约化与展开：开普勒问题

广义MICZ-Kepler问题与酉最高权模-- II

Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules -- II

对于每一个整数$n\ge 2$，我们证明了一个2n维广义MICZ-开普勒问题有一个$\widetilde{\mr{Spin}}(2, 2n+1)$动力学对称性，扩展了明显的$\mr{Spin}(2n)$对称性。束缚态的Hilbert空间被示出形成酉最高权$\widetilde{\mr{Spin}}(2, 2n+1)$ -模块，其发生在酉最高权模块的Enright-Howe-Wallach分类图中的第一约化点处。作为副产品，我们得到了这样一个酉最高权$\widetilde{\mr{Spin}}(2, 2n+1)$ -模的一个简单的几何实现。

开普勒的和声学及相关问题

The harmony in the Kepler and related problems

由Nucci（{ \it J Math Phys} { \bf 37}（1996）1772-1775）开发的降阶技术被用于产生除了Krause（{ \it J Math Phys} { \bf 35}（1994）5734-5748）在他对开普勒问题的完全对称群的研究中所报道的那些之外的非局部对称性。该技术被证明是适用于相关的问题，包含一个阻力项已被用来模拟地球大气层中的低空卫星的运动，并进一步概括。应用这种技术的一个结果是证明了简谐振子和开普勒之间的群论关系及相关问题。

广义MICZ-Kepler问题与酉最高权模

Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules

对于每一个整数$n\ge 1$，我们证明了一个$(2n+1)$维广义MICZ-开普勒问题有一个$\mr{Spin}(2, 2n+2)$动力学对称性扩展的明显$\mr{Spin}(2n+1)$对称性。束缚态的Hilbert空间被示出形成酉最高权$\mr{Spin}(2, 2n+2)$ -模块，其发生在酉最高权模块的Enright-Howe-Wallach分类图中的第一约化点处。作为副产品，我们得到了这样一个酉最高权$\mr{Spin}(2, 2n+2)$ -模的一个简单的几何实现。

The Sp(1)-Kepler Problems

Let $n\ge 2$ be a positive integer. To each irreducible representation $σ$ of $\mathrm{Sp}(1)$, an $\mathrm{Sp}(1)$-Kepler problem in dimension $(4n-3)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension five. The dynamical symmetry group of this system is $\widetilde {\mathrm O}^*(4n)$ with the Hilbert space of bound states ${\mathscr H}(σ)$ being the unitary highest weight representation of $\widetilde {\mathrm {O}^*}(4n)$ with highest weight $$(\underbrace{-1, ..., -1}_{2n-1}, -(1+\barσ)),$$ which oc…

The O(1)-Kepler Problems

Let $n\ge 2$ be an integer. To each irreducible representation $σ$ of $\mathrm O(1)$, an $\mathrm {O}(1)$-Kepler problem in dimension $n$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension two. The dynamical symmetry group of this system is $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with the Hilbert space of bound states ${\mathscr H}(σ)$ being the unitary highest weight representation of $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with highest weight $$(\underbrace{-1/2, ..., -1/2}_{n-1}, -(1/2+|σ|)),$$…

The reduction on the linear stability of elliptic Euler-Moulton solutions of the n-body problem to those of 3-body problems

In this paper, we consider the elliptic collinear solutions of the classical $n$-body problem, where the $n$ bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler-Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler-Moulton collinear solution of $n$-bodies splits into $(n-1)$ independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler $2$-body problem at Kepler elliptic orbit, and…