The momentum equation and the second-order differential equation condition
The equations of motion of mechanical systems with nonholonomic constraints are investigated. It is shown that the momentum equation is valid for a larger class of functions than the nonholonomic momenta of the article [\textit{A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden}, and \textit{R. M. Murray}, Arch. Ration. Mech. Anal. 136, 21--99 (1996; Zbl 0886.70014)]. It is proved that the equations of motion are uniquely characterized as the second-order differential equations compatible with the momentum equation. For a free and proper action of a symmetry group \(G\), the reduced equations on the orbit space are obtained by restricting the second-order differential equation condition and the momentum equation to \(G\)- invariant functions and vector fields respectively. This practically important approach is illustrated by a simple derivation of the reduced equations of motion in the case when the configuration space coincides with the group manifold of the symmetry group \(G\), and the action of \(G\) on itself is given by the left multiplication. If the action of \(G\) is proper but not free, the restriction of the second-order differential equation condition and the momentum equation to \(G\)-invariant functions does not give equations of motion on the whole constraint distribution \(D\), but on an open subset of \(D\). This situation is also illustrated by an example showing the limitations of the method.
scientific article; zbMATH DE number 1857371
scientific article