/* Canonical scaling Q = a*q, P = p/a, with a non-zero.
   qn and pn are independent new-coordinate atoms.
   Every printed residual must be the scalar zero. */
kill(all)$

PB_old(X,Y) := diff(X,q)*diff(Y,p)-diff(X,p)*diff(Y,q)$
Q_old : a*q$
P_old : p/a$
print("fundamental Poisson-bracket residual =",
      ratsimp(PB_old(Q_old,P_old)-1))$

/* Type-2 generating function F2(q,pn) = a*q*pn. */
F2 : a*q*pn$
print("type-2 momentum relation residual =",
      ratsimp(diff(F2,q)-a*pn))$
print("type-2 coordinate relation residual =",
      ratsimp(diff(F2,pn)-a*q))$

/* Inverse map q = qn/a, p = a*pn. */
q_of_new : qn/a$
p_of_new : a*pn$
lagrange_qn_pn :
  diff(q_of_new,qn)*diff(p_of_new,pn)
  -diff(q_of_new,pn)*diff(p_of_new,qn)$
print("mixed Lagrange-bracket residual =",
      ratsimp(lagrange_qn_pn-1))$

/* Direct invariance test for two non-trivial phase-space functions. */
A_old : q^2*p$
B_old : q*p^2$
old_bracket_in_new :
  subst([q=q_of_new,p=p_of_new],PB_old(A_old,B_old))$
A_new : subst([q=q_of_new,p=p_of_new],A_old)$
B_new : subst([q=q_of_new,p=p_of_new],B_old)$
PB_new(X,Y) :=
  diff(X,qn)*diff(Y,pn)-diff(X,pn)*diff(Y,qn)$
print("Poisson-bracket invariance residual =",
      ratsimp(old_bracket_in_new-PB_new(A_new,B_new)))$