= Solution
The central pair potential is invariant under simultaneous spatial translations, simultaneous rotations and time translations. The corresponding time-independent <Noether charges> are total <momentum> $P=p_1+p_2$, total <angular momentum> about a fixed origin, and total <energy> $H$.
In one dimension this supplies \b[$P$ and $H$: two charges]. In two dimensions it supplies \b[$P_x,P_y,L_{xy},H$: four charges]. In three dimensions it supplies \b[three momentum components, three angular-momentum components and $H$: seven charges]. For generic states these listed charges are independent in those dimensions. Galilean boosts can also be expressed by explicitly time-dependent conserved combinations, but are not part of the requested translation-rotation-energy count.
Back to article page
