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I^, had prescribed values initially, they would keep them forever, and the distribution function could never become uniform over the microcanonical surface corresponding to a given value £ of the energy. Clearly, the final state would drastically depend on the preparation of the system, and concepts such as approach to equilibrium would lose their meaning. CLASSICAL DYNAMICS Ergodic Systems Because of the difficulties in using integrable systems to incorporate the approach to equilibrium, James Clerk Maxwell and Ludwig Boltzmann turned their attention to another type of dynamical system.

They introduced what is generally known today as the ergodic hypothesis. " Mathematicians have pointed out that a trajectory cannot fill " a surface " and that the statement must be altered to indicate that the system will eventually come arbitrarily close to every point of the energy surface, in accord with the quasi-ergodic hypothesis (see Farquhar, 1964). It is interesting to note that we are dealing with a prototype of dynamical systems, which is just the opposite of the point of view taken in the study of integrable systems.

If the trajectory is conditionally periodic— that is, if it is restricted to a torus—the successive intersections follow a closed curve in the q2, p2-plane. If the trajectory is "random," in the sense that it wanders erratically through the phase space, the intersection point also wanders erratically through the plane. 9. 8 A three-dimensional trajectory for the Henon-Heiles system. P2 r> ^: q. 9 Various types of trajectories; (A) periodic; (B) conditionally periodic; (C) random. 27) by "^^=577' ^"^ = 1 7 ; (^-^^^ An essential difference between thi(^ example and that of the harmonic oscillator is that, in general, HQ wnl not be hnear in the J's and these two frequencies will be action dependent.

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