# Non-linear Pendulum

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

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

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.