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

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Best Differential Equation, Partial Differential Equation and Calculus of Variations books?

Electrical Engineer here thinking of switching to physics. What are the best Differential Equation, Partial Differential Equation and Calculus of Variations books? Ideally they explain the topic ...
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what is difference between variations of the work and virtual work

virtual work part https://en.wikipedia.org/wiki/Virtual_work I really want to know that both equations are same or not.(mathematically)(I think that they are the same.) Thanks for reading.
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Pontryagin's maximum principle

So I've been doing some optimal control theory lately. It's really interesting but I've spent the whole day trying to wrap my head around pontryagin's maximum principle. There's a lot of mathematical ...
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Geodesics using Euler-Lagrange

thanks for taking a look at my question. This is a homework problem from a section covering Euler-Lagrange equations. I'm asked to consider the arc length formula: $S = \int\limits_{{t_1}}^{{t_2}} {...
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Finding solution to Calculus of Variation of linear functional whose domain consists of vector valued function

Problem Statement: Find $x^*$ such that it solves the optimization problem $$\max_{x \in \Omega} \quad f(x) = e_i^TAx$$ $$ \Omega = \{x: t \to \Delta^{n}|x \in C^1, x(0) = x_o\}$$ Where $\Delta^...
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Taylors theorem for second variation

In Hilberts Methods of mathematical physics (p. 214), a functional $$J[\varphi] = \int_{x_0}^{x_1} F(x, \varphi, \varphi') \mathrm dx$$ is expanded by Taylor's theorem $$J[\varphi + \epsilon \eta] =...
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119 views

Two Approaches Two Different Solutions: Optimal Controls vs. Different Method

If I try to solve a problem two different ways, I get two different answers which generally means I am committing some horrible sin! Given the problem, \begin{align} \min_u\ S &= \int dt\ L(x, u) ...
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curve extremizing the functional

Let $y \in C^([0,\pi])$ satisfying $y(0)=y(\pi)=0$ and $\int_0 ^\pi y^2(x)dx=1$ extremizes the functional $J(y)=\int_0^\pi (y'^2(x))dx$ then $y(x)=\frac{\sqrt{2}}{\pi} \sin x$ $y(x)=-\frac{\sqrt{2}...
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Why is it useful to show the existence and uniqueness of solution for a PDE?

Don't get me wrong, I understand that it is important in mathematics to qualitatively study the problems given. But I would like to know to what extent this helps for example, to actually solve the ...
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Corollary 2.1 in Ekeland and Temam on lower semicontinuity

Why in Corollary 2.1 on page 10 (see the picture) from Ekeland and Temam book Convex Analysis and Variational Problems there is equality in (2.11), i.e why $$\forall u\in V,\quad \overline F(u)=\...
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26 views

Help with derive geodesic equation

Let $(M,g)$ be a pseudo-riemannian manifold and $p,q\in M$. Suppose $\alpha:[a,b]\to M$ a smooth curve on $M$ such that $\alpha(a)=p$ and $\alpha(b)=q$. If we consider: $$L[H(s,\cdot)]=\int_a^b \sqrt{...
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What is $\int \frac{\mathrm{d} f(t)}{\int f(t) \, \mathrm{d} t}\, \mathrm{d} t$? [on hold]

The integral $\int \frac{\mathrm{d} x}{\mathrm{d} y}^{-1} \, \mathrm{d} x$ is $ y + c$ subject to some interesting qualifications about continuity and holes. I think I can evaluate $\int \frac{\...
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Derive geodesic equation

Let $(M,g)$ be a riemannian manifold and $(U,\psi)$ local chart on $M$. If $\alpha=(x^1,\ldots,x^n)$ is a curve on $U$ such that verify: $$\sum_{i,j=1}^n \Big(\frac{1}{2}\frac{\partial g_{ij}}{\...
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Using calculus of variations to find a curve through the end points of a family of curves?

Let $I:=[0, 1]$ and let $\alpha:I\times I\longrightarrow X$ be a family of smooth curves on a smooth manifold $X$. Let us think of $\alpha$ as a family of curves $\{\alpha_s: I\longrightarrow X\}_{s\...
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64 views

How would I solve integrals of derivatives?

Solving integrals of values like $\int f \, \mathrm{d} f $ is simple enough but how would I solve integrals of derivatives? For example consider the integral: $$ \Delta F = \int^b_a \int \frac{\...
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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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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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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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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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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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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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46 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...