0
votes
2answers
60 views

Calculus of Variations

In the Calculus of Variations there is a passage from Euler's characteristic equation: $$ \frac {\partial F}{\partial y} - \frac {d}{dx} \left(\frac {\partial F}{\partial y'} \right)=0 $$ in ...
0
votes
1answer
63 views

How to prove $\int_a^b f(x)\varphi(x)dx=0\Rightarrow f(x)=0$

I am doing some reading on the calculus of variations and one of the first examples uses the following theorem: Let $f\in C[a,b]$. If $\int_a^b f(x)\varphi(x)dx=0$ for all $\varphi\in C[a,b]$, then ...
0
votes
1answer
78 views

Fundamental lemma of calculus of variations, gradients

Let $D \subset \mathbb{R}^d$ be a smooth bounded domain. Let $C_c^\infty(D)$ denote smooth and compactly supported functions on $D$. Let $f \in [C_c^\infty(D)]^d$ be a smooth, compactly supported ...
1
vote
0answers
46 views

Extremal of functional $ I\left[ y(x) \right] = \int_{0}^{\frac{\pi}{2}} {\left((y')^2 - y^2 + 2xy\right)dy} $

I have the following functional: $$ I\left[ y(x) \right] = \int_{0}^{\frac{\pi}{2}} {\left((y')^2 - y^2 + 2xy\right)dy} $$ subject to boundary conditions: $$ \begin{align} y(0) &= 0 \\ ...
0
votes
2answers
74 views

Euler-Lagrange problem solution

Hi, Can anyone solve this question? I have no clue.
0
votes
0answers
23 views

Is there a name for this simple functional?

Given age-specific fertility rates $F(t)$ and a maximum age $T$, and assuming zero mortality before age $T$ (everyone suddenly dies at $T$), the population growth rate $r$ satisfies $$\int_0^T ...
2
votes
1answer
86 views

Maximizing score in number-guessing game.

This is inspired by a puzzle (related to the two-envelopes problem) that I've seen in several places, including unbounded generalizations. The basic premise is that Alice chooses two real numbers ...
1
vote
0answers
134 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) ...
2
votes
1answer
87 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
60 views

How can I integrate this?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $\phi_1,v,\phi\in W_0^{1,p}(\Omega)$ with $p\in (1,\infty)$. How can I evaluate the integral: $$\int_0^1F(s)ds$$ where ...
1
vote
0answers
182 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) ...
1
vote
0answers
29 views

variation of a final state due to changes in period (where the period is a parameter)

I have a simple ordinary differential equation $\frac{dx}{dt}=f(x,t,p,T)$ $x(0) = x_0$, $x(T) = x_T$ where $p$ and $T$ are constant parameters. How do I compute $\frac{dx_T}{dT}$ ? Thanks! NOTE: I ...
2
votes
1answer
233 views

Separation of variables and substituion; first integral from the Euler-Differential Equation for the minimal surface problem

Let $P_1=(a,y_a),P_2=(b,y_b), y\in C^1 (a,b), y_a>0,y_b>0$ And the area integral: $\int^b_a y(x) \sqrt{1+y'(x)}dx$ From the Euler differential-equation we obtain: $$y'=1/\alpha ...
8
votes
0answers
390 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
187 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
102 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
1answer
122 views

Complicated “functional integral”

I came across the following "functional" at work: $$ \Pi [b]=\int_0^\infty\int_0^{\lambda b(v,\lambda)} vf(v,\lambda) \; dv \; d\lambda $$ it's part of an optimization problem that tries to find ...