Large Amplitude Pendulum Equation

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    Amplitude Pendulum
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Discussion Overview

The discussion revolves around the equation for a large-angle pendulum, specifically focusing on the patterns in the coefficients of the series expansion and the potential use of elliptic integrals. Participants explore the mathematical formulation and implications of large amplitude pendulum motion.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant notes that the equation for a large-angle pendulum can be infinitely long and questions the pattern in the coefficients following a specific term.
  • Another participant suggests that there is no discernible pattern and indicates the necessity of elliptic integrals for the solution.
  • Several participants express uncertainty about the nature of the series, with one proposing it could be a Taylor expansion, while another mentions the existence of various series solutions.
  • A participant shares a link to a pendulum calculator they developed, which calculates pendulum periods up to a certain angle.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the existence of a pattern in the coefficients of the series expansion, with some asserting there is none while others speculate on potential forms of the series.

Contextual Notes

Participants reference the complexity of the equations involved and the potential need for elliptic integrals, indicating that assumptions about the series may depend on specific mathematical definitions or contexts.

Who May Find This Useful

This discussion may be of interest to those studying pendulum dynamics, mathematical series, or anyone developing tools for calculating pendulum motion.

StevenJacobs990
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The equation for large-angle pendulum can be infinitely long. What is the pattern with the latter numbers in "..."?
pendl3.gif
 
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Sorry, your attachment won't open for me.
 
sophiecentaur said:
Does this link help?
Yeah, but what's the pattern that comes after 11/3072 (theta)^4?
 
There is no pattern. That's why you need the elliptic integrals.
 
StevenJacobs990 said:
Yeah, but what's the pattern that comes after 11/3072 (theta)^4?
AS I said before, your attachment is not readable.
 
This is the attachment

pendl3.gif
 
Here is a graphic I made.
Look at equation 3.
pendulum.png
 
pendulum.png
StevenJacobs990
I don't know the equation for generating those numbers in the formula but here is the large amplitude formula carried out to theta 20:
 
  • #10
Honestly, I wouldn't be surprised if this a Taylor expansion of some sort.
 
  • #11
rumborak said:
Honestly, I wouldn't be surprised if this a Taylor expansion of some sort.
. . . .or something else. There are (my Mathematician friends tell me) many equations that can only be solved using a series - Taylor or not so well known ones.
 
  • #12
I originally was writing a pendulum calculator and while researching the Internet, I came across this topic. Anyway, I finished the calculator and it is online: http://www.1728.org/pendulum.htm
It can calculate pendulum periods up to theta 14 and uses the arithmetic mean to calculate exact pendulum periods.
Try it out if you like.
 

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