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

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2
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1answer
31 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 ...
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0answers
21 views

specific non linear pde

I would really appreciate to hear your insights or comments about the following problem: Consider the following non linear pde: let $\Omega$ be the unit square with vertices at (0,0),(1,0),(0,1) and ...
0
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0answers
26 views

How to find the infintesimal generator and conserved current of the symmetries of the minimal surface problem

For the Lagrangian $L(x,y,z,z_x,z_y)=\sqrt{1+z^2_x+z^2_y}$ how do you find infinitesimal generator and conserved current of the six symmetries (3 translations and 3 rotations)? I was using Noether's ...
4
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1answer
339 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
85 views

How to properly take derivatives in calculus of variations (Euler-Lagrange formula)

Why is it that, in calculus of variations (specifically Euler-Lagrange), we can take the derivative of a function with respect to a function $f$ and set this derivative to $0$ if only $f'$ appears in ...
0
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1answer
58 views

Indicator function of a level set

Consider a positive definite locally Lipschitz function $V:\mathbb{R}^2\to\mathbb{R}_{\geq0}$. Fix $c\in\mathbb{R}_{\geq0}$ and consider the sublevel-set $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, ...
1
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0answers
42 views

Can the Euler-Lagrange equations be derived from a variation over a time of order $dt$ rather than $t$?

In the calculus of variations, the solution of the Euler-Lagrange equations gives those functions for which a given functional is stationary. Now all derivations I've come across up to now, carry out ...
0
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1answer
22 views

References about Nemytskii Mappings

I need some references about Nemytskii Mappings. Can anyone tell me some textbook about it? I am reading chapter 2 of this text www.math.tifr.res.in/~publ/ln/tifr81.pdf . And I need more results ...
1
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1answer
31 views

Calculus of variation Transformation

Find the transformations that transform $$X = -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$$ to $\bar{X}$ = $\frac{\partial}{\partial s}$ That is X(s)=1 and X(t)=0 I know we have to ...
2
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0answers
37 views

Minimizing a functional by variation

I have a problem at the last step of my proof. I have the following functional to be minimized on $\rho\in L^1(\mathbb R^d)$. Here $\lambda$ is a Lagrange multiplier and $\rho\geq 0$. $h(\rho) = ...
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23 views

Suggestion for a work on variational calculus

Hi dear forum :D stack I must complete the Master FISYMAT in Spain my assistant professor has told me to do about calculus of variations. What suggestions on this matter can you give me ? To get the ...
0
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1answer
73 views

How to prove $\int_a^b f(x)\varphi(x)dx=0\Rightarrow f(x)=0$

I am doing some reading on the calculus of variations and one of the first examples uses the following theorem: Let $f\in C[a,b]$. If $\int_a^b f(x)\varphi(x)dx=0$ for all $\varphi\in C[a,b]$, then ...
2
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1answer
33 views

Intuition behind Young measures

I am studying the calculus of variations. Many interesting results arise from the theory of Young measures. For a greenhorn (as I am) an intuition behind Young measures is hidden, because many texts ...
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0answers
50 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 ...
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0answers
24 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 ...
2
votes
2answers
46 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 ...
0
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1answer
37 views

Restricted function

Let $A=\{(x,y): x,y\in(-1,1)\}$. Is there a function $f:A\mapsto A$ such that $f(x,0)=(x,x^2)$ $f$ differentiable and bijective on $A$. I have tried a lot of constructions but the problem is in ...
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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 ...
2
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1answer
84 views

Extremum of functional of a complex function

consider functional $E$ defined by $$E[z]=\int F(x,z(x))dx$$ where $F$ is a complex-valued nonlinear function. How can we find the function $z(x)$ so that $$G=|E|^2=EE^*=\iint ...
4
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1answer
32 views

Optimization in $L_1$, does this make sense?

I'd like to find a probability distribution $f(x)$ on the unit interval $[0,1]$ that obeys a given set of moment constraints, e.g. $\int_0^1 xf(x) dx = \mu_0$ for some given $\mu_0$, and so forth. ...
0
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1answer
44 views

How to compute $\frac{\partial F_x}{\partial F}$?

If $F$ is a function of $x$ and $y$, and $F_x$ denotes $\dfrac{\partial F}{\partial x}$, how would you compute $\dfrac{\partial F_x}{\partial F}$? Can I use the chain rule to write this as: ...
3
votes
2answers
55 views

Minimizing a Functional with a Path Length Constraint

Say you have some functional of the form $\int_0^{t_f} L(x,\dot{x},y,\dot{y},z,\dot{z}) dt$ that you're trying to minimize. Normally one can solve this using the Euler-Lagrange equations, and when you ...
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0answers
13 views

minimyze the functional - variational methods

What i write here is in this article http://projecteuclid.org/download/pdf_1/euclid.cmp/1103922134 in the page 571. Let $N\geq 3$ and $0< \sigma < \frac{2}{N-2}$ For $f \in H^1(R^n)$ define ...
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0answers
35 views

Implicit function theorem for boundary value problems

I have a nonlinear, two point boundary value problem of the form $F(x, y(x), y'(x); \Omega ) = y''$ along with some boundary conditions of the form $y_\Omega(0) = a_\Omega, y_{\Omega}(1) = b_\Omega$. ...
1
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1answer
74 views

Simplification of Euler-Lagrange equation when integral independent of y

I'm supposed to show, that if my function $f(y,y',x)$ is independent of $y$, then the Euler-Lagrange equation turns out to be $\partial f/\partial y' = const$. Now, the Euler-Lagrange equation is ...
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0answers
86 views

How does Calculus of Variational work in Finite Element Method

I'm learning Finite Element Method. And it is said in a lot of books that Calculus of Variational is the basis of Finite Element Method. But as far as I know, Calculus of Variational is to find a ...
4
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2answers
50 views

How to take partial derivatives of functions whose inputs depend on the same variable?

I am starting to learn about the Calculus of Variations and the Euler-Lagrange equation is extremely confusing to me: The Euler–Lagrange equation, then, is given by ...
2
votes
1answer
44 views

What does “Calculus of Variations in $L^p$ spaces” deal with?

Next semester I have the possibility to attend a course with the above title. But I'm not sure if I should, because I looked on Wikipedia and Princetions Companion to Mathematics and couldn't find ...
2
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0answers
39 views

Cancelling some dx's

How can you prove $$\frac{\partial f}{\partial y} = \frac{d}{d x} \left( \left( \frac{\partial}{\partial \frac{dy}{dx} } \right) \right)f ?$$ It's tempting to cancel the two $dx$'s, but can ...
1
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1answer
22 views

Derivation of inner variations

In Giaquinta's and Hildebrandt's 1996, "Calculus of Variations 1", pages 147-148, they develop the definition of inner variations. They first fix $\lambda\in ...
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0answers
48 views

Derivation of Euler lagrange for Yang Mills

I need someone to sketch the conventional steps(from variation to vanishing of arbitrary function chosen , etc, etc) of Classical Yang-Mills. If using exterior product product could you emphasize any ...
0
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1answer
90 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 ...
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0answers
57 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
19 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
150 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 + ...
1
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1answer
69 views

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: $$y'' +ky =0,$$ ...
2
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0answers
78 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 ...
0
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1answer
51 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 ...
2
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1answer
51 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 ...
1
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0answers
60 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 ...
2
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0answers
25 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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0answers
65 views

Single Variable Calculus of Variations Question

Problem: Minimize $I(f)$ subject to the constraint $J(f)\leq 0$, where $$I(f)=\int_{x_1}^{x_2}\frac{dx}{f(x)}\tag{$f:[x_1,x_2]\to \mathbb{R}_{\geq 0}$}$$ ...
3
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0answers
70 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 $F(t,x)$, ...
0
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1answer
62 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 ...
1
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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 ...
1
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1answer
84 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 $$ J(x)=\int_1^{t_f} \dot{x}^2(t)t^3\,dt $$ ...
11
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4answers
395 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 ...
0
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1answer
42 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
votes
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 $$F_y=\frac{2y}{(y')^2}$$ $$F_{y'}=-\frac{2(1+y^2)}{(y')^3}$$ ...
2
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0answers
80 views

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 ...