# Tagged Questions

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### 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 ...
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### 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 )? $$l^2> 4πA$$
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### Isoperimetric inequality proof [duplicate]

Can someone give me a neat clear proof (the most simple but rigorous avaiable) of the isoperimetric inequality $L^2> 4πA$?
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### 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 ...
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### 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 $x(t)$: $\Phi (x)(t) = x''(t) + \int_a^b \int_a^b K(t,s_1,s_2,x(s_1),x(s_2)) ~dx$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $$X=x+\epsilon2x\ ... 1answer 62 views ### Show that the path of shortest hyperbolic length satisfies (x-c)^2+y^2=r^2 The hyperbolic length of a curve y:[a,b]\rightarrow\mathbb{RxR}_+ is given by the functional$$\lambda(y)=\int_a^b\frac{\sqrt{1+y'^2}}{y}dx$$Show that the path of shortest hyperbolic length ... 2answers 100 views ### Show that the infimum of a functional is zero, but this infimum is never achieved. Show that the infimum of the integrals$$\int_0^1(y'^2(x)-1)^2dx$$among all y(x)\in C^2[0,1] such that y(0)=y(1)=0, is zero, but is not achieved by any function in this set. What I've worked on: ... 2answers 85 views ### Solving C_1=4y^2+(y')^2+8y While working through the exercises in a book on the Calculus of Variations, I've hit a roadblock in trying to solve this differential equation: C_1=4y^2+(y')^2+8y Let me back-up a bit and fill-in ... 1answer 174 views ### Minimizing an integral with variable endpoints I am trying to minimize the following functional:$$ J[y]=\int\limits_0^T{\frac{\sqrt{1+y'(x)^2}}{y(x)}dx},  y(0)=1, ~ T-y(T)=1, $$where T is variable. Using the necessary conditions I've ... 2answers 94 views ### A given functional is path independent if k equals to the one of the following: The functional$$\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$$is path independent if k equals (A) 1 (B) 2 ... 1answer 88 views ### Applying the Lagrange Euler Formulation I was doing my tutorial on Lagrange-Euler formulation for robotic systems when i came across a slight problem. Referring to the picture in the link, I would like to know if my answer (equation 1) ... 1answer 63 views ### Use variation parameters method to solve: 4x^2y'' + y = 8x^{1/2} I've tried so hard and just get horrible, horrible equations.$$y_p = u_1(x)x^{1/2} + u_2(x)x^{1/2}\log x$$1answer 136 views ### Find an upper bound for lowest eigenvalue using calculus of variations. So I'm doing a little calculus of variations on an eigenvalue problem. The goal of this is to find an upper bound for the \omega_0 as follows: \omega_0^2 \leq ... 0answers 113 views ### Nonlinear BVP pde and variational inequality Suppose f \in L^2(\Omega) where \Omega is bounded. The problem: for a \in \mathbb{R} find u_a \in H^1_0(\Omega) s.t$$-\Delta u_a + \frac{m(u_a)}{a} = f$$where m(r) = \begin{cases} r ... 2answers 304 views ### Euler Lagrange sufficient minimization condition Suppose g \in C^1([a,b]\times\mathbb{R}\times\mathbb{R}). Let S = \{ f \in C^1([a,b]): f(a)=a_0, f(b)=b_0\}. I am trying to show that if f \in S satisfies$$ \frac{d}{dx}g_z(x,f(x),f'(x))= ...
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 ...