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Author Bloch, Anthony M. ♦ Brînzănescu, Vasile ♦ Iserles, Arieh ♦ Marsden, Jerrold E. ♦ Ratiu, Tudor S.
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Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Abstract For a given skew symmetric real n × n matrix N, the bracket [X, Y]N = XNY − Y NX defines a Lie algebra structure on the space Sym(n, N) of symmetric n × n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry, integrability, and linearizability of the Hamiltonian system ˙ X = [X 2, N], or equivalently in Lax form, the equation ˙X = [X, XN + NX] on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure of Sym(n, N). This is established by finding another compatible Poisson structure. If N is invertible, several remarkable identifications can be implemented. First, (Sym(n, N), [·, ·]) is Lie algebra isomorphic with the symplectic Lie algebra sp(n, N −1) associated to the symplectic form on R n given by N −1. In this case, the system is the reduction of the geodesic flow of the left invariant Frobenius metric on the underlying symplectic group Sp(n, N −1). Second, the trace of the product of
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Education Level UG and PG ♦ Career/Technical Study
Learning Resource Type Article