Non-holonomy, critical manifolds and stability in constrained Hamiltonian systems

We approach the analysis of dynamical and geometrical properties of nonholonomic mechanical systems from the discussion of a more general class of auxiliary constrained Hamiltonian systems. The latter is constructed in a manner that it comprises the mechanical system as a dynamical subsystem, which is confined to an invariant manifold. In certain aspects, the embedding system can be more easily analyzed than the mechanical system. We discuss the geometry and topology of the critical set of either system in the generic case, and prove results closely related to the strong Morse-Bott, and Conley-Zehnder inequalities. Furthermore, we consider qualitative issues about the stability of motion in the vicinity of the critical set. Relations to sub-Riemannian geometry are pointed out, and possible implications of our results for engineering problems are sketched.Comment: Latex, 58 page

On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies

We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$, for cubic, quintic, focusing and defocusing interactions. For both the focusing and defocusing case, and any $d\geq1$, we prove local existence and uniqueness of solutions in certain Sobolev type spaces \cH_\xi^\alpha of sequences of marginal density matrices. The regularity is accounted for by \alpha>\frac12& if d=1$,$\alpha>\frac d2-\frac{1}{2(p-1)} $if$d\geq2$and$(d,p)\neq(3,2)$, and$\alpha\geq1$if$(d,p)=(3,2)$, where$p=2$for the cubic, and$p=4$for the quintic GP hierarchy; the parameter$\xi>0$is arbitrary and determines the energy scale of the problem. This result includes the proof of an a priori spacetime bound conjectured by Klainerman and Machedon for the cubic GP hierarchy in$d=3$. In the defocusing case, we prove the existence and uniqueness of solutions globally in time for the cubic GP hierarchy for$1\leq d\leq3$, and of the quintic GP hierarchy for$1\leq d\leq 2$, in an appropriate space of Sobolev type, and under the assumption of an a priori energy bound. For the focusing GP hierarchies, we prove lower bounds on the blowup rate. Also pseudoconformal invariance is established in the cases corresponding to$L^2$ criticality, both in the focusing and defocusing context. All of these results hold without the assumption of factorized initial conditions.Comment: AMS Latex, 28 page