Light-cone Yang–Mills mechanics: $SU(2)$ vs. $SU(3)$

We investigate the light-cone $SU(n)$ Yang–Mills mechanics formulated as the leading order of the long-wavelength approximation to the light-front $SU(n)$ Yang–Mills theory. In the framework of the Dirac formalism for degenerate Hamiltonian systems, for models with the structure groups $SU(2)$ and $SU(3)$, we determine the complete set of constraints and classify them. We show that the light-cone mechanics has an extended invariance{:} in addition to the local $SU(n)$ gauge rotations, there is a new local two-parameter Abelian transformation, not related to the isotopic group, that leaves the Lagrangian system unchanged. This extended invariance has one profound consequence. It turns out that the light-cone $SU(2)$ Yang–Mills mechanics, in contrast to the well-known instant-time $SU(2)$ Yang–Mills mechanics, represents a classically integrable system. For calculations, we use the technique of Gröbner bases in the theory of polynomial ideals.