Natural Science Forum / Physics / Research / January 2008

I haven't read Carter's paper, but it sounds as though section 9.1 of
Goldstein's book, "Classical Mechanics", may provide what you are
looking for (Eq. 9.7 of my 1950 edition corresponds to the property
you mention).

In summary, consider a generating function S(q, P, t), where q and P
may be n-vectors, that generates a canonical transformation from
coordinates q,p and Hamiltonian H to coordinates Q, P and Hamiltonian
K.  Hence, from general canonical transformation theory, Q, p and K
are related to the generating function via
    Q = del S/del P,    p = del S/del q,  K = H + del S/del t .

Now, the special defining property of the particular generating
function corresponding to the Hamilton-Jacobi equation is that
     K = 0.
Clearly, this equation IS the Hamilton-Jacobi equation.  Hence,
Hamilton's equations of motion for coordinates Q and P follow
immediately as
    dQ/dt = del K/del P =  0, dP/dt = - del K/del Q = 0,
i.e., both Q and P are constants of the motion.  But, as noted above,
one has
   Q = del S/del P.
Hence, the partial derivative of S with respect to the constant of the
motion P is itself a constant of the motion, Q, as desired.

In practice, one solves the H-J equation to find S(q,P,t), where P
represents the n constants expected in the general solution of such a
partial differential equation (actually, there are n+1 constants, but
the last one is just an additive constant, which is ignored, as it
just redefines energy by an additive constant).  One can then
determine the momentum p and the constants of the motion  Q via the
corresponding partial derivatives.