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4196 results in General and Classical Physics

Page 153 of 210

  • 1 - Universal laws of nature

  • Joseph L. McCauley, University of Houston
  • Book:
    Classical Mechanics
    Published online:
    05 June 2012
    Print publication:
    08 May 1997, pp 1-44

  • Summary

    In the presentation of the motions of the heavens, the ancients began with the principle that a natural retrograde motion must of necessity be a uniform circular motion. Supported in this particular by the authority of Aristotle, an axiomatic character was given to this proposition, whose content, in fact, is very easily grasped by one with a naive point of view; men deemed it necessary and ceased to consider another possibility. Without reflecting, Copernicus and Tycho Brahe still embraced this conception, and naturally the astronomers of their time did likewise.

    Johannes Kepler, by Max Caspar

    Mechanics in the context of history

    It is very tempting to follow the mathematicians and present classical mechanics in an axiomatic or postulational way, especially in a book about theory and methods. Newton wrote in that way for reasons that are described in Westfall's biography (1980). Following the abstract Euclidean mode of presentation would divorce our subject superficially from the history of western European thought and therefore from its real foundations, which are abstractions based upon reproducible empiricism. A postulational approach, which is really a Platonic approach, would mask the way that universal laws of regularities of nature were discovered in the late middle ages in an atmosphere where authoritarian religious academics purported pseudo-scientifically to justify the burning of witches and other nonconformers to official dogma, and also tried to define science by the appeal to authority.

  • 18 - Simulations, complexity, and laws of nature

  • Joseph L. McCauley, University of Houston
  • Book:
    Classical Mechanics
    Published online:
    05 June 2012
    Print publication:
    08 May 1997, pp 441-452

  • Summary

    Integrability, chaos, and complexity

    Phase space flows divide into two basic classes: completely integrable and ‘nonintegrable’ (incompletely integrable). In the first case the motion is equivalent to a single time-translation for all finite times (chapter 3). In the second case singularities of the conservation laws prevent this degree of simplicity (section 13.2).

    Nonintegrable systems can be divided further into two classes: those that are in principle predictable for all times, and those that are not. The first class includes deterministic dynamics where there is a generating partition. An example, the asymmetric tent map, was given in chapter 14. A chaotic dynamical system generates infinitely many different statistical distributions through the variation of classes of initial conditions. Only the differentiable distribution (corresponding to the invariant density) is generated by initial data that occur with measure one. The fragmented statistical distributions, including multifractal ones, occur for the measure zero set of initial conditions.

    Whenever a dynamical system is computable (McCauley, 1993), which is the only case that can be used computationally and empirically, then the measure zero initial conditions are of dominant importance. There is, as yet, no known case in nature where the differentiable probability distribution describes the experiments or observations performed on a driven-dissipative chaotic dynamical system. The generating partition forms the support of every probability distribution that occurs mathematically, and can be discovered (when it exists) through backward iteration of the Poincaré map.

Page 153 of 210