Change of variables in 2D phase space from Cartesian to Cylindrical

Clear[ptheta]

ptheta[x_, y_, m_, r_, t_] = m r^2 D[ ArcTan[y[t]/x[t]], t]

((m (r^2) (-((y[t] x'[t])/((x[t])^2))+((y'[t])/(x[t]))))/(1+(((y[t])^2)/((x[t])^2))))/. x'[t]-> px/m /. y'[t]-> py/m /. x[t] -> x /. y[t] -> y  // FullSimplify

(((r^2) (py x-px y))/((x^2)+(y^2))) /. r -> Sqrt[x^2 + y^2] // FullSimplify

Clear[ptheta, r, x, y, px, py, pr, r, theta]
ptheta[x_, y_, px_, py_] = py x-px y ;
r[x_, y_, px_, py_] = Sqrt[x^2 + y^2] ;
pr[x_, y_, px_, py_] = (px x+py y )/r[x, y, px, py] ;
theta[x_, y_, px_, py_] = ArcTan[y/x] ;
(* This is the Jacobian syntax *)
j[x_, y_, px_, py_] = D[{r[x, y, px, py], theta[x, y, px, py], pr[x, y, px, py], ptheta[x, y, px, py]}, {{x, y, px, py}}]

jj = FullSimplify[j[x, y, px, py]] /. px -> (Subscript[p, x])/. py -> (Subscript[p, y ])
TraditionalForm[jj]
Det[jj]  
Det[jj]  // FullSimplify