Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on an infinite dimensional spaces.

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Find $\min_{y \in \mathcal{A}} J(y)$, if it exists.

Let $\mathcal{A}$ be the set of continuously differentiable functions at the interval $[a,b]$. Let $J$ be the functional $$J(y)=\int_a^b \sqrt{1+y'(x)^2}dx$$ Find $\min_{y \in \mathcal{A}} J(y)$, if ...
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1answer
26 views

How to calculate difference between two points in some values?

I have 5 values which are -2, -1, 0, 1, 2. I want to calculate difference between two variables which contains the values from these given values. Suppose I have ...
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1answer
28 views

How can I solve the following exercise

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
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1answer
15 views

How does the value of a functional change when you perturb the extremizing function?

In deriving the Euler equation for etremizing a functional \begin{equation*} J[y] = \int_a^b F(x,y,y')\,dx, \end{equation*} we look at: \begin{equation*} J[y+h]-J[y] = \int_a^b(F_yh+F_{y'}h')\,dx + ...
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0answers
20 views

Optimization with Integral Inequality constraint and nonnegativity conditions

Trying to solve this: $$\min TC(A,a,q)= \int_M f(A,a,q)\,dx\, dy$$ $$s.t.$$ $$a\le\int_M g_i(A,q)\,dx\,dy$$ $$q\le \text{constant}$$ $$A,a,q\ge0$$ $(x,y)$ is omitted in $A(x,y), a(x,y), q(x,y)$ ...
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0answers
82 views

Mathematical definition of the Hamiltonian function.

I'm reading this nice text on Calculus of Variations, by Peter Olver. In page $8$, he calls $$J[u] = \int_a^b L(x,u,u')\,{\rm d}x$$ the objective functional, and the integrand $L(x,u,u')$ the ...
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1answer
52 views

How to minimize functional?

In Bishop's book [1] they show that the optimal y(x) w.r.t. squared error loss function $$E[L]=\int \int \{y(x)-t\}^2p(x,t)dxdt $$ is given by a conditional expectation $y(x) = E_t[t|x]$. However, ...
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1answer
18 views

Combining an outcome of a score

Hey I was wondering how many are the possibilities of combining the scored points of a result such $133:75$ from a basketball game? Considering that there are fouls($1$ point), normal($2$ points) and ...
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1answer
67 views

Extremizing the following boundary value problem

Consider the functional $$J(y)=y^2(1)+\int_0^1y'^2(x)\,dx$$ with $y(0)=1$ , where $y\in C^2[0,1]$. If $y$ extremizes $J$ then find the value of $y(x)$. I tried through Bolza problem. Firstly ...
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1answer
42 views

Find the curve which together with $\gamma$ encloses the greatest area.

I'm working through Gelfand & Fomin's Calculus of Variations by myself, and could use the guidance of someone familiar with the subject. The problem I'm on now is the following: "Given two points ...
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0answers
30 views

Prove that two functionals with identical differentials differ by a constant.

I am self-studying Calculus of Variations and am struggling to prove results about the variation of a functional that are analogous to results in elementary analysis about differentials/derivatives. ...
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0answers
20 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?
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0answers
8 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
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1answer
48 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 ...
4
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0answers
47 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 ...
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0answers
21 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 ...
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2answers
46 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 ...
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1answer
30 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 ...
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1answer
85 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 ...
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0answers
41 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 ...
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1answer
30 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 ...
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1answer
29 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 ...
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1answer
76 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?
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1answer
44 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? ...
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1answer
58 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 ...
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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 ...
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0answers
22 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 ...
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0answers
25 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 ...
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0answers
38 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 ...
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1answer
23 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 $x(1)=2, \ x(2)=17$
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2answers
58 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 ...
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1answer
30 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]= ...
2
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1answer
32 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 ...
3
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1answer
30 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{ ...
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1answer
50 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): $F=\int_{x=0}^1{f(x)dx}$ = ...
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0answers
20 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 ...
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0answers
39 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 ...
4
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0answers
62 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 ...
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2answers
44 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: ...
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0answers
42 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 ...
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2answers
70 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 $$\dot{\bf x} (t) = A {\bf x}(t) + B {\bf u}(t)$$ $$x(0) = x_0$$ ...
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0answers
9 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}) = ...
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0answers
24 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 ...
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1answer
108 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 ...
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4answers
127 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 ...
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1answer
70 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:= ...
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33 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 ...
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0answers
23 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 ...
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1answer
66 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 ...
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1answer
66 views

variational problem with constraints

Let me bring to your attention the following problem. Suppose we have the functional $$ F = \int\limits_{a}^{b} f(y(x))\cdot\frac{dy}{dx} dx .$$ It is easy to see that that the Euler-Lagrange ...