Haven't got space for conservation

Consider a system with NN particles whose coordinates are ri={rix,riy,riz}\mathbf{r}_i = \{r_i^x,r_i^y,r_i^z\}, and whose velocities are r˙i\dot{\mathbf{r}}_i. The energy of the system is described by the kinetic energy 12mr˙2\frac{1}{2}m\sum \dot{\mathbf{r}}^2 and an effective potential energy term V(r1,,rN).V\left(\mathbf{r}_1, \ldots, \mathbf{r}_N\right).

All we know about VV is that it depends on the positions only through their differences r1r2\mathbf{r}_1 - \mathbf{r}_2, i.e. V({ri},t)=V(r1r2,r1r3,,t).V(\{\mathbf{r}^i\}, t) = V\left(\mathbf{r}_1-\mathbf{r}_2, \mathbf{r}_1-\mathbf{r}_3, \ldots, t\right). As this system evolves in time, which of the following bulk quantities must be conserved?

Assumptions

  • All interactions of the system with the outside world are described by V.V.
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