# 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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