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Teaching Dynamics: Particles in Equivalent Universes

oliver oreilly's picture

If you have ever had to study and teach classical mechanics, then one of the challenges is to explain the equivalences of distinct formulations of the equations of motion for discrete mechanical systems. It is not transparent, particularly in the presence of constraints, how the Newton-Euler equations, Lagrange’s equations, Gibbs-Appell equations, Maggi's equations, Kane’s equations, Boltzmann-Hamel equations, and several other fomulations, are equivalent. The large number of applicable principles and the challenges of explaining the concept of virtual work to students can also be very intimidating and frustrating.   


In joint work [1, click here for graphical summary on twitter] with Theresa Honein, we exploit the idea that every discrete mechanical system can be represented by a single particle moving on a manifold in a space of appropriate dimensions. The equations of motion for this particle can be formulated using the Newton-Euler balance laws for the mechanical system. With the assistance of this construction, we are then able to show that all of the aforementioned formulations are equivalent to linear combinations of the Newton-Euler balance laws accompanied by a judicious selection of the coordinates and, in some instances, quasi-velocities. 


We illustrate our discussion using the example of the simplest possible non-holonomically constrained system of particles. This two particle system is equivalent to the celebrated (and much studied) Chaplygin sleigh. As with many non-holonomically constrained systems, such as the rattleback, this system has a preferred direction of motion - the twitter link above has animations of these motions.


In a companion work [2, click here for graphical summary on twitter] we use a recent treatment of constraint forces and constraint moments [3,4] to present a simple derivation of the Gibbs-Appell equations of motion for a constrained rigid body using the Newton-Euler balances of linear momentum and angular momentum as a starting point.


I hope these works help you with your teaching and/or studies. The references cited are given below, please feel free to reach out to me ( if you are interested in copies. 


1. T. E. Honein and O. M. O’Reilly. The Geometry of Equations of Motion: Particles in Equivalent UniversesNonlinear Dynamics, 2021


2. T. E. Honein and O. M. O’Reilly. On the Gibbs-Appell Equations for the Dynamics of Rigid BodiesASME Journal of Applied Mechanics, 2021.


3. O. M. O’Reilly. Intermediate Dynamics for Engineers: Newton-Euler and Lagrangian Mechanics. Cambridge University Press, Cambridge, second edition, 2020.


4. O. M. O’Reilly and A. R. Srinivasa. A Simple Treatment of Constraint Forces and Constraint Moments in the Dynamics of Rigid BodiesASME Applied Mechanics Reviews, 67(1):014801-014801-8, 2014.






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