The Hamilton-Jacobi equation
147
in which these repeated quantities are regarded as constant in carrying out
the differentiations. Hence
(46.4)
P=constant
q=constant
The Jacobian in the numerator is, by definition, a determinant of order s
whose element in the ith row and kth column is Representing the
canonical transformation in terms of the generating function (q, P) as in
(45.8), we have = In the same way we find that the
ik-element of the determinant in the denominator of (46.4) is
This means that the two determinants differ only by the interchange of rows
and columns; they are therefore equal, so that the ratio (46.4) is equal to
unity. This completes the proof.
Let us now suppose that each point in the region of phase space considered
moves in the course of time in accordance with the equations of motion of the
mechanical system. The region as a whole therefore moves also, but its volume
remains unchanged:
f dr = constant.
(46.5)
This result, known as Liouville's theorem, follows at once from the invariance
of the volume in phase space under canonical transformations and from the
fact that the change in p and q during the motion may, as we showed at the end
of §45, be regarded as a canonical transformation.
In an entirely similar manner the integrals
11 2 dae dph
,
in which the integration is over manifolds of two, four, etc. dimensions in
phase space, may be shown to be invariant.
47. The Hamilton-Jacobi equation
In §43 the action has been considered as a function of co-ordinates and
time, and it has been shown that the partial derivative with respect to time
of this function S(q, t) is related to the Hamiltonian by
and its partial derivatives with respect to the co-ordinates are the momenta.
Accordingly replacing the momenta P in the Hamiltonian by the derivatives
as/aq, we have the equation
(47.1)
which must be satisfied by the function S(q, t). This first-order partial
differential equation is called the Hamilton-Jacobi equation.
148
The Canonical Equations
§47
Like Lagrange's equations and the canonical equations, the Hamilton-
Jacobi equation is the basis of a general method of integrating the equations
of motion.
Before describing this method, we should recall the fact that every first-
order partial differential equation has a solution depending on an arbitrary
function; such a solution is called the general integral of the equation. In
mechanical applications, the general integral of the Hamilton-Jacobi equation
is less important than a complete integral, which contains as many independent
arbitrary constants as there are independent variables.
The independent variables in the Hamilton-Jacobi equation are the time
and the co-ordinates. For a system with s degrees of freedom, therefore, a
complete integral of this equation must contain s+1 arbitrary constants.
Since the function S enters the equation only through its derivatives, one
of these constants is additive, so that a complete integral of the Hamilton-
Jacobi equation is
Sft,q,saas)+
(47.2)
where X1, ..., as and A are arbitrary constants.
Let us now ascertain the relation between a complete integral of the
Hamilton-Jacobi equation and the solution of the equations of motion which
is of interest. To do this, we effect a canonical transformation from the
variables q, P to new variables, taking the function f (t, q; a) as the
generating function, and the quantities a1, A2, ..., as as the new momenta.
Let the new co-ordinates be B1, B2, ..., Bs. Since the generating function
depends on the old co-ordinates and the new momenta, we use formulae
(45.8): Pi = af/dqi, Bi = af/dar, H' = H+dfdd. But since the function f
satisfies the Hamilton-Jacobi equation, we see that the new Hamiltonian is
zero: H' = H+af/dt = H+as/t = 0. Hence the canonical equations in
the new variables are di = 0, Bi = 0, whence
ay=constant,
Bi = constant.
(47.3)
By means of the S equations af/dai = Bi, the S co-ordinates q can be expressed
in terms of the time and the 2s constants a and B. This gives the general
integral of the equations of motion.
t Although the general integral of the Hamilton-Jacobi equation is not needed here, we
may show how it can be found from a complete integral. To do this, we regard A as an arbi-
trary function of the remaining constants: S = f(t, q1, ..., q8; a1, as) +A(a1, as). Re-
placing the Ai by functions of co-ordinates and time given by the S conditions asidar = 0,
we obtain the general integral in terms of the arbitrary function A(a1,..., as). For, when the
function S is obtained in this manner, we have
as
The quantities (as/dqs)a satisfy the Hamilton-Jacobi equation, since the function S(t, q; a)
is assumed to be a complete integral of that equation. The quantities asida therefore satisfy
the same equation.
