Project Ideas for Graduate Classical Mechanics and Math Methods

[saybrook1]

Hi guys - I was hoping people might have some topic ideas for a project I need to do in graduate classical mechanics as well as math methods. It needs to be a solution to an advanced problem or cover an advanced topic. I'm just looking for ideas that haven't been beaten to death. Any suggestion will help, thank you.

 

[mpresic]

I have lots of good ideas for Graduate Classical Mechanics and /or Mathematical Physics, and they have not been beaten to death at all. These are highly original and probably will promote interest more than tired old projects I once was assigned in Chaos theory (when it was the flavor of the month in 1988). I am not really down on Chaos theory, it was just that I was voluntold to produce a paper, when I wanted to really do a paper on path integrals.

One project is hard and may satisfy both mechanics and math methods. Do an in-depth study of either the symmetric top, or free body (an-axisymmetric case) equations of motion. Include computer source code for the simulations solving the coupled non-linear differential equations for the spin-rates, and if possible the orientation.

This is a hard project involving elliptic functions with connections to Wierstrass and or Jacobian elliptic functions. You will learn an enormous wealth of mathematical physics for a complete exposition. You may need to examine old (and perhaps forgotten) books by Kellogg, Whittaker, and Landau Mechanics. Abramowitz and Stegun's mathematical handbook will probably also be necessary.

Easier (fun) project is a complete exposition of Foucault's pendulum. An in-depth presentation of the Foucault pendulum can include holonomy as in the Differential Geometry book by Oprea, als treated in texts by other mathematics authors. You might want to examine a paper in American Journal of Physics regarding treating the problem with the concept of beats. (I do not remember the citation but I can look it up, if you are interested.)

If you are more of an engineering bent, many problems in optimal control theory involve the Lagrangian formulation of mechanics. Bryson and Ho 's book Applied Optimal control shows a problem in Minimum time injection into Earth Orbit which involves solving coupled algebraic equations. Another problem involves Minimum time rendezvous with Mars from Earth orbit. This one involves solving a two point boundary value problem .

Trouble is all of these problems may be time consuming. The first and third may include involve computational resources and possibly math libraries or numerical recipes. The upside for 1 and 3 is you get significant experience in solving real-world problems. I actually use the both the first and third problem posed in my work years ago. The second was a project I used in partial fulfillment of a doctorate. We were called on to present a original paper, but it had to be unrelated to our research. Good Luck. I hope these suggestions are useful

 

[saybrook1]

mpresic, I apologize for taking so long to respond to you; my life is horribly busy as you might imagine. I really really appreciate the time you took to write this out and I think I will try and pursue either the pendulum or the symmetric top. Either way, I will let you know what I end up doing; it was really nice of you to write all of this.

 

[mpresic]

Best of Luck. I really like these ideas but you might propose this to you prof to see what he/she thinks.

 

[saybrook1]

Best of Luck. I really like these ideas but you might propose this to you prof to see what he/she thinks.

Thanks; already got the go ahead to work on the Foucault Pendulum project and it's relation to holonomy! Hopefully it works out, don't have too long to complete it.