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

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2
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0answers
25 views

Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= ...
1
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1answer
39 views

Why does the functional have a local minimum at $0$?

Definition: Let $J: A \to \mathbb{R}$ be a functional , where $A \subset V$ and $(V, ||\cdot||)$ a linear space with norm. Let $y_0 \in A$ and $h \in V$ such that $y_0+ \epsilon h \in A $ for ...
0
votes
0answers
5 views

Minimizing constrained functions on $l^p$

Suppose we have some functionals $H,G:l^p(\mathbb{N}^+)\to\mathbb{R}$, and we want to find some $p \in l^p(\mathbb{N^+})$ which minimize $H$, subject to the constraint that $G(p)=0$ is constant. As ...
0
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0answers
12 views

An MCQ involving Rayleigh - Ritz method for the functional $I(y) = \int_{0}^{1}(\frac{1}{2}(y^{'})^2 - y)dx$

An MCQ involving Rayleigh - Ritz method for the functional $$I(y) = \int_{0}^{1}(\frac{1}{2}(y^{'})^2 - y)dx$$ Let $y_\text{app}$ be polynomial approximation, involving only one coordinate ...
1
vote
1answer
9 views

An MCQ for finding the extremal of the functional $J = \int_{a}^{b} F(x, y, y^{'})$

Consider a functional $$J = \int_{a}^{b} F(x, y, y^{'}),$$ where $F(x, y, y^{'}) = \frac{1 + y^{2}}{(y^{'})^2}$ for admissible function $y(x).$ Which of the following are extremals for $J$? $y(x) = ...
4
votes
1answer
38 views

Intuition of weak star convergence.

Given $\Omega=(0,1)$, consider the following sequence $$ v_j(x)\colon=\begin{cases} \;a &\text{if }jx-\lfloor jx \rfloor\le\theta\\ \;b &\text{otherwise} \end{cases} $$ where ...
2
votes
1answer
60 views

Functional Maximization

So how do we solve a problem like this: Find the function $s(x)$ such that $s(x)$ maximizes $$\int_0^{s^{-1}(k)} s(x) dx $$ where $x\in[0,10]$, $s(x)\in[0,1]$, and $k\in[0,1]$ ($k$ is a constant). ...
0
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0answers
23 views

Ideal shape for underwater habitat

Is there an analytic solution to this problem or do I need to compute a discrete approximation using a relaxation procedure - or something similar? I want to find the shape of a roughly spherical ...
1
vote
1answer
26 views

Calculate the (variational) derivative of the following equation;

Consider $ E[u]= \int^1_0 \big(u'(x)\big)^2+\big(u(x)\big)^2-2f(x)u(x) dx.$ Calculate the variational derivation for a function $v$; in other words, calculate $\frac{d}{d\epsilon}E[u+\epsilon v]$ at ...
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0answers
10 views

The Curved slope of an FMX ramp [on hold]

I would like some help working out the curve of a ramp with the following dimensions are 4.5m long 1.8m high.. please see an example Thanks in advance..
0
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0answers
8 views

An MCQ to determine extremal of a functional $J(y) = y^2(1) + \int_{0}^{1}(y^{'})^2(x)dx,$ [duplicate]

The following problem occurs in an exam: Consider a functional $J(y) = y^2(1) + \int_{0}^{1}(y^{'})^2(x)dx,$ $y(0) = 1, $ where $y \in C^2 ([0, 1]).$ If $y$ extremizes $J$, then $y(x) = 1 ...
3
votes
2answers
57 views

Advice on second order non-linear ordinary differential equation

I'm currently working on some problems concerning the calculus of variations and I have come up with the following differential equation that I now want to solve: $$1 + y'(x)^2 - y''(x)(y(x)-\lambda) ...
2
votes
1answer
46 views

Divergence identity

From PDE Evans (2nd edition), page 515, we are given $$\sum_{i=1}^n \left(\left(Du \cdot x + \frac{n-p}p u \right)p|Du|^{p-2}u_{x_i}-|Du|^px_i \right)_{x_i}=0. \tag{10}$$ Then the author goes ...
1
vote
1answer
26 views

Lagrangians independent of $x$

In PDE Evans, 2nd edition, the following formula is printed as equation $\text{(9)}$ in §8.6 (on page 514): $$\sum_{k=1}^n (L_{p_i}u_{x_k}-L\delta_{ik})_{x_i}=0 \quad (k=1,\ldots,n) \tag{9}$$ ...
2
votes
2answers
56 views

Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $ F (x,y,y')=y'^2(x)$ After solving the ...
1
vote
3answers
43 views

Proving an identity

Given $a,b\in\mathbb{R}$ with $a < b$ and defining $F(z):=\int_0^z f(s) \, ds$ with $z \in \mathbb{R}$, how can one establish that $$F(a+b)=F(a)+f(a)b+ b^2\int_0^1 (1-s)f'(a+sb) \, ds,$$ which is ...
4
votes
2answers
113 views

solution of $y^{\prime \prime} + y^n = 1$ [closed]

I am not able to figure out the solution for the differential solution $$y^{\prime \prime} + y^n = 1$$ I want to specifically find an answer for $$y^{\prime \prime} + y^2= 1$$and $$y^{\prime \prime} + ...
0
votes
1answer
29 views

Potential energy of a hanging string of a prescribed length

Consider a homogeneous, flexible string of a prescribed length hanging in a vertical plane where its ends are fixed at two points P and Q. Determine the equilibrium configuration of the string by ...
3
votes
1answer
51 views

What is a functional? And how is it defined for the length?

Im reading about Calculus of varations and there is a lot of references to "the functional" i.e we want to find the minimum of the functional etc. From what i have read, "the functional" is simply the ...
2
votes
1answer
19 views

Does $\log$ minimize this functional for its Abel equation?

Suppose that we have the functional equations ("Abel equation", it is called) for a function $F: [1, \infty) \rightarrow \mathbb{R}$ given by $$F(1) = 0$$ $$F(ex) = F(x) + 1$$ where $e$ is the ...
3
votes
1answer
48 views

Calculus of variations question with two variables

If $u(x)$ and $v(x)$ satisfy $u(0)=1$, $v(0)=-1$, $u(\pi/2) =0$, $v(π/2) =0$ on extremals of functional $$ \int_0^{\pi/2}\left[\big({\frac{du}{dx}\big)^2 +\big(\frac{dv}{dx}\big)^2 +2 \,u v ...
4
votes
1answer
96 views

Calculus of variations: two integrals

I would like to find the extrema of the following integral with respect to $u\left(s\right)$: ...
1
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0answers
40 views

What's the maximum speed of snake so that the frog can escape?

Suppose there's a round pond, a frog which can swim as 1 meter / second, and a snake that moves along the pond ridge but cannot swim. If the frog can reach any point on the ridge of the pond before ...
0
votes
1answer
40 views

A weird Calculus of Variations problem

I became stuck with the following Calculus of Variations problem. The problem is related with something called as the "Nadaraya-Watson" model in statistics. We have $N$ inputs ${x_n}$ and each of ...
0
votes
0answers
33 views

What's the “real” definition of variational derivative

This seems a notational question. Given a functional $S[f]=\int L(f,f';x)\ dx$, I want to derive $\delta S[f]$. There are quite a lot of literature interchanging integrate and variation. That is, ...
0
votes
0answers
28 views

How to take derivative of integral of square matrix function

I have a function as following $$F=\int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx+ \int |A^TG(x)-C^TJ(x)|^2 (1-H(x)) \, dx+\lambda_1 A^2+\lambda_2 B^2+\lambda_2 C^2$$ where $A^T$ is transpose of vector $A$. $A$ ...
0
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0answers
22 views

Number Partitioning of summands

So, I need to partition the number 133 in 1, 2 and 3. Like $$133 = 128*1 + 1*2 + 1*3$$ $$133 = 126*1 + 2*2 + 1*3$$ $$133 = 125*1 + 1*2 + 2*3$$ Where I always must use at least one 1, 2 or 3. I ...
0
votes
0answers
14 views

Proving decoupling between generalized coordinates in the lagrangian

Say you have a lagrangian $F$ for a system: $J[y,z] = \int_a^bF(x,y,y',z,z')dx \tag{1}$ If y and z are associated with two parts of the system and the parts are distant enough that the interaction ...
1
vote
1answer
30 views

Extremizing the boundary value problem $I[y]=\int_0^1y'^2(x)\,dx+y^2(0)-2y^2(1)$

Extremizing the boundary value problem $$I[y]=\int_0^1y'^2(x)\,dx+y^2(0)-2y^2(1)$$ My Thought: First, we use Euler-Lagrange equation and solving we get , $y(x)=C_1x+C_2$. Then we put it in ...
0
votes
2answers
39 views

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 ...
0
votes
1answer
16 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 ...
1
vote
1answer
24 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 : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
0
votes
1answer
13 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 + ...
1
vote
0answers
10 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 $A(x,y), a(x,y), q(x,y)$ ...
2
votes
0answers
65 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 ...
1
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1answer
34 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, ...
0
votes
1answer
17 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 ...
1
vote
1answer
44 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 ...
1
vote
1answer
34 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 ...
3
votes
0answers
23 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. ...
0
votes
0answers
19 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
votes
1answer
31 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 ...
-1
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0answers
22 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
46 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
14 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
38 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
27 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
75 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 ...
4
votes
0answers
32 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 ...