Haven't got space for conservation

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

All we know about \(V\) is that it depends on the positions only through their differences \(\mathbf{r}_1 - \mathbf{r}_2\), i.e. \[V(\{\mathbf{r}^i\}, t) = V\left(\mathbf{r}_1-\mathbf{r}_2, \mathbf{r}_1-\mathbf{r}_3, \ldots, t\right).\] 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 \(V.\)
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