User user23534523 - MathOverflow

Restricted Three-Body Problem

2011-07-31T14:59:08Z

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity of solving the effect of three bodies which all pull on each other while moving, a total of six interactions. Mathematician Richard Arenstorf while at NASA solved a special case of this problem, by simplifying the interactions to four, because, the effect of the spacecraft's gravity upon the motion of the vastly more massive Earth and Moon is practically non-existent. Arenstorf found a stable orbit for a spacecraft orbiting between the Earth and Moon, shaped like an '8'

http://en.wikipedia.org/wiki/Richard_Arenstorf

Arenstorf's technical report is here

http://hdl.handle.net/2060/19630005545

Was Arenstorf's solution purely analytical, or did he use numerical mechanisms? Is the '8' shape an optimal path, meaning the route on which the spacecraft would expand the least amount of energy? If yes, how was this requirement included in the derivation in mathematical form?

If anyone has a clean derivation for this problem, that would be great, or any links to books, other papers, etc.

Thanks

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

2011-04-01T14:01:29Z

For a image denoising problem (below):

http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf

the author has a functional E defined

$E(u) = \int\int_\Omega F \ d\Omega$

which he wants to minimize. F is defined as

$F = ||\nabla u ||^2 = u_x^2 + u_y^2$

Then, the E-L equations are derived:

$\frac{\partial E}{\partial u} = \frac{\partial F}{\partial u} - \frac{d}{dx} \frac{\partial F}{\partial u_x} - \frac{d}{dy} \frac{\partial F}{\partial u_y} = 0$

Then it is mentioned that gradient descent method is used to minimize the functional E by using

$\frac{\partial u}{\partial t} = u_{xx} + u_{yy}$

which is the heat equation. I understand both equations, and have solved the heat equation numerically before. I also worked with functionals. I do not understand however how the author jumps from the E-L equations to the gradient descent method. How is the time variable t included? Any detailed derivation, proof on this relation would be welcome. I found some papers on the Net, the one by Colding et al looked promising.

References:

http://arxiv.org/pdf/1102.1411 (Colding et al)

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.1675&rep=rep1&type=pdf (ROF)

http://dl.dropbox.com/u/1570604/tmp/functional-grad-descent.pdf

http://dl.dropbox.com/u/1570604/tmp/gelfand_var_time.ps (Gelfand and Romin)

Answer by user23534523 for Thinking and Explaining

2011-04-28T20:23:54Z

Thank you for starting this discussion. I think any kind of pedagogical tool should be shared with students and collegues, especially in writing form. When I read "the classics", i.e. works of famous mathematicians I always wondered the process they went through to reach those conclusions, what imagery went through their heads while they proved a theorem, maybe that would be useful to me, or not, but I always wanted to know. I think during a lecture saying something like "here is how I do it, imagine group elements breaking into a formation organized into circular groups" is no discomfort to anyone. Maybe this explanation can help one student, or two (or all) students to grok the topic just a tad more, and that's still important. Richard Feynman used to say (paraphrasing) that he never really knew in advance how his students would understant quantum mechanics, he did not have any single method, he'd only try to explain the topic from many different angles hoping that one of those angles provide an entry point for a student, into the subject.

Comment by user23534523

2011-04-02T07:43:36Z

I am trying to work my way up to ROF paper :) I wanted to understand the simplest case first, then do ROF -- this gradient descent connection was my first hurdle. I have some sample ROF Matlab code, I think my questions there will be about the use of Lagrange multipliers, and their discretization method.

Comment by user23534523

2011-04-02T07:29:32Z

this is not a homework problem no, it's for my own learning.