Hamiltonian dynamics of classical spins

Abstract

We discuss the geometry behind classical Heisenberg model at the level suitable for third or fourth year students who did not have the opportunity to take a course on differential geometry. The arguments presented here rely solely on elementary algebraic concepts such as vectors, dual vectors and tensors, as well as Hamiltonian equations and Poisson brackets in their simplest form. We derive Poisson brackets for classical spins, along with the corresponding equations of motion for classical Heisenberg model, starting from the geometry of two-sphere, thereby demonstrating the relevance of standard canonical procedure in the case of Heisenberg model.