Chemistry Reference
In-Depth Information
z
z
y
x
y
Figure 2.1.
The Euler angle definition for a rigid rod.
r
where
is the positional coordinates and
θ
,
φ
and
ψ
are the Euler's angles
(see Figure 2.1),
p
,
p
θ
,
p
φ
, and
p
ψ
are the conjugate momenta.
The total energy of the system,
U
, is the sum of the kinetic energy of
each rigid rod
T
i
(translation and rotation) and potential energy
V
i
.
The translational kinetic energy of a rigid rod
T
ti
is simply
1
2
m
(
p
x
+
p
y
+
p
z
)
,
T
ti
=
(2.5)
where
m
and
p
(
p
x
,
p
y
and
p
z
) are the mass and kinetic momenta
(components).
The rotational kinetic energy of a rigid rod is more complicated and we
adopt the form from the literature as follows
p
ψ
2
I
2
p
θ
2
I
1
p
ψ
cos
θ
)
2
2
I
1
sin
2
θ
+
(
p
φ
−
T
ri
=
+
,
(2.6)
where
I
1
and
I
2
are principal inertial moments around the long and the
short axes, respectively. The total energy of the system is the sum of
the two contributions
N
N
T
=
(
T
ti
+
T
ri
)=
T
i
.
(2.7)
i
=1
i
