22 research outputs found
On the geometry and behavior of n
The kinematic separation of size, shape, and orientation of n-body systems is investigated together with specific issues concerning the dynamics of classical n-body motions. A central topic is the asymptotic behavior of general collisions, extending
the early work of Siegel, Wintner, and more recently Saari. In particular, asymptotic formulas for the derivatives of any order of the basic kinematic quantities are included. The kinematic Riemannian metric on the congruence and shape moduli spaces are introduced via O(3)-equivariant geometry. For n=3, a classical geometrization procedure is explicitly carried out for planary 3-body motions, reducing them to solutions of a rather simple system of geodesic equations in the 3-dimensional
congruence space. The paper is largely expository and various known results on classical n-body motions are surveyed in our more geometrical setting
Symbol correspondences for spin systems
The present monograph explores the correspondence between quantum and
classical mechanics in the particular context of spin systems, that is,
SU(2)-symmetric mechanical systems. Here, a detailed presentation of quantum
spin-j systems, with emphasis on the SO(3)-invariant decomposition of their
operator algebras, is followed by an introduction to the Poisson algebra of the
classical spin system and a similarly detailed presentation of its
SO(3)-invariant decomposition. Subsequently, this monograph proceeds with a
detailed and systematic study of general quantum-classical symbol
correspondences for spin-j systems and their induced twisted products of
functions on the 2-sphere. This original systematic presentation culminates
with the study of twisted products in the asymptotic limit of high spin
numbers. In the context of spin systems, it shows how classical mechanics may
or may not emerge as an asymptotic limit of quantum mechanics.Comment: Research Monograph, 171 pages (book format, preliminary version