Re: [Phys-L] analytic determination of the position of a non-linearized simple pendulum

[brian whatcott <betwys1@sbcglobal.net>]

On 9/29/2014 5:58 PM, Bernard Cleyet, you asked if anyone has integrated:
> d(theta) / d(t) = sort{ (2g/L) * ( cos(theta)-cos(A) ) }

I copied your equation to Wolfram Alpha (on-line) for a solution thus:

solve theta'(t) = sqrt((2×g/L) (cos(theta)-cos(A)))  for  theta

See
http://tinyurl.com/nhldcdn

Here is the solution:

theta(t) = 2 am((i (sqrt(2) sqrt(g) sqrt(L) sqrt(cos(A)-1) t+sqrt(g) sqrt(L) 
sqrt(cos(A)-1) c_1))/(2 L), csc^2(A/2))
or in Wolfram Language plain text:
{θ[t] == 2 JacobiAmplitude[((I/2) (Sqrt[2] Sqrt[g] Sqrt[L] t Sqrt[-1 + Cos[A]] 
+ Sqrt[g] Sqrt[L] Sqrt[-1 + Cos[A]] Subscript[c, 1]))/L, Csc[A/2]^2]}

Brian Whatcott