Noether theorem and scaling, ex.: 1-D Harmonic Oscillator

[marfi11]

Hello,

if I think of the harmonic oscillator action S = ∫L dt, L = 1/2 (dx/dt)^2 - 1/2 x^2, and then of the "scaling transformation" x -> x' = 1/a x (a>0, const), then x = a x', and in new coordinates x', S' has the same form as in x except for multiplication by the constant a, formally S'[x'(t)] = a.S[x'(t)]. Thus in scaled coordinates, the action functional S' (and its functional derivative) will "simply" be multiplied by a (positive) constant.
(I understand that stationary "points" and extrema "points" of the functional S'[x'] remain (formally) "unchanged" compared to the functional S[x])

Does Noether's theorem apply to a such transformation ?

Thank you

I wish you a pleasant day

 

[Valerie Park]

.Yes, Noether's theorem does apply to a such transformation. The theorem states that for any continuous symmetry of the action (in this case, the scaling transformation), there is a corresponding conserved quantity associated with that symmetry. In this case, the conserved quantity would be the momentum associated with the scaling transformation, which can be calculated using the formula p = a * x - (dx/dt).