Hamiltonian dynamics of classical spins

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.

💡 Research Summary

The paper “Hamiltonian dynamics of classical spins” is written for senior undergraduate students who have not taken a formal course in differential geometry, yet wish to understand the geometric underpinnings of the classical Heisenberg model. The authors deliberately restrict themselves to elementary algebraic tools—vectors, dual vectors, tensors, Hamilton’s equations, and Poisson brackets—in order to derive the full dynamical structure of a system whose phase space is the two‑sphere (S^{2}).

The exposition begins with a concise review of real vector spaces (V) and their duals (V^{*}). By introducing tensors of type ((0,2)), ((1,1)), and ((2,0)) the authors define the metric tensor (G_{ij}) and show how it provides an isomorphism between vectors and covectors (index raising and lowering). This groundwork is essential because, on a non‑Euclidean manifold such as (S^{2}), the gradient of a scalar field is naturally a covector rather than a vector—a point that will later affect the coordinate‑free formulation of Hamilton’s equations.

Next, the familiar symplectic structure on the flat phase space (\mathbb{R}^{2}) is recast in a basis‑independent language. The symplectic 2‑form (\omega = dq\wedge dp) is represented by the antisymmetric matrix
(A = \begin{pmatrix}0 & 1 \ -1 & 0\end{pmatrix}).
Hamilton’s equations (\dot q = \partial H/\partial p,; \dot p = -\partial H/\partial q) become the single vector equation (\dot X = A,\nabla_{X} H). By explicitly transforming to a new basis (U) (and its dual (V)) the authors verify that the form of the equation is invariant, confirming that (A) encodes an intrinsic geometric object: the area‑measuring 2‑form.

The crucial step is to replace the Euclidean phase space with the curved manifold (S^{2}). Using spherical coordinates ((\theta,\phi)) the metric is (g_{ab}= \mathrm{diag}(1,\sin^{2}\theta)) and the associated symplectic form is (\Omega = \sin\theta, d\theta\wedge d\phi). The inverse of this 2‑form, the Poisson tensor (\Lambda^{ab}), defines the Poisson bracket for any two functions (f,g) on the sphere:
({f,g}= \Lambda^{ab}\partial_{a}f,\partial_{b}g).
Because the spin vector (\mathbf{S}=(S_{x},S_{y},S_{z})) lives on a sphere of fixed radius (S), the components satisfy the Lie‑Poisson algebra
({S_{i},S_{j}}= \varepsilon_{ijk}S_{k}),
which is precisely the classical counterpart of the su(2) commutation relations. This result emerges directly from the symplectic geometry of (S^{2}) without invoking any coordinate‑dependent tricks.

Armed with the Poisson structure, the authors write the classical Heisenberg Hamiltonian for a lattice of spins:
(H = -\frac{J}{2}\sum_{\langle n,m\rangle}\mathbf{S}{n}!\cdot!\mathbf{S}{m}).
Applying the Poisson brackets yields the equations of motion for each site:
(\dot{\mathbf{S}}{n}= J,\mathbf{S}{n}\times\sum_{\lambda}\mathbf{S}_{n+\lambda}).
The cross product reflects the underlying su(2) Lie algebra and shows that each spin precesses around the effective field generated by its nearest neighbours.

To connect with the familiar concept of spin waves (magnons), the authors linearize the dynamics around a ferromagnetic ground state (\mathbf{S}=S\hat{z}). Small deviations (\delta\mathbf{S}{n}) obey a discrete wave equation, which in the continuum limit becomes (\partial{t}^{2}\delta\mathbf{S}=c^{2}\nabla^{2}\delta\mathbf{S}) with propagation speed (c\propto JS). Thus the classical spin wave is identified as the low‑amplitude limit of the full nonlinear precession equations.

Finally, an alternative definition of the Poisson bracket is presented: ({f,g}= \Omega^{-1}(df,dg)). This formulation emphasizes that the bracket is the contraction of the inverse symplectic 2‑form with the exterior derivatives of the functions, making the definition manifestly coordinate‑free and directly applicable to any symplectic manifold.

In summary, the paper demonstrates that the entire classical Heisenberg model—its phase‑space geometry, Poisson algebra, Hamiltonian dynamics, and linear spin‑wave excitations—can be derived using only elementary linear‑algebraic concepts and the notion of a symplectic form. By avoiding the heavy machinery of differential geometry, the authors provide a pedagogically accessible route for students to see how classical spin dynamics naturally arise from the geometry of the two‑sphere, thereby bridging the gap between classical and quantum descriptions of magnetism.