at equilibrium (in MOND as in Newtonian gravity) is the virial theorem $$\label{eqvirtheo}
2 K + W=0,$$ where $K$ is the total kinetic energy and $W=\Tr W_{ij}$ is the trace of the Chandrasekhar potential energy tensor $$\label{eqwij}
W_{ij}\equiv-\int \rho({{\bf x}}) x_i {\partial \phi({{\bf x}}) \over \partial x_j} \dxcube$$ (e.g., Binney & Tremaine 1987). Note that in MOND $K+W$ is [*not*]{} the total energy, and is not conserved. However, [*$W$ is conserved in the limit of dMOND*]{}, being $W=-(2/3)\sqrt{G{a_0}M_*^3}$ for [*all*]{} systems of finite total mass ${{M_*}}$ (see Appendix \[appw\] for the proof). As a consequence, in dMOND the virial theorem writes simply $\sgv^4=4G{{M_*}}{a_0}/9$, where $\sgv \equiv \sqrt{2 K / {{M_*}}}$ is the system virial velocity dispersion (this relation was proved for dMOND spherical systems by Gerhard & Spergel 1992; see also Milgrom 1984). In our simulations we also tested that equation (\[eqvirtheo\]) is satisfied at equilibrium, and that $W$ is conserved in the dMOND case (see Sections \[secic\] and \[secres\]).

Numerical simulations {#secsim}
=====================

  ----------------------------------------------- ---------------------------------------------
  ${t_{\rm *n}}={r_*}^{3/2} (G {{M_*}})^{-1/2}$   ${t_{\rm *d}}={r_*}(G {{M_*}}{a_0})^{-1/4}$
  ${v_{\rm *n}}=(G {{M_*}})^{1/2}{r_*}^{-1/2}$    ${v_{\rm *d}}=(G {{M_*}}{a_0})^{1/4}$
  ${E_{\rm *n}}=G {{M_*}}^2{r_*}^{-1}$            ${E_{\rm *d}}=(G{a_0})^{1/2}{{M_*}}^{3/2}$
  ----------------------------------------------- ---------------------------------------------

  : Time, velocity, and energy units for Newtonian and MOND (subscript n), and dMOND 