Hamilton Vlasov系统的对称约化与周期解
Symmetry reduction and periodic solutions in Hamiltonian Vlasov systems
摘要：
本文讨论了从经典Vlasov系统的平衡点分支出周期解的一般方法。这个问题的主要途径是通过任何弗拉索夫系统的哈密顿表示来选择，首先由Fröhlich，Knowles和Schwarz提出，最近由作者推广。该方法将问题转化为一组具有相位等变哈密顿量的复值$\mathcal{L}^2$函数。通过Marsden-Weinstein对称性约化，将问题映射到商流形$\mathbb{S}^{\mathcal{L}^2}/\mathbb{S}^1$上的一个哈密顿系统上，这实际上证明了闭合动力学的许多轨迹是必要的.作为一个玩具模型来应用的方法，我们使用谐波Vlasov系统，一个非相对论性的Vlasov方程与吸引的谐波两体相互作用势。该模型的简单结构允许直接计算其所有解，因此可以测试Vlasov系统中Hamilton形式主义和对称性约化的好处。
英文摘要：
In this paper, we discuss a general approach to find periodic solutions bifurcating from equilibrium points of classical Vlasov systems. The main access to the problem is chosen through the Hamiltonian representation of any Vlasov system, firstly put forward by Fröhlich, Knowles, and Schwarz, and generalized more recently by the author. The method transforms the problem into a setup of complex valued $\mathcal{L}^2$ functions with phase equivariant Hamiltonian. Through Marsden-Weinstein symmetry reduction, the problem is mapped on a Hamiltonian system on the quotient manifold $\mathbb{S}^{\mathcal{L}^2}/\mathbb{S}^1$ which actually proves to be necessary to close many trajectories of the dynamics. As a toy model to apply the method we use the Harmonic Vlasov system, a non-relativistic Vlasov equation with attractive harmonic two-body interaction potential. The simple structure of this model allows to compute all of its solutions directly and therefore test the benefits of the Hamiltonian formalism and symmetry reduction in Vlasov systems.
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