Morin 4.23 --- Corrections to the Pendulum
The simple pendulum satisfies the nonlinear differential equation
In[3]:=
''
t
0^2 Sin
t
Out[3]=
t
0
2
Sin
t
We will agree to launch this pendulum at rest from its maximum angle
a
(also known as
q
0), and we seek the period as a
function of the input parameters
w
0 and
a
.
On dimensional grounds we know that the answer will be of the form T[
a
] = T0 f[
a
] where T0 is defined as 2
p
/
w
0, i.e.
the period in the limit of small angles, where
one can approximate Sin[
q
]=
q
and get the good old simple harmonic oscillator.
With this definition, we know f[
a
=0]=1,
and we seek the rest of the terms in the series
expansion for f[
a
].
By the way, since the period for -
a
should be exactly the same as for +
a
, f[
a
] is an even function,
and the expansion will contain only even powers of
a
.
Morin's problem 4.23 asks you to find the coefficient "c" in the expansion f[
a
] = 1 + c
a
^2 + ...
Numerical Interlude
As a first pass, we'll just find the coefficient numerically.
We'll choose
w
0=1, for which T0=2
p
In[4]:=
T0
2
Out[4]=
2
In[5]:=
de
''
t
Sin
t
Out[5]=
t
Sin
t
In[6]:=
bc1
'
0
0
Out[6]=
0
0
Choosing a smallish angle, say
a
=
p
/10
In[7]:=
10
Out[7]=
10

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