An equation is derived for the energy of a small disturbance in a system that is generated by a distribution function (DF) of the form f(J) – most galaxies and star clusters can be closely approximated by such a DF. The theory of van Kampen modes is extended to such general systems. A bilinear form on the space of DFs is defined such that the energy of a disturbance is its norm under this form. It is shown that van Kampen modes which differ in frequency are then orthogonal, with the consequence that the energies of van Kampen modes are additive. Consequently, most of the insight into the dynamics of ergodic systems that was gained in a recent paper on the van Kampen modes of ergodic systems applies to real clusters and galaxies.

中文翻译：

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恒星系统模式 II：非遍历系统

一个方程是由 f(J) 形式的分布函数 (DF) 生成的系统中小扰动的能量推导出来的——大多数星系和星团都可以用这样的 DF 来近似。van Kampen 模式的理论被扩展到这样的一般系统。定义了 DF 空间上的双线性形式，使得扰动的能量是其在该形式下的范数。结果表明，频率不同的范坎彭模式是正交的，结果范坎彭模式的能量是相加的。因此，在最近一篇关于遍历系统的 van Kampen 模式的论文中获得的对遍历系统动力学的大部分见解都适用于真实的星团和星系。