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Via compression ([11, 7]) we write the $n$-dimensional Chaplygin spheresystem as an almost Hamiltonian system on $T^*SO(n)$ with internal symmetrygroup $SO(n-1)$. We show how this symmetry group can be factored out, and passto the fully reduced system on (a fiber bundle over) $T^*S^{n-1}$. Thisapproach yields an explicit description of the reduced system in terms of thegeometric data involved. Due to this description we can study Hamiltonizabilityof the system. It turns out that the homogeneous Chaplygin ball, which is notHamiltonian at the $T^*SO(n)$-level is Hamiltonian at the $T^*S^{n-1}$-level.Moreover, the 3-dimensional ball becomes Hamiltonian at the $T^*S^2$-levelafter time reparametrization, whereby we re-prove a result of [4, 5] insymplecto-geometric terms. We also study compression followed by reduction ofgeneralized Chaplygin systems.

$G$-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball