
288 Classical Mechanics
© 2010 Taylor & Francis Group, LLC
xt
t
n
() cos(
=
∞
∑
ω
. (9.38)
The symmetry of the potential function means that both even and odd n are present in Equation
9.38. The determination of b
n
can be made in the same way that we did for the symmetric case, with
the result
b
mn k
xk kx nt t
n
=
−
+
∫
1
22
2
23
0
2
πω
π
()
()cos( )(
d
.
(9.39)
If third-order accuracy is maintained, Equation 9.39 leads to
b
k
k
bb
mk
kbbkbb kb
0
1
3
1
2
1
2
1
201212 31
3
1
2
3
4
=− =
−
++
,
ω
(9.40a)
b
mk
kb b
mk
kb
b
2
2
1
21
2
3
2
1
21
1
3
1
24
1
9
1
4
=
−
=
−
+
()
,
ωω
(9.40b)
and the approximate Fourier series solution is
xt
k
k
bb t
k
k
k
k
k
() co
os=− ++
++
2
1
1
2
1
2
1
1
2
2
2
1
3
6
2
48
332
3
1
1
3
k
co