12
2 Fundamental Concepts
first called hidden symmetries by Moser (1979). When there are as many global
symmetries as numbers of degrees of freedom, the dynamical system is said to be
integrable.
A second concept that is important for understanding the dynamics of nonlinear
systems is nonlinear resonance. As Kolmogorov (1954), Arnol’d (1963), and
Moser (1962) have shown, when a small symmetry-breaking term is added to
the Hamiltonian of system, most of the phase space continues to behave as if
the symmetries still exist. However, in regions where the symmetry-breaking term
allows resonance to occur between otherwise uncoupled degrees of freedom, the
dynamics begins to change its character. When resonances do occur, they generally
occur on all scales in the phase space and give rise to a fractal structuring of the
phase space.
The third concept that is essential for understanding conservative nonlinear
dynamics is chaos or sensitive dependence on initial conditions. For the class of
systems in which symmetries can be broken by adding small symmetry-breaking
terms, chaos first appears in the neighborhood of the nonlinear resonances. As the
strength of the symmetry-breaking term increases and the size of the resonance
regions increases, ever larger regions of the phase space become chaotic.
The dynamical evolution of systems with broken symmetry cannot be determined
using conventional perturbation theory, because of the existence of nonlinear
resonances. In Sect. 2.2, we show that nonlinear resonances cause a topological
change locally in the structure of the phase space, and that conventional perturbation
theory is not adequate to deal with such topological changes.
In Sect. 2.3, we introduce the concept of integrability. A system is integrable if it
has as many global constants of the motion as degrees of freedom. The connection
between global symmetries and global constants of motion was first proven for
dynamical systems by Noether (1918). We will give a simple derivation of Noether’s
theorem in Sect. 2.3.
It is usually impossible to tell if a system is integrable just by looking at the
equations of motion. As we show in Sect. 2.4, the Poincaré surface of section
provides a very useful numerical tool for testing integrability and will be used
throughout the remainder of this book. We will illustrate the use of the Poincaré
surface of section for the classic model of Henon and Heiles (1964) and for a model
of the HOCl molecule.
In Sect. 2.5, we introduce the concept of nonlinear resonances and illustrate
their behavior for some simple models originally introduced by Walker and Ford
(1969). These models are interesting because they show that resonances may appear
or disappear as parameters of the system are varied and the overlap of nonlinear
resonances leads to the onset of chaos.
Conventional perturbation theory does not work when nonlinear resonances are
present. But Kolmogorov, Arnol’d, and Moser (collectively called KAM) have
developed a rapidly converging perturbation theory that can be used to describe
non-resonant regions of the phase space, precisely because it is constructed to avoid
the resonance regions. KAM perturbation theory will be described in Sect. 2.6.
2 Fundamental Concepts
first called hidden symmetries by Moser (1979). When there are as many global
symmetries as numbers of degrees of freedom, the dynamical system is said to be
integrable.
A second concept that is important for understanding the dynamics of nonlinear
systems is nonlinear resonance. As Kolmogorov (1954), Arnol’d (1963), and
Moser (1962) have shown, when a small symmetry-breaking term is added to
the Hamiltonian of system, most of the phase space continues to behave as if
the symmetries still exist. However, in regions where the symmetry-breaking term
allows resonance to occur between otherwise uncoupled degrees of freedom, the
dynamics begins to change its character. When resonances do occur, they generally
occur on all scales in the phase space and give rise to a fractal structuring of the
phase space.
The third concept that is essential for understanding conservative nonlinear
dynamics is chaos or sensitive dependence on initial conditions. For the class of
systems in which symmetries can be broken by adding small symmetry-breaking
terms, chaos first appears in the neighborhood of the nonlinear resonances. As the
strength of the symmetry-breaking term increases and the size of the resonance
regions increases, ever larger regions of the phase space become chaotic.
The dynamical evolution of systems with broken symmetry cannot be determined
using conventional perturbation theory, because of the existence of nonlinear
resonances. In Sect. 2.2, we show that nonlinear resonances cause a topological
change locally in the structure of the phase space, and that conventional perturbation
theory is not adequate to deal with such topological changes.
In Sect. 2.3, we introduce the concept of integrability. A system is integrable if it
has as many global constants of the motion as degrees of freedom. The connection
between global symmetries and global constants of motion was first proven for
dynamical systems by Noether (1918). We will give a simple derivation of Noether’s
theorem in Sect. 2.3.
It is usually impossible to tell if a system is integrable just by looking at the
equations of motion. As we show in Sect. 2.4, the Poincaré surface of section
provides a very useful numerical tool for testing integrability and will be used
throughout the remainder of this book. We will illustrate the use of the Poincaré
surface of section for the classic model of Henon and Heiles (1964) and for a model
of the HOCl molecule.
In Sect. 2.5, we introduce the concept of nonlinear resonances and illustrate
their behavior for some simple models originally introduced by Walker and Ford
(1969). These models are interesting because they show that resonances may appear
or disappear as parameters of the system are varied and the overlap of nonlinear
resonances leads to the onset of chaos.
Conventional perturbation theory does not work when nonlinear resonances are
present. But Kolmogorov, Arnol’d, and Moser (collectively called KAM) have
developed a rapidly converging perturbation theory that can be used to describe
non-resonant regions of the phase space, precisely because it is constructed to avoid
the resonance regions. KAM perturbation theory will be described in Sect. 2.6.