
135Hamiltonian Formulation of Mechanics
© 2010 Taylor & Francis Group, LLC
or
() ()δθ δθ×
∂
∂
+×
∂
∂
=
=
∑
r
H
r
p
H
p
j
j
j
j
j
N
1
3
0
from which we obtain
δθ
⋅×
∂
∂
+×
∂
∂
=
r
H
r
p
H
p
j 11
3
0
N
∑
=
(5.20)
where we have used the vector identity ()
ABCC
×⋅=×⋅ .
Using Hamilton’s equations, Equation 5.20 can be further simplied:
δθ δθ
⋅×+×
{}
=− ⋅
=
∑
()()rp pr
jj
j
N
1
3
d
d
tt
rp
L
t
j
j
N
()
×=−⋅ =
=
∑
1
3
0δθ
d
d
where
is the total angular momentum of the system. Because
is arbitrary, we have
d
,or(aconstant vector
L
L==0 α .
5.4 CANONICAL TRANSFORMATIONS
As shown in the previous section, there is some advantage in using cy ...