# A minimal-variable symplectic method for isospectral flows

## TL;DR

This paper introduces a simple, second-order numerical integrator for isospectral flows that is intrinsically defined on quadratic Lie algebras and symmetric matrices, preserving spectral properties and Lie--Poisson structure.

## Contribution

It presents a minimal-variable, structure-preserving integrator that is computationally efficient and applicable to a broad class of isospectral flows, improving upon existing methods.

## Key findings

- The integrator is isospectral for general flows.
- It preserves Lie--Poisson structure when Hamiltonian.
- It is simple, second-order, and structure-preserving.

## Abstract

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.