Tagged Questions
1
vote
0answers
72 views
How to take the limit of some integral?
$$ f\left( x^{\prime },t+\varepsilon \right) = \int_{-\infty }^\infty dx\int_{-i\infty }^{i\infty }
\frac{d\tilde{x}}{2\pi i} \left(1+\varepsilon \left[
\tilde{x}D_{1}\left( x,t\right)
...
3
votes
1answer
39 views
Representation for function (“null-Lagrangian”)
Let $L(t,x,p) \in C^m([0,1] \times \mathbb{R}^n \times \mathbb{R}^n;\mathbb{R})$, $m\geqslant1$ and define for any $u \in C^1([0,1];\mathbb{R}^n)$
$$
\mathcal{L}u = ...
1
vote
0answers
106 views
How to find $\kappa$ to minimize integral $I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) \mathrm{d}x$
I am trying to find such value $\kappa \in (0,1)$ that would minimize the integral
\begin{equation}
\begin{aligned}
I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) ...
8
votes
0answers
347 views
Finding a proper solution of a given functional
It's my first post here, but I worked very hard to find solution and I failed.
Hereinafter, I skip physical background and directly proceed to my mathematical problem.
No matter how, you know the ...
1
vote
1answer
119 views
Find function to make maximum value
Let ${f : [0, 1] \rightarrow [-1, 1] }$ is a continuous function such that ${ \int_{0}^{1} x f \left(x\right) dx =0}$
Find $f(x)$ such that ${ \int_{0}^{1} \left(x ^{2 } + \frac{1}{4} \right) f ...
0
votes
1answer
77 views
Volume integral and Variations
Suppose I wish to find the Euler-Lagrange equation for an integral $\int_V f(u,\mathop{\mathrm{grad}} u)\,dV$ where $V$ is a volume given by some equation, for example say $x^2+y^2+z^2\le 1$, and ...
1
vote
0answers
89 views
Complicated “functional integral”
I came across the following "functional" at work:
$$
\Pi [b]=\iint_0^{\lambda b(v,\lambda)} vf(v,\lambda) \; dv \; d\lambda
$$
it's part of an optimization problem that tries to find $b$, subject ...
