The geometry of nonholonomic Chaplygin systems revisited

We consider nonholonomic Chaplygin systems and associate to them a $(1,2)$ tensor field on the shape space, that we term the gyroscopic tensor, and that measures the interplay between the non-integrability of the constraint distribution and the kinetic energy metric. We show how this tensor may be naturally used to derive an almost symplectic description of the reduced dynamics. Moreover, we express sufficient conditions for measure preservation and Hamiltonisation via Chaplygin's reducing multiplier method in terms of the properties of this tensor. The theory is used to give a new proof of the remarkable Hamiltonisation of the multi-dimensional Veselova system obtained by Fedorov and Jovanovi\'c in J. Nonlinear Sci. 2004, and Reg. Chaot. Dyn. 2009.