  We study relations between vakonomically and nonholonomically constrained
Lagrangian dynamics for the same set of linear constraints. The basic idea is
to compare both situations at the level of variational principles, not
equations of motion as has been done so far. The method seems to be quite
powerful and effective. In particular, it allows to derive, interpret and
generalize many known results on non-Abelian Chaplygin systems. We apply it
also to a class of systems on Lie groups with a left-invariant constraints
distribution. Concrete examples of the unicycle in a potential field, the
two-wheeled carriage and the generalized Heisenberg system are discussed.
