Small oscillations in nonholonomic systems

In summary, the participants discussed the lack of discussion about non-holonomic systems in classical mechanics courses. One participant shared their experience with non-holonomic systems in graduate level courses, which mainly involved solving problems involving cars and unicycles with various restraints. The conversation then shifted to a specific problem from a reference book and the potential value of studying non-holonomic systems. However, it was noted that while non-holonomic mechanics is a popular field of research, it is not as relevant or useful for practical purposes.
  • #1
zwierz
334
62
I wonder why nobody discuss this topic in classical mech. courses
 
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  • #2
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.
 
  • #3
zwierz said:
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?
 
  • #4
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
 
  • #5
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.
 
  • #6
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

 
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  • #7
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?
 

1. What are small oscillations in nonholonomic systems?

Small oscillations in nonholonomic systems refer to small, periodic movements of a system around its equilibrium state, where the constraints of the system are not holonomic (i.e. they cannot be expressed as simple geometric relationships between the coordinates). These oscillations can occur in a variety of physical systems, such as mechanical systems, electrical circuits, and biological systems.

2. How do small oscillations differ from large oscillations?

The main difference between small and large oscillations is the amplitude of the oscillations. Small oscillations have amplitudes that are much smaller compared to the system's equilibrium state, while large oscillations have amplitudes that are comparable to or larger than the equilibrium state. Small oscillations can often be approximated linearly, while large oscillations may require nonlinear analysis.

3. What is the importance of studying small oscillations in nonholonomic systems?

Studying small oscillations in nonholonomic systems can provide valuable insights into the behavior and stability of the system. By analyzing the system's linearized equations of motion, we can determine the system's natural frequencies and modes of vibration, which can help us understand how the system responds to external forces and perturbations.

4. How are small oscillations in nonholonomic systems mathematically represented?

Small oscillations in nonholonomic systems can be described mathematically using the theory of linear dynamical systems. This involves writing the system's equations of motion in terms of the system's generalized coordinates and constraints, and then linearizing these equations around the equilibrium state. This results in a set of linear differential equations that can be solved to obtain the system's natural frequencies and modes of vibration.

5. What are some real-world applications of small oscillations in nonholonomic systems?

Small oscillations in nonholonomic systems have a wide range of applications in various fields, including mechanical engineering, electrical engineering, and physics. For example, the suspension system of a car can be modeled as a nonholonomic system undergoing small oscillations, which can help engineers design a more stable and comfortable ride. In physics, small oscillations in complex systems such as protein molecules can provide insights into their structural and functional properties.

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