Year: 1991
Type: article
Abstract: Newton, Lagrange, Hamilton, Poisson.These pioneers who formulated classical mechanics in ever more elegant terms were unaware of the trouble lying ahead.Poincaré was the first to see the abyss, but the warning he expressed received little attention during his lifetime.Two years after his death in 1912, chaos became the new paradigm in Europe . . . in everything except mechanics.The formulation of quantum mechanics in the 1920's by Heisenberg, Schrödinger and Dirac further delayed an upheaval in classical mechanics by provoking a revolution in physics.It swept classical dynamics with its potentially interesting but yet little understood peculiarities into limbo for some 50 years.Recently the long-ignored trouble spots of classical mechanics have come back with a vengeance.The profound implications of chaos in nonlinear dynamics are apt to undermine our understanding of quantum mechanics.Our goal is to illustrate classical and quantum manifestations of Hamiltonian chaos with some simple models of nonlinear dynamics.The emphasis is on the use of modern computational tools both to visualize the connection between theoretical integrability criteria and phenomenological manifestations of chaos and to establish a link as best we can between classical Hamiltonian chaos and quantum nonintegrability effects."Where the action is, there is no chaos."This paradoxical statement is true in Hamiltonian dynamics.The integrability of systems with two degrees of freedom depends on the existence of a second integral of the motion I(q 1 , q 2 , p 1 , p 2 ) in addition to the Hamiltonian H(q 1 , q 2 , p 1 , p 2 ).There are far-reaching geometric and analytic consequences of the existence of this second integral.1 The entire four-dimensional (4d) phase space is densely foliated by 2d invariant tori, and the Hamiltonian is expressible as a function H(J 1 , J 2 ) of two global action variables J 1 , J 2 .There exists a canonical transformation under which the canonical equations
Source: Computers in Physics
DOI: 10.1063/1.4822981
Authors: Niraj Srivastava, Charles Kaufman, Gerhard Müller
Institution: University of Rhode Island
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4211036069