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- Title
Geometry of Discrete-Time Spin Systems.
- Authors
McLachlan, Robert; Modin, Klas; Verdier, Olivier
- Abstract
Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space $$(S^2)^n$$ . In this paper, we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on $$T^*\mathbf {R}^{2n}$$ for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions, and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with Kähler geometry, this provides another geometric proof of symplecticity.
- Subjects
HAMILTONIAN systems; DYNAMICAL systems; DISCRETE-time systems; RIEMANNIAN manifolds; RIEMANNIAN submersions
- Publication
Journal of Nonlinear Science, 2016, Vol 26, Issue 5, p1507
- ISSN
0938-8974
- Publication type
Article
- DOI
10.1007/s00332-016-9311-z