Integrability versus chaos

Besides the one little regular island at 0 and — 2 there are undoubtedly more regular islands in the phase space of the double pendulum at E = 2. We missed them by our rather coarse choice of initial conditions. As indicated in Fig. 3.3(b), their total area in phase space is probably very small. Nevertheless, Fig. 3.3(b) illustrates an important feature of the phase space of most physical systems the phase space contains an intricate mixture of regular and chaotic regions. The system is said to exhibit a mixed phase space. 

As illustrated by Figs. 3.3(a) and (b), Poincare sections are a very powerful tool for the visual inspection and classification of the dynamics of a given Hamiltonian. The double pendulum illustrates that for autonomous systems with two degrees of fireedom a Poincare section can immediately suggest whether a given Hamiltonian allows for the existence of chaos or not. Moreover, it tells us the locations of chaotic and regular regions in phase space. 

Constants of the motion were already discussed briefly at the end of Section 3.1. Here we attempt to eliminate some commonly encountered misconceptions surrounding the topic of the very existence of constants of the motion. Also, we present some examples for the occurrence of nontrivial constants of the motion in mechanical systems of current interest. 

Suppose we investigate an autonomous Hamiltonian system with / degrees of freedom. Then, the solution of the 2/ first order canonical equations (3.1.21) contains 2/ integration constants Ck, fc = 1. 2/ according to 

Following Landau and Lifschitz (1970) we solve (3.3.1) for the 2/ constants Ck as functions of pa and Qa. This defines 2/ functions Ckipa i), 

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