Behavior of a constrained particle on superintegrability of the two-dimensional complex Cayley–Klein space and its thermodynamic properties

Abstract

In this paper, we study the motion of a particle on a family of superintegrable Hamiltonians of the unitary groups in Cayley–Klein space. This possibility is provided through the general commutator relations between generators from its respective group. Hence, by converting coordinates from Cartesian space to Eulerian parameterization, the obtained metric in terms of the new coordinates is equal to the same Lagrangian of geodesic motion. On the other hand, by reducing the SUκ1κ2(3) group manifold to the SUκ1κ2(3)∕SUκ1(2)≃Ω5 quotient space and considering the three Fourier transformations, the particle is reduced to two-dimensional space (Ω2=S2,H2ordS1+1) depending on the contraction parameter κ. Then with overlapping the reduced coordinates in the Eulerian parameterization with the canonical coordinates, we incorporate them into the Lagrangian. Therefore, we find three constant integrals of motion from the vector fields and we construct the superintegrable Hamiltonian. However, we obtain the wave function and the energy spectrum of the system by separating the variables and their solutions. Finally, by applying the resulting energy, for the two special cases of the produced topological space, we investigate and compare their thermodynamic properties.