  We show that there is a manifestly covariant version of the Pauli Hamiltonian
with equations of motion quadratic on spin and field strength. Relativistic
covariance inevitably leads to noncommutative positions: classical brackets of
the position variables are proportional to the spin. It is the spin-induced
noncommutativity that is responsible for transforming the covariant Hamiltonian
into the Pauli Hamiltonian, without any appeal to the Thomas precession
formula. The Pauli theory can be thought to be $1/c^2$ approximation of the
covariant theory written in special variables. These observations clarify the
long standing question on the discrepancy between the covariant and Pauli
Hamiltonians. We also discuss the transformational properties of spin axis in
the passage from laboratory to comoving and instantaneous frames, and reveal
the role of Thomas spin-vector in the covariant scheme.
