The momentum equation and the second order differential equation condition

It is shown that equations of motion of mechanical systems with nonholonomic constraints are equivalent to the momentum equation for all vector fields on the configuration space with values in the constraint distribution, and the second-order differential equation condition for all smooth functions on the configuration space. For a free and proper action of the symmetry group on the configuration space, the reduced equations of motion are equivalent to the momentum equation for all invariant vector fields on the configuration space with values in the constraint distribution and the second-order differential equation condition for all invariant functions on the configuration space. Applications to singular reduction are discussed.