Equivalent problems in rigid body dynamics — I

The general problem of motion of a rigid body about a fixed point under the action of stationary non-symmetric potential and gyroscopic forces is considered. The equations of motion in the Euler-Poisson form are derived. An interpretation is given in terms of charged, magnetized gyrostat moving in a superposition of three classical fields. As an example, the problem of motion of a satellite — gyrostat on a circular orbit with respect to its orbital system is reduced to that of its motion in an inertial system under additional magnetic and Lorentz forces.When the body is completely symmetric about one of its axes passing through the fixed point, the above problem is found to be equivalent to another one, in which the body has three equal moments of inertia and the forces are symmetric around a space axis. The last problem is well-studied and the given analogy reveals a number of integrable cases of the original problem. A transformation is found, which gives from each of these cases a class of integrable cases depending on an arbitrary function. The equations of motion are also reduced to a single equation of the second order.