Nonintegrable systems

MSN. 129.1. Prigogine, Time, dynamics and chaos Integrating Poincare s nonintegrable systems, in XXVI Nobel Conference", J. Holte, ed., Gustavus Adolphus College, St. Peter, Minnesota, pp. 55-88. 

MSN. 148.1. Prigogine, Why irreversibility The formulation of classical and quantum mechanics for nonintegrable systems, Lecture Notes, Department of Physics, Keio University, Yokohama, 1994. 

Classical dynamics and orthodox quantum mechanics are constructed along the model of integrable systems in the sense of Poincare. Our aim is to construct dynamics for nonintegrable systems. As far as we know, this is a new attempt, which has its roots in the early work of the Brussels School [1-9]. The main result is that we have to replace the unitary transformation f/ by a nonunitary... 

An important point is that AA = 1. So we can go from the initial representation to the A representation and come back by A . We have verified this property. The transformation to the A representation leads to a number of new properties hidden in the initial representation. The equations for density matrices become irreducible to classical trajectories or wave amplitudes. For nonintegrable systems, amplitudes have to be replaced by probabihties. 

The situation changes drastically for nonintegrable systems. As we will see, the transformation U = U is replaced by a nonunitary transformation At A. ... 

QUANTUM AND CLASSICAL DYNAMICS OF NONINTEGRABLE SYSTEMS 141 and the relations for... 

The fluctuations are the consequence of nondistributivity of the A transformation. We need a new mathematical framework (i.e., nondistributive algebra) to analyze nonintegrable systems. This fact reminds us that whenever we found new aspects in physics, we needed new mathematical frameworks, such as calculus for Newton mechanics, noncommutative algebra for quanmm mechanics, and the Riemann geometry for relativity. 

We believe that our approach is only the starting point for new investigations on physics of nonintegrable systems. The horizon seems widely open for new results. 

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