Small oscillations in nonholonomic systems

[zwierz]

I wonder why nobody discuss this topic in classical mech. courses

 

[Mmm_Pasta]

Non-holonomic systems are not covered in much detail to begin with. I haven't had to encounter non-holonomic systems in my other courses. Although, I guess you could say the same with Hamilton-Jacobi theory and Poisson Brackets with the exception of quantum mechanics. In my graduate course in classical mechanics we only solved problems involving cars and unicycles with various restraints imposed on the steering wheel, direction of motion (forward/reverse). etc. I'm actually not sure how knowledgeable physicists are in general about non-holonomic systems. What are your experiences?

I know that there is one professor in my department who loves classical mechanics. I never took a course from him, but my peers who did showed and talked about some interesting problems that I didn't get to see when I took classical mechanics.

 

[Dr.D]

I wonder why nobody discuss this topic in classical mech. courses

If you have something in particular in mind, why not post it for discussion here?

 

[zwierz]

I just copy my answer to the letter of one of the participants:
Example: a hemisphere is rolling about its equilibrium on a horizontal plane. There is no slipping. It is proposed to write equations of motion in the linear approximation.
Reference: E. T. Whittaker A Treatise On The Analytical Dynamics Of Particles And Rigid Bodies

 

[Dr.D]

I am familiar with that problem, although i do not recall just where it is in Whittaker (my copy is on the shelf, and it stays there most of the time).

Let's consider the answer to your original question in terms of this specific example, the hemisphere rolling on a horizontal plane without slipping. What actual physical system does this model? Can you think of any? I cannot. This is the crux of the matter as to why such problems are not discussed very much; they do not represent actual systems of interest.

Perhaps there is something in that problem (or a similar nonholonomic problem) that could be of great value, but no one has recognized it as yet. In a parallel situation, I am sure that things hung from vines, ropes, and such long before anyone thought to use a pendulum for time keeping. If problems of the sort motivating this thread are of interest to you, perhaps you will be the one to discover some aspect of great use to mankind. I do not think you will find this a crowed field for research.

 

[zwierz]

Actually nonholonomic mechanics is a fundamental and very popular field of modern research. You can make sure of that even with the help of google; not to mention such classical effects as

 

[Dr.D]

What you have shown under the name Celtic Stone I know as a Rattleback.

It may be a very popular research topic for those who simply want to know, but it is not very popular at all with those that have a specific task to accomplish. When you understand this matter and can completely model it mathematically, what are you then able to do?