The gravitational Vlasov-Poisson system on the unit 2-sphere with initial data along a great circle




Lind, Crystal

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The Vlasov-Poisson system is most commonly used to model the movement of charged particles in a plasma or of stars in a galaxy. It consists of a kinetic equation known as the Vlasov equation coupled with a force determined by the Poisson equation. The system in Euclidean space is well-known and has been extensively studied under various assumptions. In this paper, we derive the Vlasov-Poisson equations assuming the particles exist only on the 2-sphere, then take an in-depth look at particles which initially lie along a great circle of the sphere. We show that any great circle is an invariant set of the equations of motion and prove that the total energy, number of particles, and entropy of the system are conserved for circular initial distributions.



Vlasov, Poisson, Curved spaces, 2-sphere, Kinetic equations, Collisionless Boltzmann equation, Gauss's Law, Conservation laws, Non-Euclidean, Potential, Gravitational Force, Gravitational potential, Gravity, Equation of motion