111
2.7. Hamilton–Jacobi theory
at time zero. Assuming for the moment that such a transformation
can be found, we express the new set
q i = q i (Q, P),
p i = p i (Q, P),
i = 1, . . . , n,
(2.303)
where the new coordinates q i and canonical momenta p i are as
yet unspecified functions of the old Q i and P i . Here we adopt a
different notation from the earlier sections, for reasons which will
become clear in the following. The p i are not to be confused with
the components of kinetic momentum described earlier. In order
for the motion to be physically possible, we require that the new
q i , p i also obey Hamilton’s equations
∂K
∂K
= q˙ i ,
= −p ˙ i ,
i = 1, . . . , n,
(2.304)
∂p i
∂q i
where K(q, p; t) is the Hamiltonian in the new system. Any transformation for which Hamilton’s equations of motion (2.300, 2.304)
are satisfied in both the old and new systems is called a canonical
transformation.
Hamilton’s principle (2.8) can be written separately in the two
systems (2.19) as
t 2 4
˙
δ
P i Q i − H(Q, P; t) dt = 0
t 1
i

t 2 4

δ
p i q˙ i − K(q, p; t) dt = 0.
(2.305)
t 1
i
In order for both equations to hold, the integrands can differ at
most by the total time derivative of an arbitrary function F as
follows:
4
4
d
˙
P i Q i − H =
p i q˙ i − K + F (Q, q; t),
(2.306)
dt
i
i
where this represents a necessary condition relating the Hamiltonians H and K in the two systems. The function F is called a