$$
L = \sum _ { i = 1 } ^ { N } \frac { 1 } { 2 } m _ { i } \left| \dot { \vec { x } _ { i } } \right| ^ { 2 } – \sum _ { i < j } V \left( \vec { x } _ { i } – \vec { x } _ { j } \right)
$$
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?
Best Answer
In general, there is no reason to expect that there exist conserved quantites for a symmetry which is not a symmetry of the action, but merely of the equations of motion.
The case of the non-relativistic Lagrangian and Galilean transformations is a special case. As Qmechanic works out in this answer, the Galilean transformations are quasi-symmetries of the Lagrangian, i.e. only change it by a total time derivative. In this case, Noether's theorem still applies and yields a conserved quantity (for the free Lagrangian) $$ Q = m(\dot{x}t - x),$$ which is Galilean invariant. Note that Qmechanic's third example shows that a symmetry of the equation of motion does not always imply a quasi-symmetry of the Lagrangian, and therefore there is no conserved quantity associated to it in the general case.