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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

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Euler Lagrange equation for finding extremal value for $\int_0^1 xw'(x)dx$ given $w(0)=0,w(1)=10$

Euler Lagrange equation for finding extremal value for $\int_0^1 xw'(x)dx$ given $w(0)=0,w(1)=10$ Based on Euler-Lagrange(EL) equation, we have \begin{align} \frac{\partial xw'(x)}{\partial w(x)}-\frac{d}{dx}(\frac{\partial xw'(x)}{\partial…
Ethanabc
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Bolza problem - coercivity: I do not understand the solution I got

Consider $$ F(u)=\int\limits_0^1 (1-u'(x)^2)^2+u(x)^2\, dx, u\in W^{1,4}(0,1), u(0)=u(1)=0. $$ Show, that $F$ is coercive. To do so use the Young inequality $$ 2ab\leq\varepsilon a^2+\frac{b^2}{\varepsilon}~\forall~a,b,\varepsilon > 0. $$ Solution…
user34632
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Canonical equations from action integral

I'm going through Lanczos Variational Principles of Classical Mechanics and on page 169 it says that we can form the action integral $$A=\int_{t_1}^{t_2} \left[\sum p_i\dot q_i-H(q_1,...,q_n;p_1,...,p_n;t)\right]dt$$ from which we can get do the…
DS08
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General variations of a functional with a second order derivative?

I have seen how to derive the general variation of a functional of first order. However, when I try to apply the methods to higher order functional, things break down. How does one derive the boundary conditions/ equations of motion for the general…
Richard Wilde
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a question about chapter9 in EVANS pde

I'm reading chapter 8 of the pde book written by EVANS. It's about variation method of pde. In page 433. what's the mean of $C_c^\infty(U)$? why the test function $v$ is in this space? enter image description here thank you!
hongjinwu
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Help understanding a passage in Gelfand and Fomin's Calculus of Variations

Page 19 of Gelfand and Fomin's Calculus of Variations considers the Euler equation arising from functionals of the form $\int_a^b f(x, y)\sqrt{1+y'^2}\, dy$. Letting the integrand be $F$, they give the $\frac{d}{dx} F_{y'}$ term of the Euler…
Connor Harris
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Additional natural boundary conditions determined by $G(y(b))$ outside of Lagrangian.

Here is a problem I am not quite sure how to approach: Determine the natural boundary condition at $x=b$ for the variational problem defined by $$J(y) = \int^b_a L(x,y,y')dx + G(y(b)),$$ where $ y\in C^2[a,b]$ and $y(a) = y_0$. I know the solution,…
CrypticParadigm
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How to solve this calc. of variations problem with inequality constraint?

I have the following calculus of variations problem: $$\min_{f(\cdot)}\int_0^1(f(x)+g(x))^\phi dx \quad \text { s.t. } \int_0^1f(x)dx=A, \quad f(x)\geq 0 \quad \forall x\in [0,1]$$ For some given function $g(x)\geq0$ and a given constant $A>0$.…
user56834
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Euler Lagrange equation for $I(u)=\frac{\int_{(a,b)}u'^2}{\int_{(a,b)}u^2}$.

Let $$I(u)=\frac{1}{2}\int_{(a,b)} u'^2\quad \text{and}\quad J(u)=\int_{(a,b)} u^2,$$ with $u\in W_0^2(a,b)$. I want to minimize $$H(u)=\frac{I(u)}{J(u)}.$$ Way 1 : Using Lagrange multiplier, I know that there is a constant $\lambda $ s.t.…
idm
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Derivation First variation formula of a functional

I did not understand two transitional stages in the red rectangulars. (Please look question marks) Could you help me?
HD239
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Why $\int f_u(x,u,u')v+f_\xi(x,u,u')v'=0\implies \frac{d}{dx}f_\xi=f_u$ a.e.?

I have the following lemma : Let $f\in \mathcal C^1([a,b]\times \mathbb R\times \mathbb R)$, $f=f(x,u,\xi)$ satisfy some hypothesis (but they are not important for my question). Then any solution $u\in W^{1,p}(a,b)$ of $$\int_a^b…
user380364
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Why in calculus of variation why always want to minimize $\int_\Omega f(x,u(x),u'(x))dx$?

I'm studying a course of calculus of variation, and I was wondering why we always want to minimise $$I(u)=\int_\Omega f(x,u(x),u'(x))dx\ \ ?$$ Indeed, as I read on the internet, we are interested on the minimisation of functional. So why in in…
user330587
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Why $\forall \varphi\in W_0^{1,2},\int_\Omega \Delta u\varphi=0\implies \Delta u=0$

Let $\Omega \subset \mathbb R^n$ open with Lipschitz boundary. Let $u\in W^{2,2}(\Omega )$. In a course I have that : by the fundamental lemma of variation calculus, if $$\int_\Omega (\Delta u)\varphi=0$$ for all $\varphi\in W_0^{1,2}$ then $\Delta…
user349449
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General variation of a functional

In Calculus of variations, I. M. Gelfand, S. V. Fomin, Section 13 they are showing how to variate functional with variable endpoints. How do they get from (4) to the next formula of the differential of the functional? How do they simplify the…
Blake
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existence of $\bar u\in u_0+W_0^{1,p}(\Omega )$ s.t. $\inf\{I(u)=\int_\Omega f(x,u(x),\nabla u(x))dx\mid u\in u_0+W_0^{1,p}(\Omega )\}=I(\bar u)$

Let consider the problem $$\inf\left\{I(u)=\int_\Omega f(x,u(x),\nabla u(x))\mathrm d x\mathrm dx\mid u\in u_0+W_0^{1,p}(\Omega ) \right\}\tag{P},$$ where $u_0+W_0^{1,p}(\Omega )=\{u\in W^{1,p}(\Omega )\mid u|_{\partial \Omega }=u_0\}$ and $\Omega…
user330587
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