Chjan Lim

The general Jacobi symplectic variables generated by a combinatorial algorithm from the full binary tree $T(N)$ are used to formulate some nonheliocentric gravitational $N$-body problems in perturbation form. The resulting uncoupled term $H_U$ for $(N-1)$ independent Keplerian motions and the perturbation term $H_P$ are both explicitly dependent on the partial ordering induced by the tree $T(N)$. This leads to suitable conditions on separations of the $N$ bodies for the perturbation to be small. We prove the Herman resonance for a new approximation of the 5-body problem. Full details of the derivations of the perturbation form and Herman resonance are given only in the case of five bodies using the caterpillar binary tree $T_c(5)$.

A low temperature relation $R^2=\Omega\beta/4\pi\mu$ between the radius $R$ of a compactly supported 2D vorticity (plasma density) field, the total circulation $\Omega$ (total electron charge) and the ratio $\mu/\beta$ (Larmor frequency), is rigorously derived from a variational Principle of Minimum Energy for 2D Euler dynamics. This relation and the predicted structure of the global minimizers or ground states are in agreement with the radii of the most probable vorticity distributions for a vortex gas of $N$ point vortices in the unbounded plane for a very wide range of temperatures, including $\beta = O(1)$. In view of the fact that the planar vortex gas is representative of many 2D and 2.5D statistical mechanics models for geophysical flows, the Principle of Minimum Energy is expected to provide a useful method for predicting the statistical properties of these models in a wide range of low to moderate temperatures.