0
votes
1answer
29 views

Infimum and supremum of $\int_{0}^{1} e^{x(t)}(x'(t))^{2} \; dt$

Find infimum and supremum of $$\phi[x]=\int_{0}^{1} e^{x(t)}(x'(t))^{2} \; dt$$ where $x \in C^{1}[0,1]$ and $x(0)=0$ and $x(1)=\log 4$. It's easy to show that $\sup \phi[x]=\infty$, but what about ...
0
votes
1answer
36 views

Determining the Euler-Lagrange equations for a minimizataion problem

I'm working on a problem in computer vision and I've ended up trying to minimize the functional $$\int \left[\lambda(S''(x))^2 + (f(x) - S(x))^2 \sum_k \delta (x - x_k)\right]dx$$ where $\lambda$ is ...
1
vote
1answer
21 views

Deriving a high ordered Euler-Lagrange equation.

I've been able to derive the Euler-Lagrange equation for $$\int_a^b F(x,y,y')dx$$ relatively easily by using the total derivative and integration by parts. However, I was unable to apply the same ...
-1
votes
0answers
24 views

Converting partial DE to integral Equation [closed]

Can anybody help me solving the below problem: What would be the functional corresponding to the following problem: $$ \frac{\partial ^{2}u}{\partial x^{2}}+ \frac{\partial ^{2}u}{\partial y^{2}} = ...
1
vote
1answer
67 views

how to solve the system of differential equations for this particle?

I'm trying to solve this problem A particle of mass m moves under the action of gravity on the inner surface of a paraboloid of revolution $x^2+y^2=az$ which assumed frictionless. Obtain the ...
0
votes
0answers
8 views

Volterra derivative of utility function in economics

I have difficulties concerning the calculus of first variation (which is also nominated as Volterra derivative I suppose.) Except the explanation on wikipedia, I did not see any useful explanation on ...
1
vote
0answers
44 views

Denominator of a function

I have a function $S(x,y)$ which satisfies the following PDE $$\frac{\partial S(x,y)}{\partial y}=-H\left(x,\frac{\partial S(x,y)}{\partial x}\right)$$ where the known function ...
1
vote
0answers
22 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
1
vote
1answer
45 views

Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
1
vote
0answers
49 views

Brachistochrone Problem to find out the path by which a bead travels in least time

The question is to find the shape of the curve down whcih a bead sliding from rest and accelerated by gravity will slip(without friction) from one point to another in the least time. So I proceeded ...
0
votes
0answers
68 views

Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Euler–Lagrange ...
0
votes
1answer
43 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
29 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
83 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
33 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
53 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
121 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
16 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 ...
6
votes
1answer
470 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
55 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
79 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
64 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
179 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
83 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
79 views

Euler-Lagrange problem solution

Hi, Can anyone solve this question? I have no clue.
1
vote
1answer
631 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: $\frac{d}{dx}(py') + (q-\lambda r)y = 0$. ...
1
vote
0answers
102 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)) - ...
5
votes
1answer
348 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
86 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
242 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
90 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. $$y_p = u_1(x)x^{1/2} + u_2(x)x^{1/2}\log x$$
2
votes
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
281 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
126 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
178 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? $$ I(u)=\int_\Omega |u|^{p+1} dx , \quad 1<p<\frac{n+2}{n-2} $$ ...
5
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
94 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
97 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 ...
4
votes
0answers
87 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
456 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
131 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}: ...
2
votes
1answer
79 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
242 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
233 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
428 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, ...
5
votes
3answers
346 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
254 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 ...
8
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 + ...