On the Theory of Superfluidity in Two Dimensions

The superfluid phase transition of the general vortex gas, in which the circulations may be any non-zero integer, is studied. When the net circulation of the system is not zero the absence of a superfluid phase is shown. When the net circulation of the vortices vanishes, the presence of off-diagonal long range order is demonstrated and the existence of an order parameter is proposed. The transition temperature for the general vortex gas is shown to be the Kosterlitz---Thouless temperature. An upper bound for the average vortex number density is established for the general vortex gas and an exact expression is derived for the Kosterlitz---Thouless ensemble.Comment: 22 pages, one figure, written in plain TeX, published in J. Phys. A24 (1991) 502

Constrained Dynamics: Generalized Lie Symmetries, Singular Lagrangians, and the Passage to Hamiltonian Mechanics

Guided by the symmetries of the Euler-Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation\textemdash called second-order, Euler-Lagrange vector fields (SOELVFs)\textemdash with integral flows that have this symmetry are determined. Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always possible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to the Hamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space needed for their evolution. In particular, the primary Hamiltonian constraints can be constructed from vectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show that the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this way equivalent.Comment: 45 pages. Published paper is open access, and can be found either at the Journal of Physics Communications website or at the DOI belo