Article
Nearly Hamiltonian Systems in Nonlinear Mechanics: Averaging and Energy Methods
Indiana University Mathematics Journal
(1976)
Abstract
Among the approaches to differential equations, perturbation
theory has tended to concentrate on local aspects-continuation of a
particular periodic solution or invariant torus as a parmeter is varied-while
the global approach has dealt principally with "generic" systems, so that its
results are seldom applicable to a concrete perturbation problem. This paper is
addressed to the global side of perturbation theory. In problems arising from
perturbation of Hamiltonian systems in action/angle variables, or from large
assemblies of weakly coupled rotators or oscillators, or from weak quasiperiodic
forcing, there are an infinite number of possible periodic and quasiperiodic
solutions of different frequencies, and it is impossible to build up a global picture
by studying the existence or nonexistence of each such orbit separately. We
seek to establish nonexistence of invariant tori over large regions of space and
to learn how orbits flow between the remaining small open sets. In some cases
it is possible to obtain a fairly complete global picture in this way. Our results
are vacuous under the hypotheses of the Kolmogorov-Amol'd-Moser theory
([1], [10]), when invariant tori do in fact exist nearly everywhere in space. The
results of this paper were announced in [13]; special cases with two degrees of
freedom are contained in [11], [12]. These papers contain examples and motivation
not repeated here.
Keywords
- oscillators,
- integers,
- Mathematical manifolds,
- Hyperplanes,
- Fourier series,
- Damping,
- Degrees of freedom,
- Coordinate systems,
- Mathematical vectors,
- Matrices
Disciplines
Publication Date
1976
Publisher Statement
Copyright 1976 Indiana University Mathematics Journal
Citation Information
James Murdock. "Nearly Hamiltonian Systems in Nonlinear Mechanics: Averaging and Energy Methods" Indiana University Mathematics Journal Vol. 25 Iss. 6 (1976) p. 499 - 523 Available at: http://works.bepress.com/james-murdock/2/