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Morin4.23 - Morin 4.23 Corrections to the Pendulum The...

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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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