Is there an "invariant" quantity for the classical Lagrangian?

$$L = \sum _ { i = 1 } ^ { N } \frac { 1 } { 2 } m _ { i } \left| \dot { \vec { x } _ { i } } \right| ^ { 2 } - \sum _ { i < j } V \left( \vec { x } _ { i } - \vec { x } _ { j } \right)$$

This is just a typical classical Lagrangian for $N$ particles. Since the Lagrangian does not explicitly depend on time, the energy must be conserved. Also, the linear and angular momentum seem to be conserved too.

However, if there is a change in the coordinate by the Galilean transformation $\overrightarrow{x}_i(t) \to \overrightarrow{x}_i(t) +\overrightarrow{v}t$, then the aforementioned quantities seeem clearly "variant". So, my question is that whether there exists a quantity that is invariant under this Galilean transformation. Could anyone please present me one? Or if there is no such quantity, could anyone please explain why?