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

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3
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1answer
89 views

Analysing functionals having no local extrema

In the calculus of variations, how do we analyse functionals for which there are no local extrema? In basic calculus, functions not having local extrema can often be seen to diverge to an infinity ...
4
votes
1answer
235 views

Multiple Integral Equation

$$f(x) = 2a \int_{0}^{x}{f(t)\;dt} - \left(\frac{b^2}{2}\right)\int_{0}^{1}{|x-t|f(t)\;dt}$$ where $0<a<b$ My task is to solve for $f(x)$. I'm having difficulty solving this integral equation. ...
2
votes
1answer
404 views

Sufficient conditions for functional extrema

In the calculus of variations, we can develop a sufficient condition for a functional $J: S \to \mathbb{R}$, $$J(y) = \int_a^b f(x,y,y') \, dx$$ to have a local maximum or minimum, where $S \subseteq ...
3
votes
2answers
112 views

Derive the solution to the Lagrangian $ \mathcal L= y(x)\sqrt{1+y'(x)^2}$

I am supposed to derive the solution to the Lagrangian $$ \mathcal L= y(x)\sqrt{1+y'(x)^2}$$ Unfortunately I am unable to solve both, the Euler Lagrange equation or the Beltrami equation. It may be ...
4
votes
1answer
354 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 ...
4
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0answers
172 views

Green's function for third order boundary value problems

How to find the Green's function $G(t,x)$ for the BVP consisting of the equation : $$u'''(t)=0 , \quad t\in (0,1)$$ and BC : $$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$ where $\frac12 < ...
0
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1answer
126 views

Gateaux derivative

I have the following definition of Gateaux differentiability $f$ is Gateaux differentiable at $x_0$ if there is a continuous and linear operator $T$ so that $$ \lim_{t \rightarrow ...
0
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1answer
24 views

Identity between functions

Let $f$ and $g$ be continuous functions of one real variable. We want to show that $\frac{\mathrm{d}}{\mathrm{d}t}f = g$ on the interval $[a,b]$. I have shown that for any subinterval $[t_a,t_b] ...
0
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1answer
199 views

Lagrange Multipliers and max area

You have a straight line of length $b$. You want to connect the ends of this fence so as to enclose a maximum area. You have a cost constraint. In the area between $x=0$ and $x=\frac{b}{2}$ costs $1$ ...
2
votes
1answer
143 views

Why is the weak limit of the derivatives the derivative of the weak limit here?

In [1, chapter 8.2.1.b, p.466] the author uses the following argument: Let $U \subset \mathbb{R}^N$ be an open, bounded domain with smooth boundary. Given a bounded sequence $(u_k)_{k \in ...
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1answer
64 views

Constrained variational calculus: Are we allowed to make use of the constraint before taking variations?

Suppose that we have a variational problem, $\int_{t_1}^{t_2}f(\vec{x}(t),\vec{x}'(t),|\vec{x}(t)|)dt$ subject to the constraint: $|\vec{x}|=1$ where $\vec{x}(t)=\left\{x_1(t),x_2(t),x_3(t) ...
5
votes
1answer
468 views

Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
0
votes
1answer
60 views

Euler-Lagrange Query

Given F: $$ F(x,y,y\prime) = 2\cdot \pi \cdot y \cdot \sqrt{1+(y\prime)^2} $$ We can derive the following Euler-Lagrange equation (I know how to do this part): $$ \frac{d}{dx}\left(\frac{y\cdot ...
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1answer
99 views

Simple question on calculus of variations: critical point of functional subject to constraint

Let $V$ be the set of smooth functions $f:[0,1]\to \mathbb{R}$ such that $\int_0^1 f(t) dt =k$. If $F:V\to\mathbb{R}$ is given by $F(f) = \int_0^1 f(t)^2 dt$, show that the only critical point of $F$ ...
2
votes
1answer
68 views

Finding the critical point of $\int_0^1(f(t))^2dt$ subject to $\int_0^1f(t)dt=k$

I wish to find the critical point of the functional $F[X]=\int_0^1(f(t))^2dt$ subject to $\int_0^1f(t)dt=k$ for a constant $k$. I read something about using a Lagrange multiplier to convert it to a ...
3
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1answer
831 views

Taylor series with functions as parameters (as opposed to variables)

I'm doing my own research on the Euler-Lagrange equation and came across a proof in van Brunt's textbook "The Calculus of Variations". However, there is something I don't quite understand. Here is an ...
1
vote
1answer
76 views

Prove a transformation is a variational symmetry for J

The following problem is from The Calculus of Variations by B.von Brunt (page 215, Exercise 9.2.1) Let $$ J(y)=\int_a^b xy'^2\mathrm{d}x. $$ Show that the transformation $$ X=x+\epsilon2x\ ...
2
votes
1answer
98 views

Formal Variational Calculus Reference Request

I want to ask for a reference to study Variational Calculus from a formal point of view. What I mean is that many of the references that I've found are inside Physics books, and the authors do not ...
0
votes
1answer
123 views

Minimum calculus of variation

Hi I am looking for a criterion that is sufficient to prove that a solution to a functional depending on two functions y(t) and x(t) is an extremum. it is about the following functional$$ \int_0^b ...
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1answer
102 views

Calculus of Variations-question on rotating curve of max volume

My calc of variations is still rusty. I'm assuming implementation of arclength revolution formula is necessary, but how to find y(1/2a)?
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0answers
55 views

Calculus of Variations - Circles with soapy membrane problem [duplicate]

Calculus of variations is coming to me at a crawl pace. Here is a problem on my agenda that I wanted to get solved, but am not quite sure how to approach. I've been thinking about it for a while ...
11
votes
3answers
642 views

Calculus of variations: find $y(a/2)$ if $y(x)$ maximizes the volume of rotation

A curve $y(x)$ of length $2a$ is drawn between the points (0,0) and (a,0) in such a way that the solid obtained by rotating the curve about the $x$-axis has the largest possible volume. Find ...
2
votes
2answers
737 views

satisfy the Euler-Lagrange equation

Two circles of unit radius, each normal to the line through their centers are a distance d apart. A soap film is formed between themas shown below; energetic considerations require the filem to ...
2
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1answer
72 views

Show that the path of shortest hyperbolic length satisfies $(x-c)^2+y^2=r^2$

The hyperbolic length of a curve $y:[a,b]\rightarrow\mathbb{RxR}_+$ is given by the functional $$\lambda(y)=\int_a^b\frac{\sqrt{1+y'^2}}{y}dx$$ Show that the path of shortest hyperbolic length ...
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vote
2answers
147 views

Show that the infimum of a functional is zero, but this infimum is never achieved.

Show that the infimum of the integrals $$\int_0^1(y'^2(x)-1)^2dx$$ among all $y(x)\in C^2[0,1]$ such that $y(0)=y(1)=0$, is zero, but is not achieved by any function in this set. What I've worked on: ...
4
votes
3answers
442 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 ...
2
votes
2answers
215 views

In calculus of variation: why are minimizing sequences bounded?

Assume the usual variational setting: Let $\mathcal{A} \subset W^{1,q}$ be the set of admissible functions and \begin{equation} I: \mathcal{A} \to \mathbb{R} \end{equation} the functional that needs ...
0
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1answer
255 views

Weak Minimizer of a Functional

I showed that $u(x) = \frac{x^2}{2}$ is a potential minimizer for the functional $\int_0^2 \frac{n}{2}u'(x)^2-nu(x) \, dx$ in $C^2[0,2]$ with $u(0) = 0$ and $u(2)=2$ where $n$ is a positive constant ...
1
vote
1answer
130 views

How to check if stationary point is extremum?

In this question the solution of Euler–Lagrange equation is $y=x$ function. $L = (y')^3$ so $L''_{y'} = 6y'$ and is positive when $y=x$. But from the answer of Emanuele Paolini follows that it is ...
2
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0answers
133 views

Gamma Convergence of functionals on Probability measures

Would be grateful if someone could provide a hint or an appropriate reference for the following. Notation: $\mathcal{P}(\mathbb{R}^n)$- Space of probability measures on $\mathbb{R}^n$ ...
1
vote
1answer
210 views

How to check if a function is minimum to functional?

Given $\int_0^1(y')^3dx$ functional and $y(0) = 0 ,y(1)=1$ conditions. Using Euler–Lagrange equation I have got $y(x)=x$. So $y$ is a stationary point of the functional. How to check if it is the ...
3
votes
1answer
181 views

Variational calculus

In variational calculus how many ways are there to define a variation, i.e. can it only be $$ \delta F(x) = \bar{F}(x) - F(x) \mbox{ , where } \bar{F}(x) = F(x + \delta x)$$ or is there another form? ...
3
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0answers
76 views

Different functional brachystochrone

Until today I thought that $$ \int_0^b \sqrt{\frac{1+y'(x)^2}{2gy(x)}} dx$$ would be the only functional to derive the brachystochrone, but in the textbook Variational Methods in Mathematical Physics ...
2
votes
1answer
118 views

Maximum likelihood for $(\mu,\sigma)$ and other related questions

$$f(x)=\frac{1}{2\sigma}\exp\left(\frac{-|x-\mu|}{\sigma}\right)$$ $$\mu\in,\sigma>0$$ When trying to calculate the maximum likelihood for $(\mu,\sigma)$, I got as far as: $\log L(\mu,\sigma)=-n ...
0
votes
1answer
65 views

Convert line integral between different metrics?

If I have $$ \int\limits_0^T \frac{\sqrt{\dot{x}(t)^2+\dot{y}(t)^2}}{\sqrt{2 y(t)}}dt $$ I can convert this problem of finding the solution to the brachistochrone problem to a geometric problem by ...
0
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1answer
114 views

Brachistochrone problem

I am trying to solve the Brachistochrone problem. First of all I was asking myself whether there is any good reason that the curve $ x $ we are looking for is in $C^2[0,t_\text{end}] $. Then most ...
2
votes
1answer
359 views

variational calculus

$f,g$ are continuous function $[a,b]$.Suppose $$\int_a^b{f(t)h(t)+g(t)h'(t)}dt = 0$$ for every $h$ belonging to $C^1[a,b]$ with $h'(a)=h'(b)=0$. Why it is true that (1)$\int_a^bf(t)dt=0$---->This is ...
1
vote
1answer
49 views

Some ideas about $H=W$

Meyers and Serrins theorem says that $H=W$. ie $H^j_p(\Omega) = W_p^j(\Omega)$ . Here the norm of $$\|u\|_{H^j_p(\Omega)} = (\int_\Omega\sum_{|\alpha|\le j}|D^\alpha u|^pdx )^{1/p}$$ where ...
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vote
1answer
145 views

How to write the following expression in index notation?

I would like to know how can I write $ ||\vec{a} \times(\nabla \times \vec{a})||^2 $ and $(\vec{a} \cdot (\nabla \times \vec{a}))^2$ in index notation if $\vec{a}=(a_1,a_2,a_3)$ Thank you for ...
3
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1answer
156 views

Computing the Euler Lagrange equations

Let $F(u) = \int_0^1(u'')^2+u^2dx $ on $C^2[0,1]$ satisfying $u(0)=a,u(1)=b,u'(0)=c,u'(1)=d$ where $a,b,c,d \in \mathbb{R}$. If $u_*$ is a minimizer, for $\phi \in C^2[0,1],\ \frac{d}{ds}| _{s=0} ...
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87 views

Finding a weak minimum

I have been struggling with the following problem I came across in a textbook. I believe that it is necessary to use the Euler-Lagrange Equation. Any help would be greatly appreciated. Let $F$ be the ...
2
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1answer
117 views

Show that the arc length integral is continuous in $C^1$

I came across this question and I am not sure how to prove it. Show that the arc length integral is continuous in $C^1$.
4
votes
2answers
182 views

Finding an explicit expression for a minimizer

Suppose $f$ is a continuous function on the interval (0,1). We consider the energy functional $F(u) = \int^1_0\frac{1}{2}((u')^2+u^2)\,dx - \int^1_0fu\,dx$ which is well defined for continuously ...
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vote
0answers
184 views

How to take the limit of some integral?

$$ f\left( x^{\prime },t+\varepsilon \right) = \int_{-\infty }^\infty dx\int_{-i\infty }^{i\infty } \frac{d\tilde{x}}{2\pi i} \left(1+\varepsilon \left[ \tilde{x}D_{1}\left( x,t\right) ...
8
votes
1answer
356 views

Minimizing Lagrangian with two functions

I read this problem where I have to minimize a functional $E[L]$ using calculus of variations, but I'm not sure what is the procedure to follow. The functional is the expected loss: $$E[L] = ...
5
votes
1answer
391 views

Geometric proof of the geodesics of a sphere?

I have seen the standard variational proof that great circles are the geodesics on the $2$-sphere. Do you know a purely geometric proof of this fact, not involving calculus of variations or ...
4
votes
1answer
259 views

Step in derivation of Euler-Lagrange equations of motion

From http://www.mathpages.com/home/kmath523/kmath523.htm Variations in $x,y,z$ and $X$ at constant $t$ are independent of $t$ (since each of these variables is strictly a function of $t$), so we ...
2
votes
1answer
122 views

Representation for function (“null-Lagrangian”)

Let $L(t,x,p) \in C^m([0,1] \times \mathbb{R}^n \times \mathbb{R}^n;\mathbb{R})$, $m\geqslant1$ and define for any $u \in C^1([0,1];\mathbb{R}^n)$ $$ \mathcal{L}u = ...
2
votes
2answers
2k views

Derivative of $f(x,y)$ with respect to another function of two variables $k(x,y)$

Suppose that we have a function $f(x,y)$ of two variables: $$f(x,y) = g(x) + h(y) + 5(x-y) = x^2 + y^2 + 5(x-y)$$ where $g(x) = x^2$ and $h(y) = y^2$ are also functions of $x$ and $y$, respectively. ...
11
votes
2answers
374 views

Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.

I am working through a book titled "An introduction to mechanics and symmetry" by Marsden and Ratiu. I have written up a brief summary trying to solidify my understanding of the general principles. ...