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Questions tagged [calculus-of-variations]

This tag is for problems relating to the calculus of variations that deal with maximizing or minimizing functionals. This problem is a generalization of the problem of finding extrema of functions of several variables. In fact, these variables will themselves be functions and we will be finding extrema of “functions of functions” or functionals.

The calculus of variations seeks to minimize or maximize an entire functional's worth of parameters instead of changing just one parameter. It achieves this by applying standard calculus techniques to the integral of a functional, thereby reducing $\mathbb R \to \mathbb R$ to just one parameter $\in\mathbb R$.

In symbols one considers $\displaystyle\max \int f(x) \ker(x)$ $dx$ rather than $\max f(s)$.

Two famous applications of the calculus of variations are the brachistochrone problem and deriving the catenary shape of a rope hanging between two poles.

Some basic problems in the calculus of variations are:

$(i)$ find minimizers

$(ii)$ find necessary conditions which minimizers must satisfy

$(iii)$ find solutions (extremals) which satisfy the necessary conditions

$(iv)$ find sufficient conditions which guarantee that such solutions are minimizers

$(v)$ qualitative properties of minimizers, like regularity properties

$(vi)$ how do the minimizers depend on parameters?

$(vii)$ stability of extremals depending on parameters.

Application: A huge number of problems in the calculus of variations have their origin in physics where one has to minimize the energy associated to the problem under consideration. Nowadays many problems come from economics.

References:

https://en.wikipedia.org/wiki/Calculus_of_variations

http://mathworld.wolfram.com/CalculusofVariations.html

http://www.math.uni-leipzig.de/~miersemann/variabook.pdf

2968 questions
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Hamiltonian versus Euler-Lagrange theorem.

If a function $f$ is $\mathcal C^2([a,b]\times \mathbb R\times \mathbb R)$, $f=f(x,u,\xi)$, and $$I(u)=\int_a^b f(x,u(x),u'(x)dx,$$ then, Euler Lagrange equation is given by $$\frac{d}{dx}f_\xi=f_u.$$ Let now $$J(u,v)=\int_a^b \Big(u'(x)v(x)- H(x,…
user330587
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Calculus of variations condition

Given this: $$\int_0^1\left(\frac{1}{2}y'^2+yy'+y'+y\right)dx$$ where $$y(0)=1$$ I'm to show that the extremal can be found by imposing this condition: $$y'+y+1=0$$ at $x=1$ Beltrami's identity…
Number 34
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Isoperimetric problem involving infinite set of extrema

Find the extremum of the following isoperimetric problem. \begin{align}\int_0^\pi y'(x)^2dx\end{align} with $y(0) = y(\pi) = 0$ subject to the constraint \begin{align}\int_0^\pi y(x)^2dx = \frac{\pi}{2}.\end{align} Show that there is an…
clocktower
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Determine whether the functional has weak or strong extrema?

The functional $$J[y]=\int_{0}^{1}((y')^2+x^2)dx$$ where $y(0)=-1$ and $y(1)=1$ on $y=2x-1$, has weak minimum weak maximum strong minimum strong maximum I have searched similar questions on this site, so I found that if $F_{y'y'}>0$ or $<0$…
Kushal Bhuyan
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Find the extremal of $J[y]=\int_1^2(y'^2+2yy'+y^2) dx $

I have to find the extremal for the following functional: $$J[y]=\int_1^2(y'^2+2yy'+y^2) dx $$ such that $y(1)=1$ and $y(2)$ is arbitrary. I got it to be equal $e^{x+1}$. Is that correct?
blabla
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First variation of a functional $J(y)$

Some basic question: The first variation of a functional $J(y)$ is defined to be (see here, f.e.) $$ \delta J(y,h)=\lim_{\varepsilon\to 0}\frac{J(y+\varepsilon h)-J(y)}{\varepsilon}=\frac{d}{d\varepsilon}J(y+\varepsilon h)_{|\varepsilon=0}. $$ Hope…
mathfemi
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Maximize volume of rotation given perimeter

Let $b > a > 0$. Fixing $y(a) = y(b) = 0$ and the total length $L = \int_a^b \sqrt{1+y'^2} dx$, I want to find the curve $y(x)$ that maximizes the volume of the (roughly toroidal) volume between the $x-z$ plane and the surface of revolution of…
Elena
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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 $y(x)=\frac{\sqrt{2}}{\pi} \sin x$ $y(x)=-\frac{\sqrt{2}}{\pi} \sin x$ $y(x)=\frac{\sqrt{2}}{\pi} \cos…
slow but keen learner
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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)? $
Narasimham
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calculus of variation

Among all curves joining a given point (0, b) on the y-axis to a point on the x-axis and enclosing a given area S together with the x-axis, find the curve which generates the least area when rotated about the x-axis
Nikhil sunam
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Minimizing a functional of two functions with three boundary conditions

What are the natural boundary conditions for the following calculus of variations problem: Minimize: $$J[y] = \int_0^b (1+(y_1')^2 + (y_2')^2)) \,dx$$ subject to the boundary conditions $$y_1(0) = 0 = y_2(0)$$ and $$b + y_1(b) − y_2(b) = 1.$$ So I…
3141
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Fundamental lemma of calculus of variation, about hypothesis

We can find on the web several forms of the fundamental lemma of calculus of variation, the simplest one I could find (French wikipedia ) is: for $f\in C^1([a, b])$ $$ \int_a^b f(x) g(x) dx = 0, \quad \forall g\in C^1([a, b]), \quad g(a)=g(b)=0…
KevMoriarty
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Calculus of variation for geodesic

I need to minimize $$J[v]=\int\sqrt{P(x)+R(x)(v')^2}dx$$ By Euler equation, I get $$\frac{d}{dx}\frac{Rv'}{\sqrt{P+Rv^{'2}}}=0$$ Then I need to solve a complex ODE, but I don't know how to deal with it.
89085731
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test function and boundary condition

For example, if we consider the Dirichlet energy $\int\frac 12 |\nabla u|^2$ and the solution space as follows: $$X=\{u\in W^{1,2}(\Omega) \text{ | } u = 0 \text{ on } \partial\Omega \}$$ , then the test function is $C_0^\infty$. My question is…
jakeoung
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Does Constant factor rule in integration hold for functionals?

The constant factor rule in integration states the following relation is valid $$\int a f(x)dx=a \int f(x) dx$$ for all constants $a$(or $a$ that are constant functions of x, that is $\dfrac{da}{dx}=0$ ) My question is: Given a functional…
Omar Nagib
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