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Unformatted text preview: Lecture 43: Chaos III (14 Dec 09) A. Extend: mechanics models 1. Refs: Goldstein Chapter 11, P. Cvitanovic, Universality in Chaos (reprint collection, 1984), FW Supplement. 2. Much of the language of modern turbulence theory arose from a few idealized models that have varying basis in physical theory but all share the feature that complex (rich) behaviors result from simple rules. 3. Integrable model: wellbehaved constants of the motion additional to the total energy. Examples with phase space that has integrable solutions in some regions and nonintegrable in others. 4. FPU (FermiPastaUlam) system of coupled linear oscillators with a little cubic or quartic anharmonicity. 5. H enonHeiles Hamiltonian (4 DOFs) based on a gravitational problem [G Sec. 11.6] H = 1 2 ( p 2 1 + q 2 1 + p 2 2 + q 2 2 ) + q 2 1 q 2 1 3 q 3 2 with stability of trajectories degrading as the energy E increases. Poincar e surface of section (G Sec. 11.5). The phase space has 4 dimensions ( q 1 p 1 q 2 p 2 ), with 3D energy surface (4  1 constraint). Project the tra jectory on the ( q 2 ,p 2 ) plane by marking point with q 1 = 0; p 1 0. (a) For low enough energy, E = 1 / 12, every trajectory followed in the computation led to smooth curves in the Poincar e surface of section ( q 2 ,p 2 ). Trajectories that are initially close propagate with separation that increases linearly with time....
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 Fall '09
 BRUCH
 mechanics

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