
140 Classical Mechanics
© 2010 Taylor & Francis Group, LLC
Hence,
pPQq
4
co
sin, . (5.38)
Now, we can evaluate the new Hamiltonian K. Because the generating function F
1
does not
depend on time explicitly, we have
KH
m
pkq
kP
Q
mk
Q
== +
=+
1
2
1
2
2
4
2
2
2
µ
µ
si
os
.
If
µ= mk
, this reduces to
KkPP
==
2µ
(5.39)
which is of a particularly simple form. Because the new coordinate Q is a cyclic coordinate, the
new momentum P conjugated to Q is a constant of the motion:
PKQ=−∂∂=
and
P = β (a constant of the motion). (5.40)
Hamilton’s equations of motion for the new coordinate Q gives
QK
m=∂ ∂=
from which we obtain
Qkmt
/ α
(5.41)
where α is the ...