Some peculiar properties of the relativistic oscillator in the post-Galilean approximation

In the post-Galilean approximation, the Lagrangians are singular on a submanifold of the phase space. A general analysis of these peculiarities, which differ by the ones considered by Dirac, is, up to day, lacking. The analysis of the dynamics of the one-dimensional and two-dimensional relativistic oscillators of the second tensor rank, is reported. A comparison with the case of scalar and vector relativistic oscillators, described by a regular Lagrangian, is accomplished. Some equilibrium statistical properties of the relativistic oscillators are also analysed.