  The motion of a rolling ball actuated by internal point masses that move
inside the ball's frame of reference is considered. The equations of motion are
derived by applying Euler-Poincar\'e's symmetry reduction method in concert
with Lagrange-d'Alembert's principle, which accounts for the presence of the
nonholonomic rolling constraint. As a particular example, we consider the case
when the masses move along internal rails, or trajectories, of arbitrary shape
and fixed within the ball's frame of reference. Our system of equations can
treat most possible methods of actuating the rolling ball with internal moving
masses encountered in the literature, such as circular motion of the masses
mimicking swinging pendula or straight line motion of the masses mimicking
magnets sliding inside linear tubes embedded within a solenoid. Moreover, our
method can model arbitrary rail shapes and an arbitrary number of rails such as
several ellipses and/or figure eights, which may be important for future
designs of rolling ball robots. For further analytical study, we also reduce
the system to a single differential equation when the motion is planar, that
is, considering the motion of the rolling disk actuated by internal point
masses, in which case we show that the results obtained from the variational
derivation coincide with those obtained from Newton's second law. Finally, the
equations of motion are solved numerically, illustrating a wealth of complex
behaviors exhibited by the system's dynamics. Our results are relevant to the
dynamics of nonholonomic systems containing internal degrees of freedom and to
further studies of control of such systems actuated by internal masses.
