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

learn more… | top users | synonyms

0
votes
1answer
24 views

Question about calculus of variation.

What is the difference between finding maxima or mimima i.e. critical point of a function and calculus of variation?
0
votes
1answer
20 views

Question about trace operator

From the general trace theorem we know for instance that if $f\in W^{1-\frac{1}{p},p}(\partial\Omega)$, then there exists a function $f\in W^{1,p}(\Omega)$ such that $f|_{\partial\Omega}=f$. But is it ...
0
votes
0answers
8 views

Momentum Potential Term in Optimization Problem for Implicit Euler Solver

I'm trying to understand the explanation of the implicit Euler solver (Section 3.1) set forth in this paper: Projective Dynamics: Fusing Constraint Projections for Fast Simulation For the purposes of ...
1
vote
1answer
27 views

Vainberg Theorem in measure theory

In a lecture notes about Variational Methdos, I found the following theorem: THEOREM: Let $(f_n)$ a sequence in $L^{p}(\Omega)$ and $f \in L^{p}(\Omega)$, such that $f_{n} \rightarrow f$ in $L^{p}(\...
0
votes
0answers
12 views

gradient of total variation norm in total variation denoising

I am learning total variation denoising. The gradient of TV norm need calculated. From the link: http://www.numerical-tours.com/matlab/denoisingsimp_4_denoiseregul/ It says that the gradient is ...
1
vote
2answers
105 views

Curve enclosing the maximum area

the curve of fixed length $l$ that joins the points $(0,0)$ and $(1,0)$ lies above the $x-axis$ and encloses the maximum area between itself and the $x-axis$, is a segment of A straight line A ...
1
vote
0answers
26 views
+100

Minimum of $F$ over Finite Perimeter Sets in $\mathbb R^N$

Problem: Let $G$ be a bounded Borel set. Let $X$ be the set of finite perimeter sets in $\mathbb R^N$ and $F: X \to \mathbb R \cup \{+\infty\}$ defined as \[ F(E)= \begin{cases} Per(E) \hspace{1,...
5
votes
2answers
62 views

How to solve this functional problem?

I am new to calculus of variations, till now I know how to get the extremal functions for a given functional using Euler-Lagrange equation. What if I have a functional but I am not looking for ...
2
votes
0answers
22 views

More precise trail function in Rayleigh–Ritz method

In order to obtain displacement field of an elasticity problem, say a plate structure, we approximate the solution using trigonometric series with unknown coefficients which satisfy the essential ...
2
votes
1answer
44 views

The plane curve of maximum average speed under constant gravitational force

A line gives us the minimum distance from $A$ to $B$. A cycloid gives us the minimum traveling time of a point mass from $A$ to $B$ (under constant gravitational acceleration $g$). What about the ...
1
vote
0answers
28 views

Derivative of one functional by another functional

I have two functionals, $F(h, \nabla h)$ and $G(h, \nabla h)$. I'd like to calculate $\frac{\delta F}{\delta G}$ Since functional derivatives also follow the chain rule, would I be correct in ...
3
votes
1answer
48 views

Find the Minimum of $ F(u)= \int\limits_{-2}^{+2}|u(x) - \chi_{[0,2]}(x)|dx + |Du|(\mathbb R)$.

Let $F: BV(\mathbb R) \to \mathbb R$ be a functional defined as: \[ F(u)= \int\limits_{-2}^{+2}|u(x) - \chi_{[0,2]}(x)|dx + |Du|(\mathbb R). \] Show that there is no minimum on $W^{1,1}$, but the ...
1
vote
2answers
52 views

Maximising $\int_{0}^{T} v(t)\, dt$, subject to constraints $|v(t)| \leq a; v(0)=0; v(T)=0$

Besides those constraints, we know nothing else about $v(t)$. Interpreting the integral as the distance travelled by a particle, a little geometry tells us that the answer should be $aT^{2}/4$ ...
4
votes
3answers
66 views

Extremizing a functional subject to an equality constraint

Question at hand is: Let $y\in\cal C^2([0,\pi])$ satisfying $y(0)=y(\pi)=0$ and $\int_0^\pi y^2(x)dx=1$ extremize the functional $$J(y)=\int_0^\pi\left(y'(x)\right)^2dx$$ It's an MCQ, and ...
0
votes
0answers
21 views

Interplanetary Optimisation using a simulator with PyGMO or SciPy

I am currently trying to use a N-body gravity simulator to model a spacecraft trajectory and using the simulator as a BlackBox to optimise the trajectory. I am thinking of using basin hopping/ ...
2
votes
1answer
710 views

DuBois-Reymond Lemma

I know thats the following statement is true. $f,g$ are continuous function $[a,b]$.Suppose $\int\limits_a^bf(t)h(t)+g(t)h'(t) \, dt=0$ for every $h$ belonging to $C_0^{\infty}[a,b]$, then $g$ is ...
1
vote
1answer
22 views

Rings of same gravity center

Using calculus of variations or otherwise, how do we find all non-circular ovals of loop length $ 2\pi $ in the plane with its center of gravity of arc at $ (0,0)? $
1
vote
2answers
22 views

Variational Inequalities - What excatly does the definition say? Why are they useful?

I am having issues understanding the definition of variational inequalities. We have the following definition: Given a set $X \subset \mathcal{R}^n$ and a mapping $F: X \rightarrow \mathcal{R}^n$ a ...
16
votes
3answers
6k 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}) &= f(\pmb{x}),\...
1
vote
0answers
30 views

Is this the correct way of using Variational Principle (Minimization Principle)?

I am constructing a smooth function $f(x)\equiv f(u(x),v(x))$, such that $u(x)$ and $v(x)$ are some trial parameters. I have the following integral $$G=\int_{x_i}^{x_f} f(u(x),v(x)) \mathrm{d}x.$$ My ...
0
votes
0answers
15 views

Is it possible to extend Jacobian, gradient, divergence and curl operators to the calculus of variations?

In the calculus of variations one can extend the concept of the vector gradient to functionals or functions of the type $ \left(\mathrm{R} \rightarrow \mathrm{R}\right)\rightarrow \mathrm{R}$ by using ...
1
vote
0answers
24 views

Minimizing the functional $\int (|\nabla u|^2- u^{2}V)$ on the Sobolev space $H^1$

I have a question about a function defined on a Banach space. Let $\Omega$ be a bounded open subset of $\mathbb{R}^{n}$ and $V:\Omega \to [0,\infty]$ a bounded function on $\Omega$. Let $H^{1}(\...
1
vote
2answers
27 views

Optimization inside integral

I want maximize the integral $$\int_a^b \left( 2 cx y(x) - e y(x)^2 \right) \, \mathrm{d}x$$ with respect to to $y(x)$. If I discretize the problem, I get $$ \frac{b-a}{n}\sum_{i=1}^n 2c(i/n(b-a)+...
0
votes
0answers
11 views

What is the constant of integration for the functional antiderivative?

Suppose I have the equation: $$ Q = \varepsilon_0 \int_{\vec{s} \in \partial C} \vec{E} \cdot \hat{n} \, \mathrm{d} A $$ Then the functional derivative is: $$ \oint_{\vec{s} \in \partial C}\frac{\...
1
vote
1answer
525 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 ...
1
vote
0answers
34 views

finding curve along which a function extremizes via theory of calculus of variations [closed]

Consider $$ I(y)= \int \limits _0 ^1 [y'(x)]^2dx \ +y(1)^2$$ with $y$ subsjected to the initial condition $y(0)=1$. Find the equation of curve along which $y$ extremizes.
1
vote
0answers
17 views

A curve in the first quadrant joins (0,0) and (1,0) and has a given area beneath it. Show that the shortest such curve is an arc of a circle. [closed]

This is an isoperimetric problem that I am not sure how to approach. Any insight would be appreciated. Not sure how to find the function that I am trying to maximize or the constraint.
2
votes
0answers
31 views

Calculus of variation problem.

The functional $$\int_{0}^{1}(1+x)(y')^{2}dx,y(0)=0,y(1)=1$$ Possesses $1.$ Strong maxima. $2.$ Strong minima. $3.$ Weak maxima but not a strong maxima. $4.$ Weak minima but not a strong minima. ...
0
votes
0answers
91 views

Determining whether the extremal problem has a weak minimum or strong minimum or both

The extremal of the functional $\int_{0}^{\alpha}{\left((y')^2 - y^2\right)dx}$ that passes through (0,0) and (${\alpha}$,0) has a weak minimum if ${\alpha}$ < $\pi$ strong minimum if ${\alpha}$ ...
1
vote
0answers
24 views

Probability if variable has $15\%$ CV

I have a relatively simple question, but I am not sure if I understand it right. I have estimated through my calculations the value $X$. $X$ depends on many things, but one of them is $Y$ and I know ...
0
votes
0answers
24 views

Intuition for second frechet derivative

I am now used to thinking of the first derivative of a map between vector spaces $f:V\to W$ in the "proper" Frechet sense, as being "the assignment to each point $v$ of $V$ of the linear map $f'(v):V\...
1
vote
2answers
135 views

Calculus of variation with inequality constraints

I want to find the function $y$ which maximizes the functional $J[y] = \int_0^1 g(x) y(x) dx$ subject to $0 \leq y(x) \leq 1$ for all $x\in [0,1]$ and $\int_0^1 y(x) dx = k$ where $g$ is a strictly ...
7
votes
2answers
224 views

Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$ Called the "energy" of $f$? I am ...
1
vote
1answer
48 views

Calculus of variations with inequality and non-integral constraints

I have a question on solving an optimization problem with calculus of variations. I am attempting to maximize the functional $$ J[y] = \displaystyle\int_a^b F(x,y,y') \, \mathrm{d}x, \tag{1}$$ ...
5
votes
3answers
812 views

Find the extremal to the functional $J(y) = \int_{0}^{1} ((y')^2 -y)dx$ and discuss whether they provide a max/min

I am having a hard time getting my head around Functionals and Calculus of Variations, My question is: Given a functional and using the Euler-Lagrange equation to find an extremal, how do we show ...
7
votes
2answers
110 views

Proving that a sphere has a minimal surface to volume ratio using Calculus of Variations

I know the problem is traditionally solved via the isoperimetric inequality, but I was hoping to solve it by minimizing a surface of revolution subject to a volume constraint. The surface area of a ...
1
vote
1answer
40 views

Solution of a sublinear elliptic problem.

In a lecture notes, the author showed the problem $\tag{$P$}$ $\begin{cases} -\Delta u = |u|^{q-2}u \textrm{ in } \Omega, \\ u(x)= 0 \textrm{ in } \partial\Omega, \end{cases}$ where $\Omega \...
4
votes
1answer
60 views

Show that $\eta(z) \det P$ is a null Lagrangian

This is problem 8.7.4 from Evans' PDE book. Assume $\eta: \mathbb{R}^n \to \mathbb{R}$ is $C^1$. Show that $L(P,z,x) = \eta(z) \det P$ is a null Lagrangian. Here $P$ is a $n \times n$ matrix and $z \...
1
vote
2answers
59 views

Alternative proof of Fundamental Lemma of Variational Calculus?

I am confused by one of the proof in the Calculus of Variations by Gelfand and Fomin. On page 9, we have Lemma: If $\alpha(x)$ is continuous on $[a,b]$, and if $\int_a^b \alpha(x)h(x)=0$ for every ...
1
vote
0answers
14 views

Euler-Lagrange Single function of single variable with higher derivatives

Here is the page on Wikipedia: So it says the fixed boundary conditions for the function itself as well as for the first $n-1$ derivatives. You can fix the boundary points physically say $y(a)=a'$ ...
0
votes
0answers
17 views

Obscure first order approximation

I don't understand this first order approximation from Gelfand, Fomin "Calculus of Variation": $$ \int_{x_0}^{x_0 + \delta x_0} F(x, y + h, y'+h') dx \sim F(x,y,y')\big|_{x = x_0}\delta x_0$$ where $...
1
vote
1answer
98 views

How do I determine a cricital point of an area functional?

The orientated area $A(\gamma)$ of a regular closed plane curve $(\gamma, \tau)$ is defined as $$A(\gamma) :=\frac{1}{2}\int_{0}^\tau \det (\gamma,\gamma')$$ Now how can I determine the cricital ...
1
vote
1answer
26 views

When the Euler-Lagrange equation reduces to 0=0

I've gotten the functional $$\int_a^b(y^2+2xyy')dx$$ with Dirichlet boundary conditions. Applying the Euler-Lagrange equation I get: $$0=\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\...
2
votes
0answers
46 views

$\int_a^b M(x)\eta''(x)dx=0$ for all $\eta$ implies $M(x) = c_0 + c_1x$ proof

Here's a lemma that my book proves: If $M(x)\in C[a,b]$ and $$\int_a^b M(x)\eta'(x)dx = 0$$ for all $\eta(x)\in C^1[a,b]$ such that $\eta(a)=\eta(b)=0$, then $$M(x)=c,$$ a constant, for all $x\in [...
0
votes
1answer
41 views

Weak solution in $\mathbb{R}^{N}$

I'm bit confusing about definition of weak solution. If I have the following problem: $\begin{cases} \tag{P} -\Delta u = f \textrm{ in } \Omega, \\ u = 0 \textrm{ in } \partial\Omega, \end{cases}$ ...
0
votes
1answer
31 views

Explanation of this proof of the Euler-Lagrange

(Please note that I am a physics student, so please try to avoid rigorous mathematics in the explanation.) I have the following proof for the Euler-Lagrange: Consider the integral $$I=\int F(x,y,y')....
1
vote
1answer
32 views

Find critical points of the functional $I[y] = c \int_0^L y(y')^3 dx$

Find critical points of the functional $I[y] = c \int_0^L y(y')^3 dx$ with $y(0)=0$ and $y(L)=R$ Euler-Lagrange equation: I arrive at $(y')^3+3yy'y''=0$ and so solve $y'=0$ and $(y')^2+3yy''=0.$...
1
vote
0answers
31 views

Calculus of Variations Open ball constraint

I'm stuck on this question, Let $V$ be an open ball in $\mathbb R^3$ in $ { x,y,z \text{ st }x^2 +y^2 +z^2<1}$ I need to minimise the integral $$ \iiint_V (u_x^2+u_y^2+u_z^2) \, dx \, dy \, dz $$ ...
2
votes
2answers
74 views

How to find the minimum value of following integral?

Let $A$ be the set of twice differentiable functions on the interval $[0,1]$, and $$B=\{f \varepsilon A: f(0)=f(1)=0, f'(0)=2\}.$$ What is $${\rm Min}_{f\varepsilon B}\int_{0}^{1} (f''(x))^2 dx~?$...
3
votes
2answers
243 views

Intuition behind variational principle

Hofer-Zehnder, in section 1.5, proves that every Hamiltonian field on a strictly convex compact regular energy surface carries a periodic orbit. I have understood the proof. What I am wondering about ...