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116
Chapter 2. Geometrical optics
identical with the first equation (2.310). Taking the partial derivative with respect to time, we obtain
4
t
∂S
∂
n
∂
=
P i dQ i −
H(Q, P; t
� ) dt
� .
(2.331)
∂t
∂t
∂t
i=1
The first term on the right is identically zero, since the integral
has no explicit time dependence. The second term on the right is
H(Q, P; t). This leads to
∂S
H +
= 0,
(2.332)
∂t
identical with (2.313). We assumed in (2.327) that S is an indefinite integral, and is therefore defined only to within an additive
constant. This integration constant can always be chosen in principle so that (2.314) is satisfied, remembering that the α i and β i
are constants of the motion. This completes the justification of
our postulate (2.327) for the form of S. We have thus identified
Hamilton’s principal function S with the indefinite integral corresponding to the action integral in Hamilton’s principle of least
action (2.8).
Next we consider the case where the Hamiltonian H(Q, P) has
no explicit time dependence. We rewrite (2.328) as
S(Q, α; t) = p · dQ − H t,
(2.333)
remembering that H = α 1 is the constant total energy. From
(2.326), it follows that
W (Q, α) = P · ds,
(2.334)
where the right side is the indefinite path integral along the physical trajectory. We have thus identified Hamilton’s characteristic
function W with the indefinite integral corresponding to the action
integral in the mechanical equivalent of Fermat’s principle (2.41).