Chemistry Reference
In-Depth Information
equality we also need the Taylor expansion proof mentioned above and available in the
references of the Further Reading section in this appendix.
Euler's equality fits with Equation (A9.85) because the right-hand side is a sum of
complex conjugates; so, if we use the earlier identity
exp( i
φ
)
+
exp(
−
i
φ
)
=
cos(
φ
)
+
isin(
φ
)
+
cos(
φ
)
−
isin(
φ
)
=
2 cos(
φ
)
(A9.95)
then Equation (A9.89) with the boundary conditions
φ
=
0
=
d
y
d
=
y
(0)
2
and
2
(A9.96)
φ
along with the trial functions
y
=
A
sin(
αφ
)
and
y
=
B
[exp(
βφ
)
−
exp(
−
βφ
) ]
(A9.97)
can also be used to show that
2i sin(
φ
)
=
exp( i
φ
)
−
exp(
−
i
φ
)
(A9.98)
Euler's analysis provided us with the complex exponential to use in the place of trigono-
metric functions in problems giving periodic functions. After a little practice it is often
much easier to manipulate than sines and cosines. A useful excercise is to obtain Equa-
tion (A9.98) from the boundary conditions in Equation (A9.96) and trial functions in
Equation (A9.97) following the same route.
Euler's problem is very close to the equation for the function of
φ
we obtained by
separating variables for the angular equation:
∂
∂φ
2
+
m
l
=
0
(A9.99)
2
The only difference is that now we have a coefficient for the
term. Looking back at
Equation (A9.93) shows that our choice of
β
would stem from
β
2
=−
m
l
, so the solutions
that will now be found are
=
exp (i
m
l
φ
)
(A9.100)
In the main text, the boundary conditions are then imposed to stipulate allowed values
of
m
l
.
Further Reading
The classical mechanics of rotating bodies is fully covered in:
Goldstein H, Poole C, Safko J (2002)
Classical Mechanics
, 3rd edition. Pearson (ISBN 0321-
188977).
