  The classical and quantum dynamics of two particles constrained on $S^1$ is
discussed via Dirac's approach. We show that when state is maximally entangled
between two subsystems, the product of dispersion in the measurement reduces.
We also quantify the upper bound on the external field $\vec{B}$ such that
$\vec{B}\geq\vec{B}_{upper}$ implies no reduction in the product of dispersion
pertaining to one subsystem. Further, we report on the cut-off value of the
external field $\vec{B}_{cutoff}$, above which the bipartite entanglement is
lost and there exists a direct relationship between uncertainty of the
composite system and the external field. We note that, in this framework it is
possible to tune the external field for entanglement/unentanglement of a
bipartite system. Finally, we show that the additional terms arising in the
quantum Hamiltonian, due to the requirement of Hermiticity of operators,
produce a shift in the energy of the system.
