How would you define 'equations of motion' (in context of symmetries and Noether's theorem)

Question

I'm studying variational principles and my notes say (while introducing Noether's theorem) that if a transformation q(t)->Q(t) is a "symmetry" then the equations of motion for Q will be identical to those for q. I'm not entirely sure what this means as I would have thought equation of motion was something like q(t)=vt but then obviously the symmetrical transformation Q(t)=q(t)+s would have equation Q(t)=s+vt, which is not identical...does this just mean that the laws of motion (e.g. F=ma) are identical in both frames? Thanks :)

Answer 1

The equations of motion for $q$ are the second-order ordinary differental equations that you have to solve in order to find $q(t)$.