Invariance of subbundles and nonholonomic trajectories

Abstract

In the comparison of nonholonomic mechanics and constrained variational mechanics, invariant affine subbundles arise in the determination of the initial conditions where the two methods yield the same trajectories. Motivated by this, differential conditions are considered for invariant affine subbundles as they arise in the comparison of nonholonomic and constrained variational mechanics. First of all, the formal integrability of the resulting linear partial differential equation is determined using Spencer cohomology. Second, iterative formulae are provided that permit the determination of the largest invariant affine subbundle invariant under an affine vector field. Finally, the problem of a disc rolling on an inclined plane with no-slip is considered as an example to illustrate the theory.