# Questions tagged [calculus-of-variations]

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

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### How do I apply Brouwer fixed-point theorem in this claim?

Let $\zeta:\mathbb{R}\to [0,+\infty)$ be a continuous non-negative function such that $\zeta(0)=0$ and $\tau\mapsto \zeta(\tau)\tau$ is a non-decreasing differentiable function whose derivative is ...
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### Hamilton equations for Classical Field Theory

This is a second part of my previous question. I'm trying to figure it out by myself how to deduce Hamilton's equations in classical field theory as it is usually obtained in physics books. Notation: ...
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### Prove/disprove $(\int_{0}^{2 \pi} \!\!\cos f(x) d x)^{2}+(\int_{0}^{2 \pi}\!\!\! \sqrt{(f'(x))^{2}+\sin ^{2} f(x)}dx)^{2}\ge 4\pi^{2}$

This problem has been posted on Math.SE but didn't receive any correct answer after a long time. Let $f(x)$ be a differentiable function on $[0,2\pi]$ s.t. $0\leq f(x)\leq 2\pi$ and $f(0)=f(2\pi)$. ...
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### Maxwell equations as Euler-Lagrange equation without electromagnetic potential

In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) ...
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I am interested in calculating the least area of a surface spanning the boundary of an octant on the unit sphere; and short of precise values I am looking for upper bounds for this area. In $\mathbf{S}... 0answers 69 views ### Calculus variation question Assume that we have to minimize the integral$I[y]=\int_0^1 L(x,y,y'(x))dx$for smooth diffeomorphic mappings$y:[0,1]\to [0,1]$with$y(0)=0$and$y(1)=1$, where$L\in C^\infty(R\times R\times R)$, ... 0answers 75 views ### Convergence of infinite linear programming Suppose we have the following linear program (LP1),$$\min_{f \in \mathcal{C}} \int_{\mathbb{R}} f(t) \cos(2 \pi x_0 t) dt \\ \text{subject to } \int_{\mathbb{R}}f(t)dt = 1 \\\forall x\in [0,1]: \int_{... 1answer 152 views ### Fréchet derivative of evaluation-like functional (multivariate) I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across this post I could not help but wonder the following. Let$H$be ... 0answers 38 views ### Variational problem where the end point cannot be held constant I want to solve a variational problem for paths$y:[0,1]\to \mathbb{R}^n$where$F$is a functional on the paths and additional parameter, and the end points are fixed,$y(0)=0$and$y(1)=b$. However, ... 2answers 200 views ### If all points of a real function with positive values would be local minimum, can one say it is constant function? During my studies I faced a function$f:\mathbb{R} \to \mathbb{R}^+ $with the property: for all$x \in \mathbb{R} $and all$y$in open interval$(x-\frac{1}{f(x)} ,x+\frac{1}{f(x)}) $we have$f(x) \...
In Theorem 14.37 of Variational Analysis by Rockafellar and Wets, it shows that for any normal integrand $f:T\times \mathbb{R}\to\mathbb{R}$, the function $p:T\to \mathbb{R}$ given by \$p(t):=\inf f(t,⋅...