Conformal mechanics

A one-dimensional particle with $H=p^2/2+g/x^2$ admits conserved time-translation, dilation and special-conformal charges. Their Poisson brackets form the sl2R Lie algebra. The homogeneity of the inverse-square potential is what allows the symmetry.

For the extended canonical variables with $\{t,E\}=-1$ and $\{x,p\}=1$, the charge $K=t^{2}E-txp+x^2/2$ generates a special conformal transformation. Its time derivative is $2te(E-H)$, so it is conserved on the constraint surface. With a time-dependent parameter $\beta$, the multiplier varies as $\delta e=-2te\beta$ and the action variation is $\dot{\beta }K-d(\beta x^2/2)/d\lambda$.