Andoyer's variables and phases in the free rigid body

Using Andoyer's variables we present a new proof of Montgomery's formula by measuring Delta u when v has made a rotation. Our treatment is built on the equations of the differential system of the free rigid solid, together with the explicit expression of the spherical area defined by the intersection of the surfaces given by the energy and momentum integrals. We also consider the phase Delta v of the moving frame when it has made a rotation around the angular momentum vector, and we give the formula for its computation.