infty((a,b))$ is positive (I take the one ...
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0answers
61 views

Test functions on a compact interval

Consider a functional $E:C([0,1]) \rightarrow \mathbb{R}$ of the form $$E(g) = \int_0^1 g(s)ds$$ In dealing with such functionals one often needs test functions. If one talks about the space ...
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27 views

Position-dependent Lagrange multipliers for functionals

I'm trying to extremise the functional $$ S = \int L(Q_i) \, dx\,dy $$ Where the $Q_i$ are functions of $x$ and $y$, subject to the constraint $$ \vec{\nabla} \cdot \vec{Q} = A(x,y) \,.$$ My ...
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66 views

Single Variable Calculus of Variations Question

Problem: Minimize $I(f)$ subject to the constraint $J(f)\leq 0$, where $$I(f)=\int_{x_1}^{x_2}\frac{dx}{f(x)}\tag{$f:[x_1,x_2]\to \mathbb{R}_{\geq 0}$}$$ ...
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73 views

Does the implicit function theorem imply Peano existence theorem

In The implicit function theorem written by Krantz & Parks, it's said that the implicit function theorem implies the following existence theorem of ODE: Theorem 4.1.1 If $F(t,x)$, ...
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1answer
63 views

Is it possible to solve or approximate this second order nonlinear system of differential equations.?

Given initial values $d[0]$ and $k[0]$, I would like to solve for the initial rate of change, $\dot d[0]$, and compare this value against some data. I have the following profit function, which I ...
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1answer
52 views

Convergence of integral means of the gradient of a Sobolev function

Let $B_R(x_0)\subset\mathbb{R}^n$ with $R<1$ for $n\geq3$ and suppose $u\in H^1\big(B_R(x_0);\mathbb{R}^N\big)\cap L^{\infty}\big(B_R(x_0)\big)$. Define, \begin{equation} \phi(R)\equiv ...
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1answer
87 views

Calculus of variations: Euler equation

Can someone please give me a hint on this problem? I don't know how to write Euler equation for this case: Find the extremal for the functional $$ J(x)=\int_1^{t_f} \dot{x}^2(t)t^3\,dt $$ ...
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417 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 ...
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1answer
45 views

Total variation for functions, Meaning of supremum as used here?

On Wikipedia article, here: http://en.wikipedia.org/wiki/Total_variation, on definition 1.1 there says, "where the supremum runs over the set of all partitions ..." AFAIK supremum is defined for a ...
2
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1answer
69 views

Extremals of functional

What i have done is to consider Euler Poisson condition. $$F_y-\frac{d}{dx}F_{y '}=0$$ Here $$F_y=\frac{2y}{(y')^2}$$ $$F_{y'}=-\frac{2(1+y^2)}{(y')^3}$$ ...
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0answers
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Calculus of variations: Isoperimetric and holonomic constraints.

A functional $$J(y)=\int_a^b F\left(x,y(x)\right)dx, \tag{1}$$ subject to an isoperimetric constraint $$\int_a^b K(x,y)dx=l, \tag{2}$$ and a holonomic constraint $$g(x,y)=0. \tag{3}$$ Most ...
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59 views

Calculus of variations: the inside function has an integral

It is known that if the functional $$J=\int_a^b L(x,f(x))dx \tag{1}$$ has an extremum, then the Euler equation $\frac{\partial{L}}{\partial{f(x)}}=0$ holds. My question is, for example, what if ...
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1answer
49 views

how to handle a gradient expression

How to prove in a rigorous way that: $$|u|=1 \implies \nabla|u|^2 = 0 \implies (\nabla u)^Tu=0$$ and then $\forall v$ $$\nabla u : \nabla((u.v)\cdot u)= |\nabla u|^2 (u \cdot v)$$
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1answer
49 views

Find the derivatives to transformed variables

Let $\theta \in \mathbb{R}$ and consider the rotational action $X = x \cos\theta - y \sin\theta$ ; $Y = x \sin\theta + y \cos\theta$. Find the transformed derivatives $Y'$ and $Y''$. How do I ...
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1answer
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a variational problem involving $L^p$ norm

Is there any way to prove that there are only finitely many maximizers to the following variational problem: $$\max\left\{||f||_{L^p[0,1]}-\int_0^1(f'(t))^2dt\right\}$$ over all functions $f$ which ...
2
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1answer
114 views

Euler Lagrange equation derivation and application of the fundamental lemma of the calculus of variations

Say we have: (1) $J(x) = \int_{\textit{to}}^{\textit{tf}} g(x(t),\dot{x}(t),t) dt$. We go through the general derivation and arrive at: (2) $\delta J(x,\delta x) = ...
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5answers
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Can this ant find its way back to the nest?

So the puzzle is like this: An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to ...
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1answer
140 views

Maximum Entropy (The existence of a Calculus of Variations problem)

Take maximum differential entropy as an example: Gaussian achieves the maximum differential entropy when the second order moment is fixed. The calculus of variation form: \begin{equation} ...
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0answers
28 views

Optimization problem for integrable functions

For the following optimization problem: find the extremal values of $$ I(x) = \int_a^b F(t,u,x) dt$$ where $x:[a,b]\rightarrow\mathbb{R}$ is a continuous function and $u$ is the primitive of $x$, ...
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Euler-Lagrange equation: no fixed endpoints

My aim is to maximize the objective $J(f) = \int_{0}^{\infty}{ F(f(x),x) p(x) dx}$, where $p(x)$ is a fixed probability density. However, the endpoints are not fixed since the class of functions I ...
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0answers
51 views

Lemma of calculus of variation for Green's function!

OK, I know the title is fundamentally wrong! But I guess you know where I'm going with it! Basically, I'm wondering if it's possible to prove that if $\int_{x\in\Omega}G(x,x')h(x')dV=0$ for any ...
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0answers
33 views

Derivative of infimum in variational problem

Let $\mathcal{E}(\phi,\alpha), \phi\in \mathcal{D}$ be a functional on some domain $\mathcal{D}$ that depends on a parameter $\alpha$. In the expression $$\frac{\partial}{\partial \alpha} \inf_{\phi ...
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Exchange limit and infimum in variational problem

Let $\{\mathcal{E}_n\}_{n\in \mathbb{N}}$ be a sequence of functionals over the same domain $\mathcal{D}$. What are sufficient conditions on the sequence and possibly the domain such that ...
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1answer
48 views

Prove differentiability of functional.

In $C[0;1]$ space let's consider following functional: $$\phi(f) = \int_{0}^{1}(1+f(t))^{3}dt.$$ Prove differentiability of $\phi$ and find $\mathrm{D}\phi(f)$ for: $f(t)=0$, $f(t)=t$, $f(t)=\cos t$. ...
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1answer
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Euler-Lagrange equation: integral over positive real line of a perturbed functional

My goal is to minimize the functional $I[f] = \int_{0}^{\infty}{L(x,f(x),f'(x)) e^{-x} dx }$ However, the solution of the Euler-Lagrange equation is usually stated as minimizing a functional of the ...
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1answer
62 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, \ \ ...
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0answers
282 views

Using Rayleigh-Ritz Method to approximate solutions to extremum problem.

I know how to use the Rayleigh-Ritz method when given a sturm-Liouville problem. But I am not sure where to start when asked questions like the one above. I know i need to plug the trial functions ...
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67 views

Transformation laws for functional derivative with different sets of variable?

We consider a one-to-one transformation $ x \rightarrow x' $, for scalar functional F(x) and F'(x'), we expect the simple transformation behavior $$ F(x) = F'(x'(x))$$ If we set $ x = {\rho, ...
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1answer
107 views

How can Hotelling reduce the Euler-Lagrange equation in his calculus of variations mine problem?

In a 1931 paper Hotelling gives the discounted profit of a mining operation as: $$P=\int_{0}^{\infty} \dot{x} p(x,\dot{x},t) e^{-rt} \:\:dt$$ Note that this is, for the most part, a typical calculus ...
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2answers
125 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. ...
3
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1answer
47 views

Euler-Lagrange on restricted set

I am reading a chapter of Evan's book on weak convergence methods for nonlinear PDE's p.49 and it states that the Euler Lagrange equation for the functional \begin{equation} I[w]:=\int_U|Dw|^2 ...
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Variations Theory

I would like to know like what we have about an increment of a function of an independent variable x which is as follows: What would be correct about functional? I think if we have a functional ...
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1answer
82 views

Variational calculus applied to the strain energy functional in solid mechanics

The question is basically about when to apply the variational operator... Given the general functional representing the strain energy of a solid under a given stress state $\sigma$ and strain state ...
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1answer
161 views

Differential Equations for a Teardrop Shape

My research has led me to a nonlinear system of differential equations which should yield a teardrop shape in the $x-y$ plane. The equations, parameterized by $t$ are ...
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1answer
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Variational Principle for an Elliptic equation

I wish to find the functional whose minimisation yields the follwoing equation on the vector function u $(\lambda + \mu) \nabla (\nabla \cdot u) + \mu \nabla^{2} u = 0$, the Navier equation of linear ...
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46 views

Euler-Lagrange Equation for a functional involving symmetric gradients

I struggle to compute the Euler-Lagrange equation for the following functional $\int_{\Omega} (\nabla^{s} u) D \nabla^{s} u \mathrm{d}\Omega$, where u is a vector valued function u = (u1 (x,y), u2 ...
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1answer
42 views

Prove that $a(u-u_{h},u-u_{h})\ge 0$

Assume that $a$ is bilinear, symmetric and positive definite form, $u\in X$ and $u_{h}\in X_{h}\subset X$. I know the following fact: $$a(u-u_{h},u_{h})=0$$ Frm positive definiteness ...
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1answer
228 views

Parabola & Area Proving (Integral)

This is not a homework question. I am a new teacher (just graduated) and a student asked me this question. The points A(3,9) and B(-2,4) lie on the parabola y=x^2. The line y=x+6 joins A and B. The ...
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Calculus of variations: Lagrange multipliers with functions depend on only one variable

The problem: \begin{align} \min & \iint k(x_1,x_2,y_1,y_2) \, dx_1 \, dx_2 \tag{1}\\ \mathrm{s.t. } & \iint h(x_1,x_2,y_1,y_2) \, dx_1 \, dx_2=l \tag{2}\\ & g(x_1,x_2,y_1,y_2)=0 ...
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1answer
74 views

Is this functional differentiable?

A functional $\Phi$ is differentiable if there exist $F$ and $R$ such that $\Phi(f+h)-\Phi(f)=F(f,h)+R(f,h)$, where $F$ depends linearly on $h$ and $R(f,h) = O(h^2)$. Define a functional $\Phi(f) = ...
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29 views

Multivariable transversality conditions for infinite time horizon in variational calculus?

I would like to find the optimal paths of $d[t]$ and $k[t]$ that maximize the following function: $$J(x)=\int_0^\infty e^{-pt}(d[t]-d[t]^2-d'[t]^2+k[t]-k[t]^2-k'[t]^2+d[t]k[t])$$ Using Mathematica ...
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1answer
37 views

Finding extremal of function J

Find a curve passing through $\left(0,0\right)$ and $\left(1,1\right)$ that is an extremal for the functional $\displaystyle{{\rm J}\left(x,y,y'\right) = \int\left\{\left[y'(x)\right]^{2} + ...