The Equations of Motion of a Rigid Body with Constraints Explicitly Depending on Time

The general equations of motion of a body for an observer in S₀ (an inertial frame) or for an observer in S₁ (an accelerated frame) are derived. They allow us to determine, in any case, the inertial and gyroscopic forces and to find the difference between them. If the vector that determines the position of the body in S₀ depends explicitly on time in S₀, the work-energy principle yields a supplementary condition and these equations can be shown to be equivalent to the Painleve integrals in the Lagrange formulation. Since we are dealing with inertial frames, gyroscopic forces rather than inertial forces are taken into account. If another reference frame is used, we can choose it so that the position vector depends implicitly on time in S₁ and another set of equations can be obtained for the motion of the body in S₁. The work-energy principle yields a supplementary condition, but inertial forces should be added. Since an explicit time dependence does not exist in S₁, gyroscopic forces do not exist as well and instead we have Coriolis forces that behave like gyroscopic forces