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The complete integrability of a Lie-Poisson system proposed by Bloch and Iserles

Carlos Tomei·Luen-Chau Li
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Abstract

We establish the Liouville integrability of the differential equation $\dot S(t)= [N,S^2(t)],$ recently considered by Bloch and Iserles. Here, $N$ is a real, fixed, skew-symmetric matrix and $S$ is real symmetric. The equation is realized as a Hamiltonian vector field on a coadjoint orbit of a loop group, and sufficiently many commuting integrals are presented, together with a solution formula for their related flows in terms of a Riemann-Hilbert factorization problem. We also answer a question raised by Bloch and Iserles, by realizing the same system on a coadjoint orbit of a finite dimensional Lie group.

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Carlos Tomei, Luen-Chau Li. 2005-12-12. The complete integrability of a Lie-Poisson system proposed by Bloch and Iserles. https://arxiv.org/abs/math/0512246

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