# I Non-linear Pendulum

#### Saim

If I calculate the time period of a non linear pendulum using elliptical integral equation, then how can I find out the angular displacement.

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#### berkeman

Mentor
If I calculate the time period of a non linear pendulum using elliptical integral equation, then how can I find out the angular displacement.
Welcome to the PF.

What reading have you been doing on this question so far? I did a Google search on the phrase from your post and got a lot of useful hits: nonlinear pendulum using elliptical integral equation

#### Saim

The differential equation which represents the motion of a simple pendulum is

If it is assumed that the angle is much less than 1 radian

and
above equation becomes

Given the initial conditions θ(0) = θ0 and /dt(0) = 0, the solution becomes

The period of the motion, the time for a complete oscillation is

For amplitudes beyond the small angle approximation,time period can be calculated using formula:

K is the complete elliptic integral of the first kind defined by

So, If somehow I am able to calculate the Time period for non linear Pendulum, then how can I plot the angular displacement θ(t) against time.

#### Baluncore

I'm not certain exactly what you want but there are some interesting articles here on the period of extreme precision pendulum clocks. The articles include useful references.
http://www.leapsecond.com/hsn2006/
This is a discussion of the circular error correction of period using AGM.
http://www.leapsecond.com/hsn2006/pendulum-period-agm.pdf
You may be able to invert the equation in his conclusion.
Unfortunately they appear to be transcendental solutions.

Following that are pendulum simulations that model displacement against time.
http://www.leapsecond.com/hsn2006/pendulum-simulation-1.pdf
Perhaps there is a solution using the model, conversion between PE and KE.

#### DrClaude

Mentor
There is no simple analytic formulation of $\theta(t)$ for the non-linear pendulum.

Check out https://en.wikipedia.org/wiki/Pendulum_(mathematics). You will find there how the period is found to be an elliptical integral, without having to know $\theta(t)$. There is also an expression for $\theta(t)$ in terms of a Fourier series.

#### DrClaude

Mentor
There is no simple analytic formulation of $\theta(t)$ for the non-linear pendulum.

Check out https://en.wikipedia.org/wiki/Pendulum_(mathematics). You will find there how the period is found to be an elliptical integral, without having to know $\theta(t)$. There is also an expression for $\theta(t)$ in terms of a Fourier series.

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