1
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0answers
20 views

Finding extremals over y in two ways

We can find the extremal of $$\int_0^1(\frac{1}{2}y'^2+yy'+y'+y)dx$$ amongst all y with $y(0)=1$ by imposing the natural boundary condition $\frac{\partial F}{\partial x}=y'+y+1=0$ at $x=1$.Solving ...
3
votes
2answers
63 views

What is the most elementary but still correct according to the most rigorous standard proof of the isoperimetric inequality?

Can you write the most elementary proof of the isoperimetric inequality (but still correct according to the most rigorous standard )? $$l^2> 4πA$$
0
votes
0answers
27 views

Isoperimetric inequality proof [duplicate]

Can someone give me a neat clear proof (the most simple but rigorous avaiable) of the isoperimetric inequality $L^2> 4πA$?
0
votes
1answer
56 views

Question about variational principles involving light rays.

In question 9 (see this link: http://view.samurajdata.se/psview.php?id=28b2e4b5&page=1 ), I've shown the light rays are follow a parabolic paths using the Euler-Lagrange equation and Fermat's ...
0
votes
0answers
32 views

The Volterra derivative of a functional with an argument being a second order derivative.

An exercise demands that we calculate the Volterra derivative of a functional $\Phi(x)(t)$ with respect to argument $x(t)$: $\Phi (x)(t) = x''(t) + \int_a^b \int_a^b K(t,s_1,s_2,x(s_1),x(s_2)) ~dx $, ...
3
votes
1answer
108 views

Calculus of variations minimum

I have a question that asks: Find the extremal of the functional $$J(x)=\int^{\pi}_02x\sin(t)-\dot x^2 dt$$ with $x(0)=x(\pi)=0$. I found $x(t)=\sin(t)$ It then asks to Show that this ...
1
vote
1answer
119 views

Determine the minimum and maximum values of an integral subject to end conditions

The question: determine the minimum and maximum values of the integral $$\int_0^1 yy'dx$$ subject to the conditions $y(0)=0$ and $y(1)=1$. There is no explicit y dependence, so our Euler-Lagrange ...
0
votes
1answer
83 views

Find the Minimum value of a Functional Constrained to end-point Conditions

The question: Find the minimum value of $\int_0^1 y'^2 dx$ subject to the conditions $y(0)=y(1)=0$ and $\int_0^1y^2dx=1$. In another question, I proved that, if we have an integral of the form ...
1
vote
1answer
71 views

Question about $C^2$ functional

i have this problem : The solutions of P correspond to critical points of the fuctional $$\phi(x)=\frac12 \int_0^{2\pi} |x'|^2 dt - \int_0^{2\pi} F(t,x) dt , x\in E $$ where $F(t,x)=\int_0^x ...
3
votes
2answers
70 views

Deducing Euler Equation

From Sydsaeter / Hammond (Further Mathematics for Economic Analysis, 2008, 2nd ed., p. 293): $$ \max \int\limits_{0}^T [N(\dot{x}(t)) + \dot{x}(t)f(x(t))] e^{-rt} dt $$ where N and f are $C^1$ ...
0
votes
0answers
63 views

Green's function

Please can someone told me how to find the Green's function $G(t,x)$ of BVP : $$u'''(t)=0 , \quad t\in (0,1)$$ and BC : $$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$ where $\frac12 < p<q<1$ are ...
3
votes
0answers
145 views

Green's function for third order boundary value problems

How to find the Green's function $G(t,x)$ for the BVP consisting of the equation : $$u'''(t)=0 , \quad t\in (0,1)$$ and BC : $$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$ where $\frac12 < ...
1
vote
1answer
61 views

Prove a transformation is a variational symmetry for J

The following problem is from The Calculus of Variations by B.von Brunt (page 215, Exercise 9.2.1) Let $$ J(y)=\int_a^b xy'^2\mathrm{d}x. $$ Show that the transformation $$ X=x+\epsilon2x\ ...
2
votes
1answer
62 views

Show that the path of shortest hyperbolic length satisfies $(x-c)^2+y^2=r^2$

The hyperbolic length of a curve $y:[a,b]\rightarrow\mathbb{RxR}_+$ is given by the functional $$\lambda(y)=\int_a^b\frac{\sqrt{1+y'^2}}{y}dx$$ Show that the path of shortest hyperbolic length ...
1
vote
2answers
104 views

Show that the infimum of a functional is zero, but this infimum is never achieved.

Show that the infimum of the integrals $$\int_0^1(y'^2(x)-1)^2dx$$ among all $y(x)\in C^2[0,1]$ such that $y(0)=y(1)=0$, is zero, but is not achieved by any function in this set. What I've worked on: ...
1
vote
2answers
85 views

Solving $C_1=4y^2+(y')^2+8y$

While working through the exercises in a book on the Calculus of Variations, I've hit a roadblock in trying to solve this differential equation: $C_1=4y^2+(y')^2+8y$ Let me back-up a bit and fill-in ...
3
votes
1answer
174 views

Minimizing an integral with variable endpoints

I am trying to minimize the following functional: $$ J[y]=\int\limits_0^T{\frac{\sqrt{1+y'(x)^2}}{y(x)}dx}, $$ $$ y(0)=1, ~ T-y(T)=1, $$ where $T$ is variable. Using the necessary conditions I've ...
0
votes
2answers
94 views

A given functional is path independent if k equals to the one of the following:

The functional $$\int_0^1 (y^{\prime 2} + (y + 2y')y'' + kxyy' + y^2) ~dx,$$ $$y(0) = 0, ~y(1) = 1, ~y'(0) = 2, ~y'(1) = 3$$ is path independent if $k$ equals (A) $1$ (B) $2$ ...
1
vote
1answer
88 views

Applying the Lagrange Euler Formulation

I was doing my tutorial on Lagrange-Euler formulation for robotic systems when i came across a slight problem. Referring to the picture in the link, I would like to know if my answer (equation 1) ...
2
votes
1answer
63 views

Use variation parameters method to solve: $4x^2y'' + y = 8x^{1/2}$

I've tried so hard and just get horrible, horrible equations. $$y_p = u_1(x)x^{1/2} + u_2(x)x^{1/2}\log x$$
1
vote
1answer
137 views

Find an upper bound for lowest eigenvalue using calculus of variations.

So I'm doing a little calculus of variations on an eigenvalue problem. The goal of this is to find an upper bound for the $\omega_0$ as follows: $\omega_0^2 \leq ...
4
votes
0answers
113 views

Nonlinear BVP pde and variational inequality

Suppose $f \in L^2(\Omega)$ where $\Omega$ is bounded. The problem: for $a \in \mathbb{R}$ find $u_a \in H^1_0(\Omega)$ s.t $$-\Delta u_a + \frac{m(u_a)}{a} = f$$ where $m(r) = \begin{cases} r ...
0
votes
2answers
307 views

Euler Lagrange sufficient minimization condition

Suppose $g \in C^1([a,b]\times\mathbb{R}\times\mathbb{R})$. Let $S = \{ f \in C^1([a,b]): f(a)=a_0, f(b)=b_0\}$. I am trying to show that if $f \in S$ satisfies $$ \frac{d}{dx}g_z(x,f(x),f'(x))= ...
2
votes
1answer
381 views

Isoperimetric problem in the calculus of variations

I'm trying to solve the following isoperimetric problem: A plane curve has length $l$ and end points at $(0, 0)$ and $(a, 0)$ on the positive $x$ axis. Show that the area $A$ under this curve is ...