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
We consider the dynamical Gross-Pitaevskii (GP) hierarchy on Rd,
d≥1, for cubic, quintic, focusing and defocusing interactions. For both
the focusing and defocusing case, and any d≥1, 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)} ifd\geq2and(d,p)\neq(3,2),and\alpha\geq1if(d,p)=(3,2),wherep=2forthecubic,andp=4forthequinticGPhierarchy;theparameter\xi>0isarbitraryanddeterminestheenergyscaleoftheproblem.ThisresultincludestheproofofanapriorispacetimeboundconjecturedbyKlainermanandMachedonforthecubicGPhierarchyind=3.Inthedefocusingcase,weprovetheexistenceanduniquenessofsolutionsgloballyintimeforthecubicGPhierarchyfor1\leq d\leq3,andofthequinticGPhierarchyfor1\leq d\leq
2,inanappropriatespaceofSobolevtype,andundertheassumptionofanapriorienergybound.ForthefocusingGPhierarchies,weprovelowerboundsontheblowuprate.AlsopseudoconformalinvarianceisestablishedinthecasescorrespondingtoL^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