lecnotes3 - 3 Conservation of volume in phase space We...

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3 Conservation of volume in phase space We show (via the example of the pendulum) that frictionless systems conserve volumes (or areas) in phase space. Conversely, we shall see, dissipative systems contract volumes. Suppose we have a 3-D phase space, such that x ˙ 1 = f 1 ( x 1 , x 2 , x 3 ) x ˙ 2 = f 2 ( x 1 , x 2 , x 3 ) x ˙ 3 = f 3 ( x 1 , x 2 , x 3 ) or d ψx = f ψ ( ψx ) d t The equations describe a “flow,” where d ψx/ d t is the velocity. A set of initial conditions enclosed in a volume V flows to another position in phase space, where it occupies a volume V , neither necessarily the same shape nor size: x 3 n V ds x V x 2 x 1 Assume the volume V has surface S . 19
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Let δ = density of initial conditions in V ; δf ψ = rate of flow of points (trajectories emanating from initial condi- tions) through unit area perpendicular to the direction of flow; d s = a small region of S ; and ψn = the unit normal (outward) to d s . Then
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This note was uploaded on 02/24/2012 for the course MECHANICAL 12.006J taught by Professor Danielrothman during the Fall '06 term at MIT.

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lecnotes3 - 3 Conservation of volume in phase space We...

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