  We obtain Mathisson-Papapetrou-Tulczyjew-Dixon equations of a rotating body
with given values of spin and momentum starting from Lagrangian action without
auxiliary variables. MPTD-equations correspond to minimal interaction of our
spinning particle with gravity. We shortly discuss some novel properties
deduced from the Lagrangian form of MPTD-equations: emergence of an effective
metric instead of the original one, non-commutativity of coordinates of
representative point of the body, spin corrections to Newton potential due to
the effective metric as well as spin corrections to the expression for
integrals of motion of a given isometry.
