The geometry of equations of motion: particles in equivalent universes

The equations of motion for the simplest non-holonomically constrained system of particles are formulated using six methods: Newton-Euler, Lagrange, Maggi, Gibbs-Appell, Kane, and Boltzmann-Hamel. The challenging tasks of exploring and explaining the relationships and equivalences between these formulations is accomplished by constructing a single representative particle for the system of particles. The single particle is constrained to move on a configuration manifold. The explicit construction of sets of tangent vectors to the manifold and their relation to the forces acting on the single particle are used to provide several helpful geometric interpretations of the relationships between the formulations. These interpretations can also be extended to help understand the relationships between different formulations of the equations of motion for more complex systems, including systems of rigid bodies and particles.

Automated summary

By constructing a single representative particle constrained to move on a configuration manifold, the relationships and equivalences between different formulations of the equations of motion can be explored and explained through several helpful geometric interpretations.