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

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Question about variational principles involving light rays.

In question 9 (see this link: http://view.samurajdata.se/psview.php?id=28b2e4b5&page=1 ), I've shown the light rays are follow a parabolic paths using the Euler-Lagrange equation and Fermat's ...
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1answer
26 views

Is this differentiation correct?

$J(x,y')=\int_1^2 xy'(x)+(y'(x))^2dx = \int_1^2{f(y,y^\prime,x)}$ Need to find $\frac{d}{dx}(\frac{\partial f}{\partial y^\prime})$ $\frac{\partial f}{\partial y^\prime}=x+2y'(x)$ ...
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24 views

Converse of Noether's (first) theorem

Noether's (first) theorem states that if a Lagrangian $L$ admits a continuous symmetry, then the following quantity are conserved. $$\left(\frac{\partial L}{\partial \dot q}\cdot\dot ...
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1answer
41 views

Elliptic partial differential equations

Consider the following elliptic PDE: $$ \Delta u=f(u), $$ where $f(u)$ is a smooth function. Which references (books, papers,...etc.) about existence of solutions for this PDE do you recommend to have ...
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1answer
42 views

How to take derivative?

Find a curve passing through (0,0) and (1,1) that is an extremal of the function $${\rm J}\left(x,y,y'\right)= \int_{0}^{1}\left[ y'^{\,2}\left(x\right ) + 12\,x\,{\rm y}\left(x\right)\right]\,{\rm ...
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2answers
100 views

Function extremal - calculus of variations

Find a curve passing through (1,2) and (2,4) that is an extremal of the function: $J(x,y')=\int_1^2 xy'(x)+(y'(x))^2dx$ I don't know what methods to use at all.
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56 views

Euler–Lagrange equation

Does this PDE $\nabla\cdot( \frac{ \nabla u}{u} )+a\, \Delta u+b\,u=0$ (*) have a variational structure? Here $a$ and $b$ are constants. In other words, the question I am asking is: Does there ...
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1answer
29 views

how to introduce time into calculus of variations for image processing?

I'm studying some topics about calculus of variation applied to image processing. I'd like to understand how to introduce time parameter to evolve an image in an iterative way. For example, let's ...
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35 views

Sufficient conditions for the use of the Beltrami identity

For reference, I shall use the notation used in the wikipedia article for the Beltrami identity in the application section (http://en.wikipedia.org/wiki/Beltrami_identity) In the article, the ...
2
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0answers
88 views

Calculating the maximum of a function

How can one determine $$\max_{f_0,f_1}\frac{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\log\left(\frac{f_1(y)}{f_0(y)}\right)\mbox{d}y}{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\mbox{d}y}$$ given ...
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2answers
77 views

Calculus of Variations

In the Calculus of Variations there is a passage from Euler's characteristic equation: $$ \frac {\partial F}{\partial y} - \frac {d}{dx} \left(\frac {\partial F}{\partial y'} \right)=0 $$ in ...
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42 views

What methods are available for this optimization problem?

I have an intermediate knowledge of the calculus of variations: I can handle constraints in functional or integral forms and extrapolate to multiple variables and functions. If I dig in my notebooks I ...
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22 views

Spectrum of operator with unknown function

I'm working on a variational problem in elasticity which involves a hefty number of Lagrange multipliers. I have calculated the second variation to be [; \int ds ~h \left[ \frac{d^4 }{ds^4 } + ...
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64 views

Stationary action functional

The last part in the derivation of the Euler-Lagrange equations for a stationary action has me confused. It's about the order of differentiation and evaluation, and whichever comes first. I'll ...
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2answers
33 views

Reading Speed for Constant Time to Finish

You open a very long new book on your e-reader and read a few pages. It helpfully informs you that based on your reading speed you have 16 hours of reading left until you are done. You read the rest ...
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1answer
49 views

Counterexample for the Chain rule for the Gateaux-derivative

I'm reading the book of Drabek, Milota - Methods of Nonlinear Analysis, and at page 121, they state: but I can't manage to find such counterexample. For clarity the Gateaux derivative is defined ...
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53 views

Calculus of variation: Reduce the order of a differential equation using a 1 parameter lie group adfmitted by it.

We are asked to reduce $y^"+y-y^{-3}=0$ using $X= \sin2x\frac{\partial}{\partial x}+y\cos2x\frac{\partial}{\partial y}$ I know we have to find the first prolongation of X and solve $X^1$F=0 using ...
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92 views

Euler-Lagrange Calculus of Variations Example

I have been working on solving Euler-Lagrange Equation problems in attempts to learn Calculus of Variations, but this one example has me stuck. I am probably making mistakes in my integration. I am ...
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31 views

Minimising line integral over a scalar field part 2

This is a continuation of this question whose general point is summarised below Say we want to find a path $y=y(x)$ in the scalar field $S(x,y)$ that finds the extrema of of its line integral. ...
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91 views

Calculus of variations for implicitly defined functional

I would like to minimize a functional of the type: $$L[\gamma]=\int_a^b F(T(\gamma(t))dt$$ on the space of paths $\gamma$, where $T=T(\gamma,t)$. Now, usually I would simply apply Euler-Lagrange's ...
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Maximizing an integral through maximum principle

Suppose that we wish to achieve $$\max\int_0^1 (1-x^2-\dot{x}^2)dt, x(0)=0, x(1)\geq 1$$ Two possible ways one can do this is by Euler-Lagrange eqn or maximum principle. Applying the Euler-Lagrange ...
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1answer
32 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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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 ...
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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 ...
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370 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 ...
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1answer
99 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 ...
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1answer
59 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 $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, ...
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43 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 ...
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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 ...
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1answer
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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 ...
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41 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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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 ...
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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 ...
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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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53 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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25 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 ...
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2answers
51 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 ...
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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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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
85 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. ...
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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: ...
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2answers
63 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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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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36 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
76 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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98 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 ...
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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 ...
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1answer
46 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 ...
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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 ...