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

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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 ...
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1answer
30 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}(\...
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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 ...
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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 ...
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25 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?
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106 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 ...
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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 ...
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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 ...
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28 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,...
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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 ...
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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 ...
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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 $aT^{2}/4$ ...
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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/ ...
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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 ...
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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)? $
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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 ...
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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 ...
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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 ...
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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}(\...
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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)+...
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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{\...
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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.
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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.
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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. ...
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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 ...
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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\...
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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 ${\alpha}$ ...
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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 $$ J[y] = \displaystyle\int_a^b F(x,y,y') \, \mathrm{d}x, \tag{1}$$ ...
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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 \...
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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 ...
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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 $y(a)=a'$ ...
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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}{\...
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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 ...
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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 $...
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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 ...
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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 $z \...
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$\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 [...
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1answer
42 views

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

I'm bit confusing about definition of weak solution. If I have the following problem: $\begin{cases} \tag{P} -\Delta u = f \textrm{ in } \Omega, \\ u = 0 \textrm{ in } \partial\Omega, \end{cases}$ ...
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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 $$ \iiint_V (u_x^2+u_y^2+u_z^2) \, dx \, dy \, dz $$ ...
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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 $(y')^2+3yy''=0.$...
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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')....
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Natural boundary and natural transient conditions of $F(x, y, y', y'')$

In view of finding the extreme points of the integral $$I=\int _a ^b F(x,y,y',y'') dx$$ what natural conditions should be used if $y(a), y(b), y'(a), y'(b)$ are not preassigned? Also, if $F$ is ...
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1answer
38 views

Integral with variation (principle of least action)

I have \begin{equation} \delta S=\delta \int_{y_1}^{y_2}\bigr[p_x\frac{dx}{dy}+(-E)\frac{dt}{dy}-(-p_y)\bigr]dy \end{equation} with fixed values at the limits on x(y) and t(y), and the teacher asks me ...
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Uniqueness of solution to Helmholtz-style equation

Let $\Omega\subset \mathbb{R}^n$ be open, bounded and connected with $C^1$ boundary. Suppose $q\in L^\infty(\Omega)$ and that $q\geq C>0$ almost everywhere. Consider the following PDE problem: $$-...
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41 views

Find the curve connecting $(x_1,y_1)$ to $(x_2,y_2)$ that minimizes the surface area of the volume of revoluion

Given two points $(x_1,y_1)$ and $(x_2,y_2)$, find the curve $\gamma$ connecting them such that the surface area of the volume obtained when rotating the curve along the $x$-axis is minimized. First ...
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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~?$...
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1answer
33 views

Simplifying trick for extremizing functionals

If I have a functional $I[y] = \int{({y^\prime}^2- 1)^2}dx$ , since $f(x) = x^2$ is an increasing function for $ x > 0$, can I make the conclusion that $I[y]$ is extremized for at the same ...
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1answer
47 views

Calculating the time of a Brachistochrone

I derived the general equation of a Brachistochrone, which is a cycloid. $y=A(1-\cos\theta)$ $x=A(\theta-\sin\theta)$ I'm now trying to calculate the time needed to go from the origin to a point $(...
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1answer
38 views

Generalization of the Beltrami identity to functionals with higher derivatives

Suppose that I have a functional $S[q]=\int_a^b L(t,q(t),q'(t))\,dt$. Such a functional is well-known to extremized by a choice of $q(t)$ satisfying the Euler-Lagrange equation $\dfrac{d}{dt}\dfrac{\...
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1answer
54 views

4th order differential equation from Euler-Lagrange

I am trying to extremise the functional $\int{[y + \frac{1}{2}y^2 - \frac{1}{2}(y^{''})^2]}dy$ and so from Euler-Lagrange I get the differential equation $1 + y + y^{(4)} = 0$ and I have no idea how ...