Consider a system of \(N\) particles whose coordinates are \(\mathbf{r}_i = \{r_i^x,r_i^y,r_i^z\}\), and whose velocities are \(\dot{\mathbf{r}}_i\).

Each pair of particles \(\left(i, j\right)\) interacts through a potential \(V\left(\mathbf{r}_i, \mathbf{r}_j\right)\) which depends on the coordinates \(\left(\mathbf{r}_i, \mathbf{r}_j\right)\) only through their difference, i.e. \[V\left(\mathbf{r}_i, \mathbf{r}_j, t\right) = V\left(\mathbf{r}_i-\mathbf{r}_j, t\right)\] As the system evolves in time, which of the following bulk quantities must be conserved?

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