This paper identifies a natural Hamiltonian on a ten dimensional complex Lie algebra that unravels the mysteries encountered in Kowalewski’s famous paper on the motions of a rigid body around its fixed point under the influence of gravity. This system reveals that the enigmatic conditions of Kowalewski, namely, two principal moments of inertia equal to each other and twice the value of the remaining moment of inertia, and the centre of gravity in the plane spanned by the directions corresponding to the equal moments of inertia, are both necessary and sufficient for the existence of an isospectral representation \(\frac{dL(\lambda )}{dt}=[M(\lambda ),L(\lambda )]\) with a spectral parameter \(\lambda \). This representation then yields a crucial spectral invariant that naturally accounts for all the integrals of motion, known as Kowalewski type integrals in the literature of the top. This result is fundamentally dependent on a preliminary discovery that the equality of two principal moments of inertia and the placement of the centre of mass in the plane spanned by the corresponding directions is intimately tied to the existence of another integral of motion on whose zero level surface the above spectral representation resides. The link between mechanical tops and Hamiltonian systems on Lie algebras is provided by an earlier result in which it is shown that the equations of mechanical tops with a linear potential, (heavy tops, in particular) can be represented on certain coadjoint orbits in the semi-direct product \({\mathfrak {g}}={\mathfrak {p}}\rtimes {\mathfrak {k}}\) induced by a closed subgroup K of the underlying group G. The passage to complex Lie algebras is motivated by Kowalewski’s mysterious use of complex variables. It is shown that the complex variables in her paper are naturally identified with complex quaternions and the representation of \(\mathfrak {so}(4,{{\mathbb {C}}})\) as the product \(\mathfrak {sl}(2,{{\mathbb {C}}})\times \mathfrak {sl}(2,{{\mathbb {C}}})\). The paper also shows that all the equations of Kowalewski type can be solved by a uniform integration procedure over the Jacobian of a hyperelliptic curve, as in the original paper of Kowalewski.

这篇论文在十维复李代数上确定了一个自然哈密顿量，它解开了 Kowalewski 关于刚体在重力影响下绕其固定点运动的著名论文中遇到的奥秘。该系统揭示了 Kowalewski 的神秘条件，即两个主转动惯量彼此相等且是剩余转动惯量的两倍，并且重心在由对应于相同转动惯量的方向所跨越的平面中惯量，是必要和充分的用于等谱表示的存在\（\压裂{分升（\拉姆达）} {DT} = [M（\拉姆达），L（\拉姆达）] \）与频谱参数\（ λ \). 然后，这种表示产生了一个关键的谱不变量，它自然地解释了所有运动积分，在 top 的文献中称为 Kowalewski 型积分。这个结果从根本上依赖于一个初步发现，即两个主要惯性矩的相等和质心在相应方向跨越的平面中的位置与另一个运动积分的存在密切相关，该积分为零平面上面的光谱表示存在。李代数上的机械顶和哈密顿系统之间的联系由较早的结果提供，该结果表明具有线性势的机械顶的方程（特别是重顶）可以在半-直接产品\({\mathfrak {g}}={\mathfrak {p}}\rtimes {\mathfrak {k}}\)由基础群G的封闭子群K诱导。向复李代数过渡的动机是 Kowalewski 神秘地使用复变量。结果表明，她的论文中的复变量自然地与复四元数和\(\mathfrak {so}(4,{{\mathbb {C}}})\) 表示为乘积\(\mathfrak { sl}(2,{{\mathbb {C}}})\times \mathfrak {sl}(2,{{\mathbb {C}}})\)。该论文还表明，所有 Kowalewski 类型的方程都可以通过超椭圆曲线雅可比矩阵上的均匀积分程序求解，如 Kowalewski 的原始论文所示。