Vlasov moments, integrable systems and singular solutions
John Gibbons, Darryl D Holm, Cesare Tronci
TL;DR
This paper explores the mathematical structure of the Vlasov equation, showing how moments preserve Lie-Poisson structure and how singular solutions relate to particle trajectories in collisionless plasma dynamics.
Contribution
It demonstrates that Vlasov moments maintain the Lie-Poisson structure and connects singular solutions to geodesic particle motion, offering new insights into plasma dynamics.
Findings
Vlasov moments preserve Lie-Poisson structure
Singular solutions correspond to particle trajectories
Geodesic motion of moments recovers individual particle dynamics
Abstract
The Vlasov equation for the collisionless evolution of the single-particle probability distribution function (PDF) is a well-known Lie-Poisson Hamiltonian system. Remarkably, the operation of taking the moments of the Vlasov PDF preserves the Lie-Poisson structure. The individual particle motions correspond to singular solutions of the Vlasov equation. The paper focuses on singular solutions of the problem of geodesic motion of the Vlasov moments. These singular solutions recover geodesic motion of the individual particles.
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