Singular perturbation approach to rattleback reversals

Lagrangian systems with constraints are common models of fundamental or idealized physical systems. Holonomic constraints, typified in the example of a freely moving rigid body, give rise to systems with certain special properties, such as symplectic Hamiltonian systems. Nonholonomic constraints arise in such systems as a disk which rolls without slipping. Holonomic and nonholonomic systems have differing mathematical structures, and they have different behaviors [1, 3, 4, 9, 21, 22, 24]. For example, when a vehicle has good contact with a road, then it is behaving as a nonholonomic system. Angular momentum is not conserved; otherwise, the vehicle could not be steered into a turn. Under icy conditions, the vehicle is essentially a holonomic system; then steering cannot change its angular momentum, the vehicle cannot be turned, and whatever spin it has will persist. Conservation of energy, however, is a dominant feature of both systems. A rattleback is a toy top in the shape of long, narrow boat, with a slight, usually imperceptible asymmetry, either in its shape, or in its mass distribution. Many people anticipate that, when spun on a table, the rattleback will behave as other tops do i.e. holonomically. And, when spun in one direction, the rattleback will behave like this. When spun in the opposite direction, rattlebacks will spontaneously reverse direction, exhibiting nonholonomic non-conservation of angular momentum. As it turns out, because of the asymmetry, the table and the rattleback are coupled nonholonomically. Angular momentum is not conserved: some time after spinning in the unstable direction, the rattleback is observed to be spinning, at nearly the same rate, in the opposite direction. The transition between the two spins is dynamically complicated: it occurs through a non-spinning longitudinal wobbling motion. This is the rattleback’s spin reversal. Rattlebacks have been observed, and the basic mathematical model obtained, for over a century [26, 27], and the have been researched off-an-on since then [2, 5, 6, 7, 8, 15, 16, 17, 19, 20, 23, 25, 28]. But the spin reversal is a global dynamical feature, and its understanding is incomplete. Part of the problem is the sheer complexity of the system. The (reduced) rattleback equations of motion, for a body with surfaceM rolling on the plane, are