There are three basic concepts which are essential for understanding the dynamical behavior of nonlinear conservative systems. The first is the concept of global symmetrieswhich serve to constrain the dynamical flow of the system to lower dimensional surfaces in the phase space. Some of these global symmetries are obvious and are related to the space-time symmetries of the system. Others are not obvious and have been called “hidden symmetries” by Moser [Moser 1979]. When there are as many global symmetries as degrees of freedom, the dynamical system is said to be integrable. The second important concept is that of nonlinear resonance. As Kolmogorov [Kolmogorov 1954], Arnol’d [Arnol’d 1963], and Moser [Moser 1962] have shown, when a small symmetry-breaking term is added to the Hamiltonian, 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 an incredibly complex structure as we shall see. The third important concept is that of chaosor sensitive dependence on initial conditions.
KeywordsPhase Space Canonical Transformation Toda Lattice Nonlinear Resonance Bernoulli Shift
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