Vlasov Dynamics

Vlasov dynamics is a generalisation of n-body dynamics. The vlasov equations describe the evolution of a measure in phase space. If this measure is a discrete finite point measure, one has an n-body problem.
For Vlasov dynamics in regions with moving boundaries, where the gas dynamics is coupled with the dynamics of the boundary and the gas density is continuous there is an existence theorem . Experiment yourself .
In the animation to the left, a one-dimensional configuration of particles is evolved in the phase space. The initial measure is on a one dimensional curve. The particles interact with the potential V(x) = exp(-x 2 )/2). The evolution defines a one parameter family of symplectic maps. to simulate this particle gas, we took 100 particles and solved numerically the Newton equations, where all particles interact with each other. Mathematica first determined the differential equation and evolved then numerically the particles with a Runge-Kutta solver. See the slide .
An other interesting example is the Calegoro-Moser system in the Vlasov case . See my Slides of Memphis AMS talk .

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