Nipoti, Londrillo & Ciotti 2003). One of these tests is described in Section \[seckin\].

We also verified that the code reproduces the Newtonian and MOND conservation laws (see Section \[secic\]): note that the conservation laws in MOND present some peculiarities with respect to the Newtonian case, so we give here a brief discussion of the subject. As already stressed by BM, equation (\[eqMOND\]) is obtained from a variational principle applied to a Lagrangian with all the required symmetries, so energy, linear and angular momentum are conserved. Unfortunately, as also shown by BM, the total energy diverges even for finite mass systems, thus posing a computational challenge to code validation. We solved this problem by checking the volume-limited energy balance equation $$\begin{aligned}
\label{eqbaltext}
{d \over d t}\int_{V_0}\left[k+\rho\phi+{{a_0}^2\over 8 \pi
G}\mathcal{F}\left({||\nabla\phi||\over
    {a_0}}\right)\right]\dxcube=\nonumber\\{1\over 4\pi G}\int_{\partial
  V_0}\mu{\partial \phi \over \partial t} <\nabla \phi,\hat{\bf n}> d a,\end{aligned}$$ which is derived in Appendix \[appetot\]. In equation (\[eqbaltext\]) $V_0$ is an arbitrary (but fixed) volume enclosing all the system mass, $k$ is the kinetic energy per unit volume, and $$\mathcal{F}(y)\equiv2\int_{y_0}^{y} \mu(\xi) \xi d \xi,
\label{eqeffe}$$ where $y_0$ is an arbitrary constant; note that only finite quantities are involved. Another important relation between global quantities for a system 