Comparing different realizations of modified Newtonian dynamics: virial
  theorem and elliptical shells
  There exists several modified gravity theories designed to reproduce the
empirical Milgrom's formula (MOND). Here we derive analytical results in the
context of the static weak-field limit of two of them (BIMOND, leading for a
given set of parameters to QUMOND, and TeVeS). In this limit, these theories
are constructed to give the same force field for spherical symmetry, but their
predictions generally differ out of it. However, for certain realizations of
these theories (characterized by specific choices for their free functions),
the binding potential-energy of a system is increased, compared to its
Newtonian counterpart, by a constant amount independent of the shape and size
of the system. In that case, the virial theorem is exactly the same in these
two theories, for the whole gravity regime and even outside of spherical
symmetry, although the exact force fields are different. We explicitly show
this for the force field generated by the two theories inside an elliptical
shell. For more general free functions, the virial theorems are however not
identical in these two theories. We finally explore the consequences of these
analytical results for the two-body force.
