Isospectral flows are abundant in mathematical physics; the rigid body, the the Toda lattice, the Brockett flow, the Heisenberg spin chain, and point vortex dynamics, to mention but a few. Their connection on the one hand with integrable systems and, on the other, with Lie--Poisson systems motivates the research for optimal numerical schemes to solve them. Several works about numerical methods to integrate isospectral flows have produced a large varieties of solutions to this problem. However, many of these algorithms are not intrinsically defined in the space where the equations take place and/or rely on computationally heavy transformations. In the literature, only few examples of numerical methods avoiding these issues are known, for instance, the \textit{spherical midpoint method} on $\SO(3)$. In this paper we introduce a new minimal-variable, second order, numerical integrator for isospectral flows intrinsically defined on quadratic Lie algebras and symmetric matrices. The algorithm is isospectral for general isospectral flows and Lie--Poisson preserving when the isospectral flow is Hamiltonian. The simplicity of the scheme, together with its structure-preserving properties, makes it a competitive alternative to those already present in literature.

中文翻译：

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等谱流的最小变量辛方法

等谱流在数学物理学中很丰富；刚体、Toda 晶格、布罗克特流、海森堡自旋链和点涡动力学，仅举几例。它们一方面与可积系统的联系，另一方面与 Lie-Poisson 系统的联系激发了对最佳数值方案的研究以解决它们。一些关于整合等谱流的数值方法的工作已经为这个问题产生了大量的解决方案。然而，这些算法中的许多算法并没有在方程发生的空间中本质上定义和/或依赖于计算量很大的变换。在文献中，只有少数几个避免这些问题的数值方法的例子是已知的，例如，$\SO(3)$ 上的 \textit{球中点法}。在本文中，我们引入了一个新的最小变量，二阶，等谱流的数值积分器本质上定义在二次李代数和对称矩阵上。该算法对于一般等谱流是等谱的，当等谱流是哈密顿流时，该算法是等谱的。该方案的简单性及其结构保留特性使其成为文献中已有的替代方案。