Nonlinear Hamiltonian Systems

In this chapter we investigate the dynamics of classical nonlinear Hamiltonian systems, which—a priori—are examples of continuous dynamical systems. As in the discrete case (see examples in Chap. 2), we are interested in the classification of their dynamics. After a short review of the basic concepts of Hamiltonian mechanics, we define integrability (and therewith regular motion) in Sect. 3.4. The non-integrability property is then discussed in Sect. 3.5. The addition of small non-integrable parts to the Hamiltonian function (Sects. 3.6.1 and 3.7) leads us to the formal theory of canonical perturbations, which turns out to be a highly valuable technique for the treatment of systems with one degree of freedom and shows profound difficulties when applied to realistic systems with more degrees of freedom. We will interpret these problems as the seeds of chaotic motion in general. A key result for the understanding of the transition from regular to chaotic motion is the KAM theorem (Sect. 3.7.4), which assures the stability in nonlinear systems that are not integrable but behave approximately like them. Within the framework of the surface of section technique, chaotic motion is discussed from a phenomenological point of view in Sect. 3.8. More quantitative measures of local and global chaos are finally presented in Sect. 3.9.