Examples of singular nonholonomic reduction
The author's main idea is to use the nonholonomic bracket for the singular nonholonomic reduction. The motivation for the presented theory was born out of the observation that the obvious symmetry group in the tippe top does not act freely and that the exceptional orbits of the group action occur when the top is in one of the most interesting positions, the vertical ones. Of course, the central idea is to use invariants. The first example is the dynamic hatchet planimeter which exhibits asymptotic stability in the reduced two-dimensional problem that is expected to be Hamiltonian. Then, however, it can not have asymptotically stable orbits by a theorem of Hamiltonian mechanics. Singular reduction explains this seeming paradox and yields a geometrically faithful framework for studying singularity in the tippe top. Simply speaking, the idea is instead of looking at the distributional symplectic geometry of the problem to investigate the related `Poisson' bracket on the space of functions. This approach to the nonholonomic bracket brings a more direct way to see the relationship between the integrability of the constraints and the Jacobi identity for the bracket and a convenient computational framework for examinig the free particle by reinterpreting the Dirac construction for constrained brackets.
scientific article; zbMATH DE number 1293242