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

learn more… | top users | synonyms

0
votes
0answers
15 views

Euler equation-Calculus of variations

How did they integrate the differential equation below to get to Esin(y/E)=+-x+c ? Shouldn't it be integrated to give Earcsin(y/E)=+-x+c?
0
votes
0answers
7 views

Euler equation for functionals

I just wanted to check for 59) the final line of E shouldn't there be a plus sign instead of a minus because when I work out the answer I get the same equation but with a plus sign in the numerator. ...
3
votes
1answer
18 views

No extremals satisfying the Euler equation - what does it mean?

Consider the functional $J[y] = \int_{0}^{1}xyy^{'}dx$. If I want to find extremals (a function $y=y(x)$ that makes the functional stationary) with boundary condition $y(0)=0$ , $y(1)=1$ for this ...
0
votes
0answers
17 views

Maximize polynomials

Hi guys I need some help. I am reading a paper and I cannot understand something simple. The author has 4 polynomials with a constrain and is trying to find the optimal solution to the problem. ...
4
votes
0answers
43 views

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$?

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$ ? To give context, this comes from: Dirac's Theory of General Relativity p19: http://imgur.com/mrkT5C7 I'm not comfortable with proofs regarding ...
1
vote
0answers
9 views

Reference Request for Penalty Method for Optimal Control?

Is there a good book or review article to read about the methods like penalty method, method of duality and method of relaxation in problems of calculus of variations and their relations to optimal ...
0
votes
2answers
27 views

Solve the following Fredholm Integral Equation

Solve the Integral Equation :$$y(x)=\frac{6}{5}(1-4x)+\lambda\int_0^1(x\ln t-t\ln x)y(t)\,dt$$ Let , $$y(x)=\frac{6}{5}(1-4x)+\lambda xC_1-\lambda\ln x C_2$$where, $$C_1=\int_0^1\ln ...
1
vote
1answer
24 views

Canonical projection of $W^{1,p}(\mathbb{R}^N)$ onto $W_0^{1,p}(\Omega)$

Suppose we have a bounded domain $\Omega \subset \mathbb{R}^N$ with sufficiently smooth boundary $\partial \Omega$. The Sobolev spaces $W^{1,p}(\mathbb{R}^N)$ and $W_0^{1,p}(\Omega)$ are defined as ...
1
vote
1answer
72 views

Proving that $\int \delta \dot{x} dt = \delta x$

Everytime I've seen this I've assumed it was true. It seems plausible. But I would like to rigorously prove it. I think this is correct, but I would like another opinion because my mathematics is very ...
0
votes
1answer
19 views

$\Gamma$-convergence (Gamma-convergence) and PDEs?

My question is about the applying calculus of variations to solving Partial Differential Equations. In particular, what is the idea behind using $\Gamma$-convergence to find weak solutions of PDEs? ...
1
vote
1answer
23 views

Congruence Property of Monotone Operators

A map $T$ is called strictly monotone if for $x\ne y$, $\langle u-v,x-y\rangle>0$ for all $u\in T(x),v\in T(y)$. Let $A$ be an $m\times n$ matrix and $b\in\mathbb R^m$. I want to prove that if ...
4
votes
0answers
30 views

How to prove an extremum existence in problems, regarding calculus of variations

Let's consider a functional $S(y)=\int_{a}^{b}{f(x, y, y') \cdot dx}$. It's known that if the function that attains minumum or maximum to $y(x)$ does exists, then it can be got from the Euler-Lagrange ...
0
votes
1answer
295 views

DuBois-Reymond Lemma

I know thats the following statement is true. $f,g$ are continuous function $[a,b]$.Suppose $\int\limits_a^bf(t)h(t)+g(t)h'(t) \, dt=0$ for every $h$ belonging to $C_0^{\infty}[a,b]$, then $g$ is ...
0
votes
1answer
18 views

Calculus of Variations - Function of y and y' only

I have the following problem: $\int^\pi_0 (4y^2-y'^2)dx$ which satisfies: $y=1$ on $x=0$ and $y'=0$ on $x=\pi$. I am to show that the solution is $y=cos(2x)$. Now, I first realised that the ...
0
votes
1answer
16 views

Palais–Smale compactness condition

Can someone explain the essence of Palais–Smale compactness condition used in the Mountain Pass Theorem, in particular its weak formulation?
-1
votes
0answers
60 views

Finding an Extermal of Hard Examples? [on hold]

Who Can show me the calculation for solving extermal for $$\int_0^1 (x^2+ \dot {x}^2+2xe^t) dt \quad \text{ when }\quad x(0)=0,\;x(1)=free.$$ My TA say a short answer and I Couldn't reach to ...
0
votes
1answer
42 views

Why using liminf instead of limsup?

In Chapter 8: Calculus of variations of Evan's Partial Differential Equations, Evan writes as follows: I am wondering about the last paragraph where he says that knowing $I[u] \leq ...
1
vote
1answer
749 views

Euler lagrange equation solving

Find the Euler-Lagrange equation for the functional $$I(y) = \int_0^1(py\,'\,^2-qy^2)\mathrm dx$$ subject to the constraint $$\int_0^1ry^2 = 1.$$ Answer: $\frac{d}{dx}(py') + (q-\lambda r)y = 0$. ...
0
votes
1answer
25 views

Why is this inequality true?

In Evan's Partial Differential Equations, he writes Then, he continues to write: But I do not understand how he gets $I[w] \geq \delta ||Dw||^q_{L^q(U)} - \gamma$. I tried to write it out and I ...
1
vote
0answers
15 views

About the definition of functional derivative and the $L^2$ inner product

There is something I do not understand well about the definition of the functional derivative. In the wikipedia page http://en.wikipedia.org/wiki/Functional_derivative it says: 1) This definition ...
3
votes
1answer
576 views

Find the extremal to the functional and discuss whether they provide a max/min

I am having a hard time getting my head around Functionals and Calculus of Variations, My question is: Given a functional and using the Euler-Lagrange equation to find an extremal, how do we show ...
-1
votes
0answers
12 views

Boundary conditions and Lagrange Constraints in Calculus of Variations

I am trying to learn about Calculus of Variations for some time now. In many problems, there are some boundary conditions defined, for example when we want to maximize a functional ...
1
vote
0answers
9 views

calculus of variations with free endpoint

I have a Lagrangian $L(x,\dot x)$ and want to solve $$\arg\min_{\gamma(t)} \int_0^\infty L(\gamma, \dot \gamma)\,dt$$ subject to holding only one of the endpoints fixed: $\gamma(0) = \gamma_0$. Now ...
4
votes
1answer
348 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 ...
0
votes
0answers
22 views

the extremals of the functional with boundary condition

This question is about the extremals of the functional J using method of variation. But I know how to calculate the extremals, the exact question is slightly different and I have no idea what title is ...
3
votes
1answer
23 views

Lower semicontinuous energy functional on compact space of Lipschitz functions

Let $\Omega \subset \mathbb{R}^{n}$ be a bounded open subset containing $0$ and let $L>0$ be some positive constant. Consider the space $A_{0}=\{f \in C^{\infty}(\overline{\Omega}) \mid f \text{ ...
1
vote
1answer
18 views

Extermal curve for specific problems?

I ran into a quiz question last month. how we can find the Extermal curve for following problem. $$ \int_1^2 \frac {\dot {x}^2}{t^3} dt $$ where $x(1)=2, \ x(2)=17$
-1
votes
0answers
51 views

Direct method in the calculus of variations

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$. $$ \mathcal{F(u)} = \int_{\Omega}\frac{1}{2}|Du|^2dx $$ $$ u \in \mathcal{A}: = \{v \in W_{0}^{1,2}: \int_{\Omega}v^2dx = 1\} $$ Does this ...
0
votes
2answers
53 views

If $\sum_{k=0}^{n}\binom nk=2^n$ then how is $2(\binom n0+\binom n2+\binom n4+…)=2^n$ [duplicate]

$$\sum_{k=0}^{n}\binom nk=2^n$$ then how is $2(\binom n0+\binom n2+\binom n4+...)=2^n$ ?? I don't think it could be because half of the members of the sum are chosen, that seems a bit intuitively ...
0
votes
0answers
17 views

How should the Calculus of Variations deal with $\delta(t-t_0)$ variations?

I'm familiar with using the Calculus of variations to find the condition for which first order variations of a functional wrt a function are zero: We start with a functional $J[x]= ...
1
vote
0answers
18 views

Gradient of the Fourier transform of a function

I am wondering if there is a simple way to express the first variation of the Fourier transform of a function as a function of said function. In other words, if $g:x\mapsto F(f)(x)$, where $F(f)$ is ...
1
vote
1answer
46 views

A maximization problem parametrized by a function

Let $f$ be a smooth positive monotonically increasing real function which is defined and finite in $[0,1]$, and define the following two quantities (see the figure below): $F=\int_{x=0}^1{f(x)dx}$ = ...
3
votes
3answers
438 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 ...
1
vote
0answers
18 views

Getting the minimum of a mixed functional

I have a functional $T$ defined on the attached picture. The functional always gives non-negative values. So it has a non-negative infinum I'm trying to figure out whether this infinum is ...
4
votes
0answers
38 views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
0
votes
0answers
34 views

Calculus Questions on Fibonacci and Length of Curve

Hi All, I have an issue trying to do part (i) and (ii). How do you go about doing it? As for the fibonacci sequence, I keep getting really big numbers, i can't seem to get the number of digits that ...
0
votes
2answers
36 views

Meaning of an Extremum of a Functional

Consider the following minimisation problem: $$\int_0^3\left(0.5\dot{x}^2-x\right)\,\mathrm{dt}$$ Subject to $x_0=0$ and $\dot{x}=0$. Using the Euler lagrange equation one can get: ...
0
votes
0answers
30 views

Variational optimization problem with several constraints

I am looking for solutions, approaches or hints to solve this variational optimization problem: Let $f:\mathbb{R}\rightarrow [0,\infty)$ be such that $\int f(x)\,dx=1$ and $\int x\,f(x)\,dx=0$ and ...
13
votes
4answers
345 views

How can $y$ and $y'$ be independent in variational calculus?

In variational calculus, functionals are written as \begin{eqnarray} F = \int f(x,y,y') dx \end{eqnarray} Where $F$ depends upon choice of $y,y'$. But for smooth regular functions specifying the $y$ ...
0
votes
2answers
48 views

Minimize a non-convex function subject to linear dynamics constraint

I want to solve the following problem: $$\min\limits_{\bf u} \frac{\bf c^T {\bf x} (T_f)}{\| \bf c\|\|{\bf x} (T_f)\|}$$ subject to $$\dot{\bf x} (t) = A {\bf x}(t) + B {\bf u}(t)$$ $$x(0) = x_0$$ ...
0
votes
0answers
8 views

How to optimize this types of problems?

Given that $min [ t_{f} - t_{0} ]$ such that $x(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $y(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $z(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $x(t_{f}) = ...
0
votes
1answer
43 views

The Euler-Poisson equation

$$\int_{0}^\pi (x''^2+4x^2) dt$$ $$ x(0)=x'(0)=0; x(\pi)=0;x'(\pi)=sinh(\pi)$$ This is The Euler-Poisson equation, i found: $$\frac {\partial f}{\partial x}-\frac {d}{dt} \frac{\partial f}{\partial ...
7
votes
4answers
102 views

Use of $L^2$ norm in calculus of variations

I am trying to make an introduction to the calculus of variations. This field has many connections with functional analysis, in which I do not have an experience. I recently learned about function ...
2
votes
1answer
123 views

Constant Speed of Geodesics

Let V be the set of smooth functions $ f : [0,1] \rightarrow \Bbb R $ such that $ \int_0^1 f(t) dt = k $. If $ F : V \rightarrow \Bbb R $ is given by $ F(f) = \int_0^1 f(t)^2 dt $, then show that the ...
47
votes
10answers
1k views

Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
1
vote
0answers
16 views

Calculus of Variations: What if the functional is an integral with boundaries at infinity?

I am trying to grasp the basics of Calculus of Variations. The problem seems to be concentrated on functionals of the form : $$ F[y] = \int_{a}^{b} G(y,y(x),y'(x))dx$$ where $y(x)$ is assumed to be ...
1
vote
1answer
48 views

Dominated convergence theorem and fundamental lemma

this is a proof of the fundamental lemma of calculus of variation. Some preparations: Let $g(x):=e^{\frac{-1}{1-||x||}} \chi_{||x||<1},$ with characteristic function $\chi,$ then $$c:= ...
1
vote
0answers
23 views

Lagrange Multipliers with Calculus of Variations

We wish to extremize $$S = \int \mathcal{L}(\mathbf{y}, \mathbf{y}', t) dt $$ subject to the constraint $$g(\mathbf{y}, t) = 0 \;.$$ We move away from the solution by $$y_i(t) = y_{i,0}(t) + \alpha ...
0
votes
0answers
16 views

How is the functional differentiation derived?

I am trying to understand how the functional derivative is derived but I consistently fail to find a convincing resource explaining it. I want to understand how it is derived from the regular ...
1
vote
1answer
44 views

Why is the Lagrangian a function on the tangent bundle?

I understand that empirically the state of a dynamical system (at a given instant in time) is determined by specifying it's position and velocity, but I'm slightly unsure as to why the Lagrangian is ...