Formulation of the Landau-Lifshitz model of ferromagnetism as a constrained dynamical system Original Research Article

We make a constraint hamiltonian analysis of the nonrelativistic CP1 model in 2 + 1 dimensions, enjoying U(1) gauge invariance. The expression of the U(1) gauge field on CP1 obtained geometrically is reproduced by the constraint hamiltonian analysis. To begin with, the CP1 model is defined weakly on S3. A subsequent gauge fixing brings down the model strongly on S2. This helps us to establish a connection between the CP1 model and the Landau-Lifshitz (LL) model using a Jordan map. The intimate connection between the gauge choices in the CP1 model leading to different spin lagrangians and the geometry of the LL model is elucidated. Stereographic projection shows the remarkable result that the different spin lagrangians simplify to a unique expression in terms of stereographic variables. Momentum generators are obtained by using Noetherʹs prescription from the CP1 model and show in a gauge independent manner that these do not commute in the nontrivial sector, agreeing with the results quoted in the literature. The angular momentum can also be defined and found to be conserved. Finally, imparting independent dynamics to the gauge field by introducing a Chern-Simons term, we find that the translational symmetry is restored, but it is no longer a CP1 model.