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Published online by Cambridge University Press: 27 June 2025
The Vlasov equation for the collisionless evolution of the singleparticle probability distribution function (PDF) is a well-known example of coadjoint motion. Remarkably, the property of coadjoint motion survives the process of taking moments. That is, the evolution of the moments of the Vlasov PDF is also a form of coadjoint motion. We find that geodesic coadjoint motion of the Vlasov moments with respect to powers of the single-particle momentum admits singular (weak) solutions concentrated on embedded subspaces of physical space. The motion and interactions of these embedded subspaces are governed by canonical Hamiltonian equations for their geodesic evolution.
1. Introduction
The Vlasov equation. The evolution of N identical particles in phase space with coordinates .(qi ; pi)
i = 1, 2,....,N, may be described by an evolution equation for their joint probability distribution function. Integrating over all but one of the particle phase-space coordinates yields an evolution equation for the single-particle probability distribution function (PDF). This is the Vlasov equation.
The solutions of the Vlasov equation reflect its heritage in particle dynamics, which may be reclaimed by writing its many-particle PDF as a product of delta functions in phase space. Any number of these delta functions may be integrated out until all that remains is the dynamics of a single particle in the collective field of the others. In plasma physics, this collective field generates the total electromagnetic properties and the self-consistent equations obeyed by the single particle PDF are the Vlasov–Maxwell equations.
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