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

[brian whatcott <betwys1@sbcglobal.net>]

You wanted  displacement steps:
solve theta'(t) = sqrt((2×g/L) (cos(theta)-cos(A)))  for  theta = 0.01
gives

(d(0.01))/(d(t)) = sqrt(2) sqrt(g/L) sqrt(0.99995-cos(A))

 ~ for theta = 0.02

(d(0.02))/(d(t)) = sqrt(2) sqrt(g/L) sqrt(0.9998-cos(A))

...and so on.

Brian

On 9/30/2014 9:32 AM, brian whatcott wrote:
> 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
>
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