Mathematics Stack Exchange
Mathematics Stack Exchange

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
2
votes
1 answer

Why is the function that minimizes Euclidean distance in a plane assumed to be of the form $y=f(x)$?

In every text I have come across that deals with minimizing the Euclidean distance (without constraints) in a 2D plane, the titular assumption seems to have been made. The standard procedure goes something like this: \begin{equation} ds^2 = dx^2 +…
SystematicDisintegration
  • 368
2
votes
1 answer

Why $m>-\infty $ if $m=\inf\{\int_\Omega f\mid u\in u_0+W_0^{1,p}\}$?

Let $\Omega $ open bounded with Lipschitz boundary. Let $$m=\inf\left\{\int_\Omega f(x,u(x),\nabla u(x))\mathrm d x\mid u\in u_0+W_0^{1,p}(\Omega ) \right\},$$ and $I(u_0)<\infty $. I recall that $u\in u_0+W_0^{1,p}(\Omega )$ means that…
user330587
  • 1,624
2
votes
0 answers

Isoperimetric problem with auxillary conditions

Find the extremals for the problem with the functional $$ I[x] = \int^1_0 [\dot{x}^2 - x^2 ] dt $$ and auxillary and boundary conditions $$ \int^1_0 \sqrt{1+\dot{x}^2} dt = \sqrt{2}\hspace{5pt} ,\hspace{5pt} x(0)=0 \hspace{5pt} ,\hspace{5pt}x(1) =…
strider
  • 371
2
votes
1 answer

Minimizing a perimeter with density

I've got the following problem in Calculus of Variations, about perimeters with density. Let $g \in C^1(\mathbb{R})$ be even and strictly convex. Define the perimeter with density of a measurable subset $E$ of $\mathbb{R}$ by: $$ P(E)=\sup \left\{…
18cyclotomic
  • 342
2
votes
1 answer

Why $\frac{d}{dx}f_\xi=f_u$ and $f(b,\bar u(b), \bar u'(b))=0$.

Let $f\in \mathcal C^2([a,b]\times \mathbb R\times \mathbb R)$. I have to show that if $\bar u\in \mathcal C^2([a,b])\cap X$ is a minimizer of $$\inf_{u\in X}\left\{\int_a^b f(x,u(x),u'(x))dx\right\},$$ for $X=\{u\in \mathcal C^1([a,b])\mid…
user330587
  • 1,624
2
votes
0 answers

The second variation of a harmonic map

For maps $ u :\mathbb{B}^n \rightarrow \mathbb{S}^{n-1}$ the second variation of the dirichlet energy \begin{align} \mathbb{E}[u] = \int_{\mathbb{B}^n}|\nabla u|^2 \,dx \end{align} is given by \begin{align} D^2\mathbb{E}[u][\phi,\phi] =…
user152723
  • 61
2
votes
0 answers

Calculus of Variation

Possible Duplicate: Calulus of variations Euler Lagrange equation Find the extremals for the following functionals and discuss whether they provide a minimum or a maximum to the functional. $$j[y]=\int_0^1 \exp(y)(y')^2 dx $$ subject to the…
George
  • 21
  • 1
2
votes
1 answer

Why do we take just the first variation?

Why, in calculus of variations, do we take just the first variation of the functional and never after that? Are we just approximating or there is a reason that we don't have to take the second variation?
user405715
  • 73
  • 4
2
votes
0 answers

Equivalence of the variation of action under the re-parametrization of the curve.

I was reading about Fermat's Principle that states the variation of the optical path $\mathrm A$ is zero. The variation of the action $\mathrm A$ was given by: $$\delta \mathrm A = \delta \int_A^B n(\mathbf r(s))\sqrt{\frac{\mathrm d\mathbf…
user142971
2
votes
1 answer

Why can't consequences of variational principle be used to modify the principle?

I was reading about the applications of Euler-Lagrange equation in Mathematics for Physics by Stone, Goldbart; they were showing the application of the variational principle for the central force problem $F = -\partial_r V(r)$ where $V$ is the…
user142971
2
votes
0 answers

Euler Lagrange equation with functional depending on function integrals instead of derivatives

Variational analysis solves the maximization with respect to the function $q(t)$ of the functional depending on $q(t)$ and its derivative $$ J[q]=\int_a^b L(t,q(t),\dot{q}(t)) \ dt \qquad (1) $$ by finding the solution of the Euler-Lagrange…
Nicola
  • 153
2
votes
0 answers

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. I tried it as $F=(1+x)(y')^{2}$ so…
neelkanth
  • 6,048
  • 2
  • 30
  • 71
2
votes
1 answer

Extremals of a Functional with two functions

Find the extremals of the functional $$J[y, z] = \int_0^\frac{π}{2} ((y')^2 + (z')^2 + 2yz) \,dx$$ subject to the boundary conditions $y(0) = 0, y(\frac{π}{2})= 1, z(0) = 0, z(\frac{π}{2}) = 1$ Do I need to convert y and z to polar coordinates so…
3141
  • 482
2
votes
1 answer

Minimizing a functional with $L^2$-norm

I want to find $\arg\!\min_u \frac 1 2 \|u-f\|^2 + \frac 1 2 \|d_x - u_x\|^2 +\frac 1 2 \| d_y - u_y \|^2 := \arg\!\min_u F(u,u_x,u_y) $ where $u$ is a function of $x$, $y$ and $u_x$, $u_y$ are partial derivatives. I saw that it uses variational…
Gobi
  • 7,458
2
votes
1 answer

Find the function that minimizes $\int_{0}^{1}e^{-(y'-x)}+(1+y)y'dx$

Suppose among all the continuously differentiable functions $y(x), x\in \mathbb{R}$, with $y(0)=0$ and $y(1)=\frac{1}{2},$ the function $y_0(x)$ minimizes the functional, $$\displaystyle\int_{0}^{1}e^{-(y'-x)}+(1+y)y'dx$$ Find the value of…
Harry Potter
  • 1,326
1 2 3
…
21 22