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
21
votes
3 answers

How can $y$ and $y'$ be independent in variational calculus?

In variational calculus, functionals are written as \begin{eqnarray} F = \int f(x,y,y') dx \end{eqnarray} Where $F$ depends upon choice of $y,y'$. But for smooth regular functions specifying the $y$ also specifies $y'$, so how can they be…
chatur
  • 1,132
10
votes
3 answers

Calculus of variations: find $y(a/2)$ if $y(x)$ maximizes the volume of rotation

A curve $y(x)$ of length $2a$ is drawn between the points (0,0) and (a,0) in such a way that the solid obtained by rotating the curve about the $x$-axis has the largest possible volume. Find $y\left(\frac{a}{2}\right)$.
user70077
  • 117
  • 3
8
votes
1 answer

Invariance of lagrangian under point transformation

A lagrangian $L(q,\dot q, t)$ is invariant under the point transformation $$q_i=q_i(s_1,...,s_n,t)$$ To prove this I show that $$\frac{d}{dt} \frac{\partial L}{\partial \dot s_i} - \frac{\partial L}{\partial s_i} = 0$$ where $$\frac{\partial…
DS08
  • 367
8
votes
3 answers

calculus of variations question

I would like to find a continuous function $y : [0,4] \to \mathbb{R}$ that minimizes the following functional $$I (y) := \displaystyle\int_{0}^4\sqrt{y\left(1+(y^{\prime})^2\right)} dx$$ subject to the boundary conditions $y (0) = 5/4$ and $y (4) =…
mimi
  • 81
8
votes
1 answer

Calculus of variations question with two variables

If $u(x)$ and $v(x)$ satisfy $u(0)=1$, $v(0)=-1$, $u(\pi/2) =0$, $v(\pi/2) =0$ on extremals of the functional $$ \int_0^{\pi/2}\left[\big({\frac{du}{dx}\big)^2 +\big(\frac{dv}{dx}\big)^2 + 2 \,u v }\,\right] dx $$ then which of the following is…
zafran
  • 904
7
votes
1 answer

Motivation for the definition of the functional derivative

We want to come up with a definition of functional derivative that allows us to harness the principle that extremal points are stationary points in order to solve problems involving optimising functionals. Let us recapitulate how derivatives are…
what a disgrace
  • 493
7
votes
1 answer

Noether's theorem

I am now reading the book Calculus of Variations written by Jost and I have a problem in the proof of Noether's theorem: Theorem 1.5.1. Let $F\in C^2([a, b]\times \mathbb R^d \times \mathbb R^d, \mathbb R)$ and a one-parameter family of maps…
user43378
  • 374
6
votes
1 answer

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 differentiable and $\dot{g}(t)=f(t),\text{ }t\in[a,b]…
John
  • 61
6
votes
1 answer

How to properly take derivatives in calculus of variations (Euler-Lagrange formula)

Why is it that, in calculus of variations (specifically Euler-Lagrange), we can take the derivative of a function with respect to a function $f$ and set this derivative to $0$ if only $f'$ appears in $L$? For example the standard distance…
Mike Flynn
  • 1,917
6
votes
1 answer

Minimizing $\int_0^T[(f')^2 + f^2]\,dx$

I've confused myself when thinking about the following variational problem: \begin{equation} \min_{f} \int_0^T \left([f'(x)]^2 + [f(x)]^2\right)\,dx \qquad f(0) = 1, f'(0) = 0. \tag{*} \end{equation} Intuitively I expect the solution to be some kind…
user7530
  • 49,280
6
votes
1 answer

Maximizing a functional of two functions.

I am trying to maximize the functional $$J[u,v] = \frac{4}{L} \int_0^L f(u,v) \ dx = \frac{4}{L} \int_0^L u(v- \bar v)^2 + v(u-\bar u)^2 \ dx, $$ where $0 \le u,v \le 1$ and, $$\bar u = \frac{1}{L} \int_0^L u \ dx, \quad \bar v = \frac{1}{L}…
Gregory
  • 3,641
6
votes
2 answers

The catenary, help with the equation for a hanging chain.

So the problem deals with a chain at a fixed length, bends but does not stretch under gravity. The total graviational potential energy is $$g\rho\int y\sqrt{1 + y^{'2}}dx$$ Where $g$ is gravity. The problem has a picture of the chain on the xy…
stack ex
  • 350
6
votes
1 answer

Why can't we construct a counter-example to the Fundamental Lemma of the Calculus of Variations?

"The fundamental lemma of the calculus of variations states that if the definite integral of the product of a continuous function $f(x)$ and $h(x)$ is zero, for all continuous functions $h(x)$ that vanish at the endpoints of the range of integration…
Gauss
  • 61
5
votes
2 answers

Geodesics of a Sphere in Cartesian Coordinates

I want to minimize $I = \int |\dot{x}|^2 dt$ subject to the constraint $|x|^2=1$ (sphere) which gives an Euler equation of $\lambda x - \ddot{x} = 0$. I have to show that the Euler equation is actually $|\dot{x}|^2 x - \ddot{x} = 0$. Is it right to…
Brooke Knight
  • 145
5
votes
1 answer

Calculus of variations minimum

I have a question that asks: Find the extremal of the functional $$J(x)=\int^{\pi}_02x\sin(t)-\dot x^2 dt$$ with $x(0)=x(\pi)=0$. I found $x(t)=\sin(t)$ It then asks to Show that this extremal provides the global maximum of $J$ I am not…
yankeefan11
  • 1,449
1
2 3
…
21 22