0
votes
1answer
34 views

Elliptic partial differential equations

Consider the following elliptic PDE: $$ \Delta u=f(u), $$ where $f(u)$ is a smooth function. Which references (books, papers,...etc.) about existence of solutions for this PDE do you recommend to have ...
0
votes
1answer
23 views

how to introduce time into calculus of variations for image processing?

I'm studying some topics about calculus of variation applied to image processing. I'd like to understand how to introduce time parameter to evolve an image in an iterative way. For example, let's ...
0
votes
2answers
57 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 ...
1
vote
2answers
28 views

Reading Speed for Constant Time to Finish

You open a very long new book on your e-reader and read a few pages. It helpfully informs you that based on your reading speed you have 16 hours of reading left until you are done. You read the rest ...
0
votes
1answer
47 views

Calculus of variation: Reduce the order of a differential equation using a 1 parameter lie group adfmitted by it.

We are asked to reduce $y^"+y-y^{-3}=0$ using $X= \sin2x\frac{\partial}{\partial x}+y\cos2x\frac{\partial}{\partial y}$ I know we have to find the first prolongation of X and solve $X^1$F=0 using ...
0
votes
0answers
42 views

Euler-Lagrange Calculus of Variations Example

I have been working on solving Euler-Lagrange Equation problems in attempts to learn Calculus of Variations, but this one example has me stuck. I am probably making mistakes in my integration. I am ...
2
votes
0answers
13 views

Maximizing an integral through maximum principle

Suppose that we wish to achieve $$\max\int_0^1 (1-x^2-\dot{x}^2)dt, x(0)=0, x(1)\geq 1$$ Two possible ways one can do this is by Euler-Lagrange eqn or maximum principle. Applying the Euler-Lagrange ...
4
votes
0answers
83 views

Euler-Lagrange Equation example

I have been working on solving Euler-Lagrange Equation problems in differential equations, specifically in Calculus of Variations, but this one example has me stuck. I am probably making mistakes in ...
0
votes
1answer
47 views

stationary function of an integral

Find the stationary function $y=y(x)$ of the integral $\int_o^4[xy'-(y')^2]dx$ satisfying the conditions $y(0)=0$ and $y(4)=3$. I don't know what a stationary function is. Can you anyone suggest me ...
3
votes
0answers
55 views

Does the implicit function theorem imply Peano existence theorem

In The implicit function theorem written by Krantz & Parks, it's said that the implicit function theorem implies the following existence theorem of ODE: Theorem 4.1.1 If $F(t,x)$, ...
0
votes
1answer
56 views

Is it possible to solve or approximate this second order nonlinear system of differential equations.?

Given initial values $d[0]$ and $k[0]$, I would like to solve for the initial rate of change, $\dot d[0]$, and compare this value against some data. I have the following profit function, which I ...
1
vote
1answer
112 views

Differential Equations for a Teardrop Shape

My research has led me to a nonlinear system of differential equations which should yield a teardrop shape in the $x-y$ plane. The equations, parameterized by $t$ are ...
2
votes
1answer
65 views

Inverse problem in calculus of variations

I am interested in knowing which differential equations follow from a variational principle. I am reading this and it provides the answer for ordinary differential equations. Is there a complete ...
0
votes
2answers
73 views

Euler-Lagrange problem solution

Hi, Can anyone solve this question? I have no clue.
1
vote
1answer
476 views

Euler lagrange equation solving

Find the Euler-Lagrange equation for the functional $$I(y) = \int_0^1(py\,'\,^2-qy^2)\mathrm dx$$ subject to the constraint $$\int_0^1ry^2 = 1.$$ Answer: $\frac{d}{dx}(py') + (q-\lambda r)y = 0$. ...
0
votes
0answers
53 views

Solving a Variational Problem with Inequalities for Boundary Conditions

The most basic type of variational problem, in one sense, is that of minimizing the functional $$ J[\phi] = \int_{x_1}^{x_2} F[x, \phi, \phi'] dx, \quad F[x_1]=a, \quad F[x_2] = b ,$$ a solution to ...
1
vote
0answers
94 views

plotting the lagrangian

From the differential equation $$\frac{\partial P}{\partial r} = \left[1+\frac{r}{\ln(1+r)}\right]D$$ I get the second-order equation $$\frac{1}{D}{P(r)}=\text{Ei}\left(2\ln(r+1)) - ...
4
votes
1answer
229 views

Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
1
vote
2answers
83 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 ...
2
votes
1answer
224 views

Euler Lagrange equation of $J[y]=\int_0^1 (yy')^2dx$ subject to the constraint $\int_0^1 y^2 dx =3$

Among all the admissible functions $y = y(x)$, find those that extremise the functional $$J[y] = \int_0^1 (yy')^2dx$$ subject to the constraint $\int_0^1 y^2 dx =3$ and the boundary conditions ...
1
vote
1answer
82 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
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
231 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 ...
1
vote
1answer
107 views

how to obtain Euler equation for smoothing spline minimization problem?

The question might be trivial, but I don't understand why this minimization problem in Sobolev space $$ \min_{g}\int_{0}^{1}\left\{ f(x)-g(x)\right\}^{2} dx+\lambda\int_{0}^{1}\left\{ ...
4
votes
1answer
165 views

Frechet Differentiabilty of a Functional defined on some Sobolev Space

How can I prove that the following Functional is Frechet Differentiable and that the Frechet derivative is continuous? $$ I(u)=\int_\Omega |u|^{p+1} dx , \quad 1<p<\frac{n+2}{n-2} $$ ...
0
votes
0answers
185 views

Pre-requisites for the Calculus of Variations

I'm interested in working through the book : "Calculus of Variations" by Gelfand and Fomin. However, I lack the pre-requisites to do so (I'm familiar with linear algebra and one-variable calculus ...
4
votes
1answer
2k views

Simple simple Euler Lagrange Equation

Just starting a course on Lagrangian Mechanics and I'm just wondering what about the Euler-Lagrange equation, and more specifically what I'm meant to be trying to do .. One of the questions from my ...
-1
votes
2answers
91 views

Am I allowed to move around an operator like this?

Can I take this product: $$\frac{dL}{dt}\frac{d L}{d \dot{x}}$$ And factor out one of the $L$'s to get: $$L\frac{d}{dt} \left( \frac{d L}{d \dot{x}}\right)$$ Where the operator $\frac{d}{dt}$ now ...
3
votes
1answer
95 views

how to solve differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$?

What's the solution of the differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$, where $y$ is a function of $x$ and $\alpha$ is a constant? Polynomial solutions don't seem to work, because the ...
3
votes
0answers
70 views

$u''+\frac{4}{x+1}u'+\frac{2}{\left(x+1\right)^{2}}u=0$ variational solution

This is a concept solution scheme derived from a particular example that I have not been able to generalise sufficiently. The objective is to find a particular solution to a certain second-order ...
4
votes
1answer
445 views

Polar coordinates, line integrals, and the Beltrami Identity

Imagine you are walking along the xy-plane. There is a landmark at the origin of the plane which distorts time at every point on the plane, such that the distortion is a function of the distance ...
2
votes
0answers
122 views

Positive rotational symmetric solution for p-Laplacian

I have the the following problem and I just can't get my head around how to solve it. Be $1<p<n$ and $q=\frac{np}{n-p}$, $u\in\mathcal{C}_{n,p}=\{f\in W^{1,p}_{loc}: ...
1
vote
1answer
76 views

Solve $I[y]=\int_{x_0}^{x_1}y^{-\frac{1}{2}}(1+(y')^2)^\frac{1}{2} \mathrm dx$ parametrically

If $$I[y]=\int_{x_0}^{x_1}F(x,y,y') \mathrm dx$$ Where $$F=y^{-\frac{1}{2}}(1+(y')^2)^\frac{1}{2}$$ Then I have shown the Euler-Lagrange equation implies that $$y(1+(y')^2)=2a$$ For some ...
2
votes
1answer
227 views

Find the extremals of $I[y]=\int_0^1(y')^2 \mathrm dt+\{y(1)\}^2$

Could anyone help me find the extremals of $$I[y]=\int_0^1(y')^2 \mathrm dx+\{y(1)\}^2$$ subject to $y(0)=1$ Most crucially I can't work out how to find the boundary $x=1$. I'm trying to go back ...
1
vote
2answers
205 views

Weak lower semicontinuity of a functional on Hilbert space?

Let $H:=\left\{u\in L^2(R^N):\nabla u \in L^2(R^N)\right\}$ and a functional $$f(u)=\int_{R^N} |\nabla u|^2dx+\left(\int_{R^N} |\nabla u|^2dx\right)^2.$$ If $\{u_n\}\subset H$ is a sequence such that ...
13
votes
1answer
339 views

Hilbert's 19th problem: Why do we care?

Hilbert's 19th problem asks: Are the solutions of regular problems in the calculus of variations always necessarily analytic? This was proven to be true (through the work of Sergei Bernstein, ...
4
votes
3answers
322 views

Treacherous Euler-Lagrange equation

If I have an Euler-Lagrange equation: $(y')^2 = 2 (1-\cos(y))$ where $y$ is a function of $x$ subjected to boundary conditions $y(x) \to 0$ as $x \to -\infty$ and $y(x) \to 2\pi$ as $x \to ...
6
votes
2answers
244 views

Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem

A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume". At that time for me to write ...
7
votes
2answers
1k views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ...