Ben Sauerwine
Qualifying Exam Study Sheet
Mechanics
(In this section, K indicates the kinetic energy of the system in terms of the variables and
V indicates the potential.)
Lagrangian:
V
K
L
−
=
Hamiltonian:
L
p
q
V
K
H
i
i
i
−
=
+
=
∑
Above, the “conjugate momentum”
of a coordinate is
p
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
=
q
L
p
&
.
Thus note that the
equations of motion take a form like
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
=
q
L
p
dt
d
which is essentially, “The time
derivative of the momentum is the force.”
Electricity and Magnetism
Radiation pressure:
c
P
Φ
=
.
Maxwell’s Equations:
Gauss:
0
ε
ρ
=
⋅
∇
E
v
v
Gauss’s Law (magnetism):
0
0
=
⋅
∇
=
⋅
∇
H
B
v
v
r
v
Faraday’s Law:
t
B
E
∂
∂
−
=
×
∇
v
v
v
Ampere’s Law:
t
E
J
B
∂
∂
+
=
×
∇
v
v
v
v
0
0
0
μεμ
The integral forms are related to these such that the cross products become loop integrals
and the dot products become surface integrals.
Note that in these may vary in matter.
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 Spring '12
 Dr.Jaouni
 Energy, Kinetic Energy, Fundamental physics concepts, Helmholtz free energy, Canonical Ensemble, vv vv, vv v vv

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