Yes, that allusion was deliberate on my part. Newton's first law is a manifestation of the symmetry principle when position "x" is the coordinate under consider

Yes, that allusion was deliberate on my part. Newton's first law is a manifestation of the symmetry principle when position "x" is the coordinate under consideration. Force is proportional to gradient(Potential) so any asymmetry in the way the Potential depends on the position produces a Force which changes the Momentum. So, Momentum is the conserved quantity corresponding to the symmetry that all positions are equivalent.