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
This Brief includes a Technical Support Package (TSP).
Extending Newtonian Dynamics To Include Stochastic Processes
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Overview
The document titled "Extending Newtonian Dynamics to Include Stochastic Processes" is a technical support package prepared under the auspices of NASA's Jet Propulsion Laboratory (JPL). It aims to explore the integration of stochastic processes into classical Newtonian dynamics, highlighting the implications of this extension for understanding complex systems.
The core premise of the document is that traditional Newtonian mechanics, which relies on deterministic equations of motion, can be enriched by incorporating stochastic elements. This approach acknowledges that real-world systems often exhibit random behavior due to various factors, such as environmental influences or inherent uncertainties in initial conditions. By introducing stochastic processes, the authors seek to develop a more comprehensive framework that can better describe the dynamics of such systems.
The document discusses the mathematical foundations necessary for this extension, including the use of ordinary differential equations (ODEs) and their relationship with partial differential equations (PDEs). It emphasizes that while ODEs provide a deterministic description of motion, the introduction of stochastic forces leads to a coupling with the Liouville equation, resulting in non-Newtonian properties. This coupling is crucial for understanding the evolution of probability densities in dynamic systems.
One significant aspect covered is the destabilizing effect of feedback mechanisms in stochastic systems. The authors illustrate how feedback can alter the stability of a system, necessitating a careful analysis of the conditions under which these changes occur. The document also references various studies and models that have contributed to the understanding of these dynamics, including works by notable researchers in the field.
Additionally, the document serves as a resource for those interested in the broader implications of these findings, suggesting potential applications in aerospace technology and other fields where understanding complex, dynamic systems is essential. It provides contact information for further inquiries and emphasizes compliance with U.S. export regulations regarding the proprietary information contained within.
In summary, this technical support package presents a novel perspective on Newtonian dynamics by integrating stochastic processes, offering insights into the behavior of complex systems and paving the way for future research and applications in various scientific and engineering domains.
