  A specific modification of Newtonian dynamics known as MOND has been shown to
reproduce the dynamics of most astrophysical systems at different scales
without invoking non-baryonic dark matter (DM). There is, however, a
long-standing unsolved problem when MOND is applied to rich clusters of
galaxies in the form of a deficit (by a factor around two) of predicted
dynamical mass derived from the virial theorem with respect to observations. In
this article we approach the virial theorem using the velocity dispersion of
cluster members along the line of sight rather than using the cluster
temperature from X-ray data and hydrostatic equilibrium. Analytical
calculations of the virial theorem in clusters for Newtonian gravity+DM and
MOND are developed, applying pressure (surface) corrections for non-closed
systems. Recent calibrations of DM profiles, baryonic ratio and baryonic
($\beta $ model or others) profiles are used, while allowing free parameters to
range within the observational constraints. It is shown that solutions exist
for MOND in clusters that give similar results to Newton+DM -- particularly in
the case of an isothermal $\beta $ model for $\beta =0.55-0.70$ and core radii
$r_c$ between 0.1 and 0.3 times $r_{500}$ (in agreement with the known data).
The disagreements found in previous studies seem to be due to the lack of
pressure corrections (based on inappropriate hydrostatic equilibrium
assumptions) and/or inappropriate parameters for the baryonic matter profiles.
