316
7 Lagrangian and Hamiltonian Mechanics
Now Hamilton’s equations are
∂ H
∂ p β
= ˙
q β ,
∂ H
∂q β
= − ˙
p β
(11)
Using (11) in (10), we obtain
dH
dt
=
∂ H
∂t
+
n
β=1
∂ H
∂q β
∂ H
∂ p β
−
∂ H
∂ p β
∂ H
∂q β
(12)
whence
dH
dt
=
∂ H
∂t
(13)
Equation (13) asserts that H changes with time only by virtue of its explicit
time dependence. The net change is induced by the fact that the variation of q
and p with time is zero.
Now in a conservative system, neither T nor V contains any explicit dependence on time.
Hence
∂ H
∂t
= 0. It follows that
dH
dt
= 0
(14)
which leads to the law of conservation of energy
H = T + V = E = constant
(15)
The Hamiltonian formalism is amenable for finding various conservation
laws.
Conservation of angular momentum: The Hamiltonian can be written as
H =
n
β=1
p β ˙
q β − L
(16)
Using the polar coordinates (r , θ )
H = p r ˙
r + p θ ˙
θ −
1
2
m ˙
r
2
+
1
2
mr
2 ˙
θ
2
− U (r )
(17)
Using (7) and (8) in (17)