Existence of axially symmetric solutions to the Vlasov-Poisson system depending on Jacobi's integral

Proof of the Jacobi Property for the guiding-center Vlasov-Maxwell Bracket

The proof of the Jacobi property of the guiding-center Vlasov-Maxwell bracket underlying the Hamiltonian structure of the guiding-center Vlasov-Maxwell equations is presented.

球对称Vlasov-Poisson系统作为保质量算法不动点的稳态

Steady states of the spherically symmetric Vlasov-Poisson system as fixed points of a mass-preserving algorithm

本研究采用曾用于数值逼近两类系统轴对称解的策略，给出球对称Vlasov-Poisson系统稳态存在性的新证明，阐明了分析该数值方案的多重理论意义。

Vlasov moment flows and geodesics on the Jacobi group

By using the moment algebra of the Vlasov kinetic equation, we characterize the integrable Bloch-Iserles system on symmetric matrices (arXiv:math-ph/0512093) as a geodesic flow on the Jacobi group. We analyze the corresponding Lie-Poisson structure by presenting a momentum map, which both untangles the bracket structure and produces particle-type solutions that are inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov moments associated to Bloch-Iserles dynamics correspond to particular subgroup inclusions into a group central extension (first discovered in arXiv:math…

Rotating, stationary, axially symmetric spacetimes with collisionless matter

The existence of stationary solutions to the Einstein-Vlasov system which are axially symmetric and have non-zero total angular momentum is shown. This provides mathematical models for rotating, general relativistic and asymptotically flat non-vacuum spacetimes. If angular momentum is allowed to be non-zero, the system of equations to solve contains one semilinear elliptic equation which is singular on the axis of rotation. This can be handled very efficiently by recasting the equation as one for an axisymmetric unknown on $\R^5$.

Hamiltonian reductions of the one-dimensional Vlasov equation using phase-space moments

We consider Hamiltonian closures of the Vlasov equation using the phase-space moments of the distribution function. We provide some conditions on the closures imposed by the Jacobi identity. We completely solve some families of examples. As a result, we show that imposing that the resulting reduced system preserves the Hamiltonian character of the parent model shapes its phase space by creating a set of Casimir invariants as a direct consequence of the Jacobi identity.

Casimir invariants and the Jacobi identity in Dirac's theory of constrained Hamiltonian systems

We consider constrained Hamiltonian systems in the framework of Dirac's theory. We show that the Jacobi identity results from imposing that the constraints are Casimir invariants, regardless of the fact that the matrix of Poisson brackets between constraints is invertible or not. We point out that the proof we provide ensures the validity of the Jacobi identity everywhere in phase space, and not just on the surface defined by the constraints. Two examples are considered: A finite dimensional system with an odd number of constraints, and the Vlasov-Poisson reduction from Vlasov-Maxwell equation…