Lecture 27: Hamilton’s equations (4 Nov 09)
Review 2 Nov homework
A. Review – Hamiltonian
1. FW Secs. 18 and 20 and L11
2. In the case that the Lagrangian has no explicit time dependence,
L
=
L
(
{
q
j
}
,
{
˙
q
j
}
), form the total time derivative, with variation through
the
q
j
and ˙
q
j
:
dL
dt
=
X
j
∂L
∂q
j
˙
q
j
+
∂L
∂
˙
q
j
¨
q
j
3. Lagrange equations give
∂L
∂q
j
=
d
dt
∂L
∂
˙
q
j
=
d
dt
p
j
;
p
j
≡
∂L
∂
˙
q
j
4. so
dL
dt
=
X
j
˙
q
j
d
dt
∂L
∂
˙
q
j
+
∂L
∂
˙
q
j
¨
q
j
=
d
dt
X
j
˙
q
j
p
j
5. Thus
H
≡
∑
j
˙
q
j
p
j

L
is a conserved quantity (the energy):
dH
dt
= 0
6. Lorentz force from L10 (Sec. 33 of FW). The Lagrangian that generates
the equation of motion for particle of mass
m
and charge
q
in electric
and magnetic fields
E
and
B
(potential and vector potential
φ
and
A
)
is
L
=
m
2
v
2

qφ
+
q
v
·
A
The canonical momentum is then (
v
= ˙
r
)
p
=
m
v
+
q
A
1
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(mechanical momentum
m
v
) and the Hamiltonian is
H
=
v
·
p

m
2
v
2
+
qφ

q
v
·
A
=
m
2
v
2
+
qφ
H
=
1
2
m
(
p

q
A
)
2
+
qφ
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 Fall '09
 BRUCH
 Work, Hamiltonian mechanics, pj, Lagrangian mechanics, dt ∂ qj, mechanical momentum mv

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