Regular and Chaotic Dynamics
What's in a name? The original title of our book, Regular and Stochastic Motion, was chosen to emphasize Hamiltonian dynamics and the physical motion of bodies. The new edition is more evenhanded, with considerably more discussion of dissipative systems a
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Overview and Basic Concepts
1.1. An Introductory Note This volume grew out of developments in dynamics aimed at understanding the behavior of an oscillator for a slow change in parameters and at understanding the behavior of coupled oscillators when the coupling is weak . These two problems, first considered independently, were found to be intimately related for multiply periodic systems. The understanding of slowly varying parameters was given its major impetus by Einstein at the Solvay Conference of 1911 , when he suggested a physical significance for the action integral. He pointed out that its "adiabatic" constancy, first demonstrated by Liouville and Green three quarters of a century earlier, was directly related to the physical notion that the number of quanta should remain constant in a slowly varying system. The resulting WK B method (Wentzel, 1926; Kramers, 1926; Brillouin, 1927) become one of the cornerstones of the treatment of wave propagation in inhomogeneous media, and of wave mechanics. The underlying theory was developed by Bogoliubov and Mitropolsky (1961) and by Kruskal (1962) and is commonly known as the method of averaging. The second development, that of treating coupled nonlinear oscillators, began with attempts to solve the three-body problem of celestial mechanics, which serves as a simplified model for the solar system . Early work dates back to the investigations of Hamilton and Liouville in the mid-nineteenth century, and stimulated the development of the Hamiltonian formalism that underlies most of our treatment of mechanics. Toward the end of the nineteenth century, many of the ideas concerning the stability of coupled nonlinear systems were investigated by Poincare (1892) and applied to the problems of celestial mechanics. It was during this period that he, Von Zeipel
A. J. Lichtenberg et al., Regular and Chaotic Dynamics © Springer Science+Business Media New York 1992
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1. Overview and Basic Concepts
(1916), and others devised the perturbative methods that have proved so fruitful in describ ing the short-time behavior of these systems. The emergence of the quantum theory greatly stimulated these developments. Secular perturbation theory, which allows the local inclusion of resonant interaction between two degrees of freedom, was formalized in the early days of quantum mechanics (Born, 1927). In Chapter 2 we treat perturbation techniques using the earlier classical methods and the more modern Lie formalism (e.g., Deprit, 1969). An important question that could not be answered by the early perturbative techniques was that of the long-time stability of the solar system . The prevailing belief was that planetary motion was "regular " (quasiperiodic) and might ultimately be "solved" by new mathematical methods. This belief was reinforced by the known solutions of the two-body problem and other simple mechanical systems and by the interpretation of the fossil record, which suggested the regularity of the earth's motion around the sun over hundreds of millions of years . At the same ti
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