Steady motions of a tetrahedral satellite
with tethered elements
The procedure of the Routh reduction is well-known for systems of
Lagrangian equations in the case, when one or few generalized
coordinates are cyclic. However, if this system admits a symmetry
group given by a vector field on the configuration space, but the
cyclic coordinates are not given explicitly, the reduction seems
difficult. Here we describe the reduction in this case.
The result is applied to the problem of motion of a system of
interacting material points moving about an attracting center. In
particular, the expression for the amended potencial is obtained by
the proposed procedure without introduction of any special
The obtained potential is used to analise the stationary motions of
a tetrahedral satellite in a central Newtonian gravitational field.
The tetrahedral structure is assumed to be regular; it is composed
by rigid and tether elements. The possibility to use flexible
tethers to provide a stationary tetrahedral configuration is
discussed. The structure possesses spherically symmetric tensor of
inertia, so we use the Routh method with the amended potential. The
reactions in the links are studied using Lagrangian multipliers. The
goal is to identify the stretched links so they could be replaced by
massless inextensible tethers.