Lagrangian formulation for Mathisson-Papapetrou-Tulczyjew-Dixon equations

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