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Can rapidity become a gauge variable? Dirac Hamiltonian method and Relativistic Rotators

The minimal Hamiltonian for a family of relativistic rotators is constructed by a direct application of the Dirac procedure for constrained systems. The Hamiltonian equations can be easily solved. It is found that the resulting motion is unique and qualitatively the same for all phenomenological rotators, only the relation between mass and spin is different. There is a critical point in the construction when such a relation cannot be established, implying that the number of primary constraints is greater. In that case the mass and the spin become unrelated, separately fixed parameters, and the corresponding Hamiltonian changes qualitatively. Furthermore, a genuine physical observable becomes a gauge variable. This paradoxical result is consistent with the fact already known at the Lagrangian level that the Hessian rank is lower than expected, and the equations of motion indeterminate on $\mathbb{R}^3\times\mathbb{S}^2$.