# 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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### Higher order Leibniz rule and ordered multiindex notation

Although I think this is probably known, I am making here a short exposition on the multiindex notations I am using to make this question self-contained. I note that there is at least two different ...
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### Sufficient and necessary conditions for decomposing the sum of random variables

Given two $n$-tuple vectors $\vec{\alpha}=(\alpha_1,\cdots,\alpha_n)$ and $\vec{h}=(h_1,\cdots,h_n)$, where $h_i\ge0$, $\sum_{i=1}^nh_i=1$, and $\alpha_i\in(0,1)$, we consider a random variable $S$ on ...
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### Why the least action principle is always (?) used in this particular form?

The least action principle in (mathematical) physics says the following. Given a system, e.g. collection of particles, whose motion satisfies a known system of differential equations (of second order)...
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### Classification of Lagrangians with given Euler-Lagrange equations

In (mathematical) physics the equations of motion of a system of particles are often interpreted as Euler-Lagrange equations for appropriate Lagrangian $L=L(x,\dot x,t)$ where $x$ is a collection of ...
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### 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 $L\in C^\infty(R\times R\times R)$, ...
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### 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_{...
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### 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 ...
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### 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, ...