Extending Newtonian Dynamics to Include Stochastic Processes

A paper presents further results of continuing research reported in several previous NASA Tech Briefs articles, the two most recent being “Stochastic Representations of Chaos Using Terminal Attractors” (NPO-41519), [Vol. 30, No. 5 (May 2006), page 57] and “Physical Principle for Generation of Randomness” (NPO-43822) [Vol. 33, No. 5 (May 2009), page 56]. This research focuses upon a mathematical formalism for describing postinstability motions of a dynamical system characterized by exponential divergences of trajectories leading to chaos (including turbulence as a form of chaos).

The formalism involves fictitious control forces that couple the equations of motion of the system with a Liouville equation that describes the evolution of the probability density of errors in initial conditions. These stabilizing forces create a powerful terminal attractor in probability space that corresponds to occurrence of a target trajectory with probability one. The effect in configuration space (ordinary three-dimensional space as commonly perceived) is to suppress exponential divergences of neighboring trajectories without affecting the target trajectory. As a result, the postinstability motion is represented by a set of functions describing the evolution of such statistical quantities as expectations and higher moments, and this representation is stable.

This work was done by Michail Zak of Caltech for NASA’s Jet Propulsion Laboratory. NPO-45594

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