The torque-free motion refers to the movement of a rigid body in space when no external torques are acting upon it. This type of motion can be observed in environments where there are no external forces or frictions, like in outer space. For example, a rotation of Mars in space is a torque-free motion. Mars is an axisymmetric object, meaning it has an axis of symmetry along which it rotates, designated as the z-axis. The rotating frame of reference is defined such that the center of mass of Mars is at the origin. This arrangement ensures that the moments of inertia about the x and y axes are equal.

An additional inertial frame of reference is used to further analyze Mars's motion. In this inertial frame, the positive Z-axis aligns with the angular momentum vector of Mars. It forms an angle theta (θ), with the positive z-axis of the rotating frame. The angular momentum of Mars can be expressed in two different ways using unit vectors. By equating these unit vectors' components, one can derive an equation for the object's angular velocity. Furthermore, by expressing the angular velocity in terms of angular displacement and equating the components once again, one arrives at the equation of motion for our torque-free, axisymmetric, rigid object living on Mars.

Throughout the motion of Mars, its angular momentum, precession, spin, and the angle θ remain constant. These constants reveal the intriguing dynamics of torque-free motion for Mars.