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

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5
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1answer
63 views

Show that $ M$ is constant on $[a,b]$ (variational calculus)

Let $F:\mathbb{R}^n\times \mathbb{R}^n \to \mathbb{R}$ be $C^2$ on $[a,b]$ and $u$ be a solution for the Euler-lagrange equations for the functional given by $$J(u) = \int F(u(t),\dot{u}(t)).dt, $$ ...
1
vote
1answer
315 views

Determine the minimum and maximum values of an integral subject to end conditions

The question: determine the minimum and maximum values of the integral $$\int_0^1 yy'dx$$ subject to the conditions $y(0)=0$ and $y(1)=1$. There is no explicit y dependence, so our Euler-Lagrange ...
0
votes
1answer
105 views

Find the Minimum value of a Functional Constrained to end-point Conditions

The question: Find the minimum value of $\int_0^1 y'^2 dx$ subject to the conditions $y(0)=y(1)=0$ and $\int_0^1y^2dx=1$. In another question, I proved that, if we have an integral of the form ...
1
vote
1answer
790 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: $\frac{d}{dx}(py') + (q-\lambda r)y = 0$. ...
1
vote
1answer
110 views

Piecewise $C^1$ function is element of $W^{1,\infty}$

Hey I'm confused about the following (apparantly) fact: Let $u:[a,b]\to\mathbb{R}$ a piecewise $C^1$ function, i.e. there exists $a=t_1<t_2<\cdots < t_n = b$ such that $u|_{[t_i,t_{i+1}]} ...
1
vote
0answers
33 views

On the controllability function (minimising a functional)

Consider a system of ODEs $$\dot{x}(t)=f(x(t))+g(x(t))u(t),$$ where $f:\mathbb{R}^n\to\mathbb{R}^n$ and $g:\mathbb{R}^n\to\mathbb{R}^{n\times m}$ are smooth. Let $L:\mathbb{R}^n\times ...
0
votes
1answer
512 views

Why Euler-Lagrange equation does not depend on the second derivative of the function?

Why Euler-Lagrange equation does not depend on the second derivative of the function? I.e. why it's $L[q, \frac{dq}{dt}]$ but not $L[q, \frac{dq}{dt}, \frac{d^2q}{dt^2}]$, neither not $L[q, ...
2
votes
1answer
131 views

Calculus of Variation: small variation in functions

I am reading a mathematical physics book, and I am trying to follow along. In the section about functionals ( $J[y] = \int_{x_1}^{x_2} f(x,y,y',\ldots,y^n)$ ), they let $y(x) \rightarrow y(x) + ...
4
votes
2answers
351 views

fundamental lemma for variational calculus

Is it possible to use the fundamental lemma of calculus of variations in some way in the following case: $F(x,y)$ is a locally integrable function on $\mathbb{R}^n \times \mathbb{R}^n$. We know that ...
0
votes
1answer
51 views

how to solve this optimization problem with functions?

Suppose $\theta\in[0,1]$ and $s(\cdot)$ is strictly increasing in $\theta$ with $0\leq(0)<s(1)\leq1$ and $s(0)+s(1)>1$ How could I solve the program, $\max_{s(\cdot)\in ...
3
votes
1answer
86 views

Motivating the Legendre Transform Mathematically

If I begin with a functional of the form $$J[y] = \smallint_a^b f(x,y,y')dx$$ & find it's Euler-Lagrange equations $$\tfrac{\partial f}{\partial y} - \tfrac{d}{dx}\tfrac{\partial f}{\partial ...
2
votes
1answer
64 views

Is a geodesic the least curved path?

It is clear that, in $\mathbb{R}^n$, straight lines are the lines with minimum possible curvature. That is, given the Frenet-Serret ($n$-dimensional equivalent) matrix, and taking its squared norm, ...
1
vote
1answer
77 views

To show that $ J(u) = \frac{1}{2}\int_\Omega |\nabla u|^2 -\frac{1}{p+1}\int_\Omega |u|^{p+1} $ is not bounded above for $1 < p < 2^*-1$

For a bounded $ \Omega\subset\mathbb{R}^n $ with smooth boundary, and for $ 1 < p < \frac{n+2}{n-2} = 2^* -1 $ where $ \frac{1}{2^*} = \frac{1}{2}-\frac{1}{n}$, I have the functional $ J : ...
0
votes
2answers
472 views

Moment of inertia about center of mass of a curve that is the arc of a circle.

Let $(x(s),y(s))$ be a smooth 2-d plane curve which is an arc of a circle of a certain radius $r$. Assume it is represented by an inelastic string $S$ of finite length, lying in a 2-d plane. Let there ...
1
vote
0answers
120 views

A maximization problem in Sobolev space

For $k>0$, let $f_k$ be a sequence of positive functions in $H_N^1(0,1)$, where $H_N^1(0,1):=\{u\in H^1(0,1)|u^{'}(0)=0=u^{'}(1)\}$, $H^1(0,1)$ is the usual Sobolev space consisting of $L^2(0,1)$ ...
2
votes
2answers
68 views

How does one evaluate this function of several variables?

In deriving the Euler-Lagrange equation, one step involves evaluating this: $$\frac{\partial f(y(x)+\alpha\eta(x), y'(x)+\alpha\eta'(x), x)}{\partial \alpha}$$ (this is from pg. 220 of 'Classical ...
2
votes
1answer
45 views

Lagrangian's arguments only up to the first derivative

A question related to a previous question I've asked. I am wondering why in QFT the arguments of the Lagrangian only go up to the first derivative? I remember hearing someone mention that it has to ...
0
votes
0answers
149 views

Linear programming with countably “infinite variables” and “finite constraints”!

Is it possible to do a linear programming with countably "infinite variables" and "finite constraints"? If not, what do you purpose? (Example Link): Maximum and minimum of an integral under integral ...
0
votes
1answer
426 views

first variation of function defined by an integral

Let $f$ be a function defined by $f(x) = \int_0^x \sin \phi(t) dt$. What is the first variation $\delta f(x)$ and how it is calculated?
3
votes
1answer
93 views

Regularity for this variational problem

The Problem. Assume $\Omega \subset \mathbb{R}^2$ bounded and $u \in H^1(\Omega,\mathbb{C})$ is some fixed function. Now consider the variational problem $$ F_\lambda(v) = \frac{\lambda}{2} ...
1
vote
1answer
84 views

Mathematical question concerning Lagrange multipliers of a Lagrangian

In Lagrangian Mechanics I have in general holonomic constraints of the form $f(q_1,...,q_n,t)=0$ and then I am able to use the method of Lagrange multipliers, where I go from a Lagrangian $L$ to a ...
6
votes
3answers
201 views

Beach Path math question

Anyone who has walked on the beach knows that walking speed is dependent upon how far away from the ocean one walks. If you walk on the wet sand you can walk much more quickly than if you walked on ...
0
votes
1answer
150 views

How to draw or construct a brachistochrone

Since the brachistochrone is such a beautiful curve in our planet, I want to build one somewhere around 1.60 m high. I need a quick way to trace the curve on the material to be cut, e.g. a wide sheet ...
5
votes
0answers
610 views

Maximum and minimum of an integral under integral constraints.

Find the maximum and minimum of the following integral in terms of $f(x),a,C$: \begin{align}I=\int_{0}^{a} \frac{x}{f(x)}p(x)dx \end{align} s.t.: 1) $\int_{0}^{a} p(x)dx=1$ 2) $\int_{0}^{a} ...
0
votes
1answer
64 views

I need clarification on $\delta$ - derivative

Please can someone tell me more about $\delta$ -derivative ($\delta=x\dfrac{d}{dx}$) as it appears in the Hadamard definition of frational derivative or elsewhere. Why, when or where we use it. Does ...
3
votes
1answer
97 views

geodesic metric

I'm trying to prove that the line segment is the minimizer of the distance $$d(x,y)=\inf l(\gamma),$$ where $x,y\in X$, $X$ is a Banach space, $\gamma$ is a path from $x$ to $y$ and ...
4
votes
1answer
369 views

Assumption that $\delta q'$ is small in the derivation of Euler-Lagrange equations.

I have never completely understood the justification of this step in the derivation of the E-L equation: $\delta L = L(q + \delta q, q' + \delta q', x) - L(q, q', x) = \partial_q L \delta q + ...
3
votes
1answer
311 views

Maximize a functional

Please help me how to deal with maximization of functional like this: $$F\{a(s)\} = \int\limits_0^t \left( g(a(s)) - \alpha\, v(s)^2 \right) ds, \ a(s) \in \left[0, \infty\right)$$ where $g(x) = x ...
1
vote
1answer
58 views

Restriction of a lower semi-continuous functional again lower semi-continuous?

Let $F: [a,b]\times \mathbb R \times \mathbb R \rightarrow \mathbb R$ be continuous, $J(u) = \int_{[a,b]} F(x, u, u') dx$ be a functional over $W^{k,p}([a,b])$. We assume that for any uniformly ...
0
votes
1answer
87 views

Why is this homotopy an isotopy?

I am trying to follow the proof of Willem's quantitative deformation lemma and I get everything except the justification for the (iv) property which states: $\eta_t(u)$ is a homeomorphism for ...
2
votes
2answers
80 views

Showing that this Coercivity condition implies uniform boundedness of a minimising sequence.

The following problem is in Dacorogna's book "Introduction to the Calculus of Variations": Let $\Omega\subset\mathbb{R}^n$ be open and bounded with a Lipschitz boundary. Let $f\in C(\mathbb{R}^n\times ...
3
votes
1answer
107 views

Maximizing score in number-guessing game.

This is inspired by a puzzle (related to the two-envelopes problem) that I've seen in several places, including unbounded generalizations. The basic premise is that Alice chooses two real numbers ...
2
votes
1answer
119 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
113 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
88 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
69 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
115 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
121 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
561 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
86 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
4k 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
429 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
42 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
115 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 ...