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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1answer
118 views

Constructing shortest interpolation curve from points in $\mathbb R^2$ with parametric equations.

Assume we are given a set of $n$ points from $\mathbb R^2$, $(x_1,y_1),(x_2,y_2)\dots(x_n,y_n)$. We want to construct a path connecting all these points using a pair of parametric equations ...
1
vote
0answers
112 views

plotting the lagrangian

From the differential equation $$\frac{\partial P}{\partial r} = \left[1+\frac{r}{\ln(1+r)}\right]D$$ I get the second-order equation $$\frac{1}{D}{P(r)}=\text{Ei}\left(2\ln(r+1)) - ...
0
votes
1answer
86 views

How to include prescribed boundary conditions in the Ritz Method

Using the Ritz method to find the displacement field in structural analysis can be done as follows. $U$ and $V$ are recpectively the internal and external energy components of a given structural ...
0
votes
1answer
67 views

Consider the following functional which is as follows:

I am stuck on the following problem: I tried using Euler's formula which is as follows: But my calculation gets complicated and I could not get the results. Can someone help me in this regard? ...
0
votes
1answer
49 views

$\mathcal{F}$ $W^{1,1}-seq.w.l.s.c.$ then $F$ convex?

I'm a little bit stuck with the following little exercise: If I know $\mathcal{F}$ is sequentially weakly lower semi-continous in $\{u \in W^{1,1}, u(0) = a, u(1) = b\}$, then I know that $F(tp_1 + ...
0
votes
1answer
111 views

Condition or Proof: Minimizer of one function is maximizing another function

I have two real functions $f(X),g(X)$ where the argument $X$ is a real matrix. The solution $X^*$ for the problem of minimizing $f$ is ending up maximizing $g$ as well. I am looking for a way to prove ...
2
votes
1answer
117 views

Does setting derivative to zero suffice always for minimization of convex functions?

I have this convex function in $X$, given by $Trace(AX^TBX)$ where $A$, $B$ are p.s.d and all entries are real. Now if I had a linear function $l(X)$ that prevents a trivial zero-matrix solution for ...
1
vote
1answer
86 views

Preserving the value of a functional after a small variation

In several (good) textbooks in calculus of variations one important step when dealing with the isoperimetric problem seems not to be properly addressed. The problem is as follows: let $G$ be a smooth ...
2
votes
1answer
80 views

Interpretation of the variational principle for the Ritz approximation, solid Mechanics

Below $U$ and $V$ are recpectively the internal and external energy components of a given structural element: $$U+V=W$$ Expressing $U$ in terms of the strains $\varepsilon$ and the material ...
2
votes
1answer
526 views

Does weak lower semi-continuity follow from convexity for complex valued functions?

I have the following problem: The definition if weak lower semicontinuity states A function $I[\cdot]$ is weakly lower semicontinuous on $W^{1,q}(U)$ provided \begin{equation} I[u] \leq ...
1
vote
1answer
74 views

Question about $C^2$ functional

i have this problem : The solutions of P correspond to critical points of the fuctional $$\phi(x)=\frac12 \int_0^{2\pi} |x'|^2 dt - \int_0^{2\pi} F(t,x) dt , x\in E $$ where $F(t,x)=\int_0^x ...
3
votes
2answers
85 views

Deducing Euler Equation

From Sydsaeter / Hammond (Further Mathematics for Economic Analysis, 2008, 2nd ed., p. 293): $$ \max \int\limits_{0}^T [N(\dot{x}(t)) + \dot{x}(t)f(x(t))] e^{-rt} dt $$ where N and f are $C^1$ ...
10
votes
2answers
3k views

Conceptual difference between strong and weak formulations

What are the conceptual differences in presenting a problem in strong or weak form? For example for a 2D Poisson problem the strong form is: \begin{split}- \nabla^2 u(\pmb{x}) &= ...
0
votes
1answer
415 views

Euler–Lagrange equation (changing variable)

Create the Euler–Lagrange equation for the following questions (if it's necessary change the variables). $$\tag{1}\int_{y_1}^{y_2}\dfrac{x'^2}{\sqrt{x'^2+x^2}}\,\mathrm{d}y$$ ...
1
vote
1answer
39 views

An Area Variation Question

I have a question on a non-typical "area" variation question. Let (M,dA) be a 2 dimensional manifold and f be a smooth function. Let $\Gamma$ be a compact 2- dimensional submanifold in M whose ...
2
votes
0answers
67 views

Find u that minimizes the integral mean

How do I find $u : [0,\infty) \to \mathbb{R}^m$ that minimizes \begin{equation} J(u(\cdot)) = \lim_{t \to \infty} \frac{1}{t} \int_0^t L(x(\tau),u(\tau)) d \tau, \end{equation} subject to ...
1
vote
1answer
40 views

Fastest path for $A$ to catch $B$

I was wondering if is there a way to attack with Euler-Lagrange equation the following problem. Suppose $B$ is moving in straight line with costant velocity $\mathbf{u}=u\,\hat{x}$. What is the ...
2
votes
1answer
110 views

Basis for solution space of Jacobi accessory equation

The Jacobi accessory equation has importance as a means of checking candidates for functional extrema. A book of mine ($\textit{Calculus of variations}$, by van Brunt) proves that we can find ...
3
votes
1answer
549 views

Geodesic and Euler - Lagrange equation

If we have a metric $g_{\mu \nu}$, defined in a Riemannian manifold we can write the equation of the geodesic: $$\frac{d^2x^\mu}{dt^2}+\Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt}$$ ...
3
votes
0answers
283 views

Transversality condition equation

I'm somewhat baffled: I have a problem in calculus of variations: $$ \int_0^T \!(x-\dot x^2)dt,\qquad x(0)=0,\qquad x(T)=T^2-2. $$ Let $ F(t,x, \dot x) =x-\dot x^2. $ I calculate all the ...
1
vote
1answer
77 views

Maximization of the product of two inner products

I have an optimization problem of the form $\max_\gamma \langle f\circ\gamma,w_f\rangle\langle g\circ\gamma,w_g\rangle$ where $f\circ\gamma$ is the composition of $f$ and $\gamma$ and the inner ...
4
votes
1answer
53 views

Existence of variation

Let $I[w] =\int_U L(Dw,w,x) dx$. Let $1<q<\infty$, and there exist constants $\alpha>0$,$\beta\ge0$ such that $$L(p,z,x)\ge \alpha |p|^q - \beta$$ This implies that if $I[w]$ exists, $$I[w] ...
8
votes
2answers
483 views

Finding Euler-Lagrange equations

Maybe you can help here. There is kind of a lengthy setup to understand what the question is asking. There is a paper I'm reading, and in one section of it I can't make heads or tails of the result. ...
1
vote
1answer
83 views

Bolza example like Question

I have to find $u$ minimizing $\int_0^1 F$ with $F(x,u(x),u'(x)) = (1-(u'(x))^2)^2+(u(x))^2$ with $u(0) = 0$ and $u(1) = 1$. I'm relatively new to CoV and got told i should try ...
2
votes
2answers
110 views

Vector Field Generating Variation Along Curve

I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following. Suppose ...
1
vote
1answer
581 views

Curvature and Torsion problem

Calculate the curvature and torsion of $$x= e^t\sin(t),\quad y= e^t\cos(t),\quad z= e^t$$ I'm not sure if I am doing this correctly since I am getting quite complicated results. But I understand ...
3
votes
2answers
104 views

Euler lagrange equation is a constant

I'm working through exercises which require me to find the Euler-Lagrange equation for different functionals. I've just come across a case where the Euler Lagrange equation simplifies to $$1=0.$$ ...
1
vote
1answer
205 views

Who was responsible for finding sufficient conditions for functional extrema?

In the calculus of variations, there is a well-known sufficient condition for weak functional extrema, involving conjugate points and the strengthened Legendre condition ($f_{y'y'} > 0$). Who was ...
2
votes
0answers
75 views

Calculus of Variations statement of a Singular Value Decomposition?

My previous question on SVD gained very little traction, so I thought I'd try a different version that hopefully has an explicit solution. As noted in the linked question, I am taking a function of ...
2
votes
1answer
687 views

Finding the Euler Lagrange equation - differentiation

I'm teachin myself the basics of Calculus of variations. So far I know how to calculate the Euler Lagrange equation for simple functionals. I'm now stuck on how to compute the total differentiation ...
3
votes
1answer
85 views

Local and global extremes

I Wrote problems and solutions, I need just few explanations. 1.Let $$J(x)=\int_{0}^{1}x'^{2}dt,\quad x(0)=0, x(1)=1. $$ Find the extrema value for $J$. I'm doing this using Euler equation ...
4
votes
0answers
109 views

Levi-Civita Connection for 2-dimensional Riemannian manifold

I'm trying to show the following. Suppose $(M, g)$ is a $2$-dimensional Riemannian manifold with connection $\nabla$. Suppose also that $\nabla$ is metric compatible, and that length extremizing ...
1
vote
1answer
147 views

Second variation positive definite but not weak local minimum?

Consider a functional $J: S \to \mathbb{R}$ where $S \subseteq C^2[a,b]$. Let $J(y) = \int_a^b f(x,y,y') \, dx$, let $y$ be an extremal (solution to the Euler--Lagrange equation) for $J$, and suppose ...
3
votes
1answer
96 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
244 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
419 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
358 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
votes
0answers
175 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
votes
1answer
129 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
votes
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
votes
1answer
206 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
147 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 ...
0
votes
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
478 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
62 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 ...
0
votes
1answer
100 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
votes
1answer
860 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
77 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\ ...