Pendulums

[Holmes]

I was a bit taken aback recently when I learned that the motion of a pendulum is complex to model, I must have last looked at this when I was in school and paid little attention, I think we used some simplified model and I never appreciated it was more complex.

My experience with differential equations is rather limited and I had no idea that an equation like Eq. 1 was so involved.

Fascinating how sometimes something that gives an impression of simplicity turns out to be far more involved.

 

[swansont]

It's actually the case for most of physics. We are able to solve a few kinds of problems under simple conditions. Outside of those few, complications are legion.

[Holmes]

14 minutes ago, swansont said:

It's actually the case for most of physics. We are able to solve a few kinds of problems under simple conditions. Outside of those few, complications are legion.

When I see this I often wonder - is there a simpler way to solve it, is our initial approach somehow inhibiting our ability to get a simpler solution.

 

 

[MigL]

The differential equation discribes the general case ( all possible cases ), and as such, has no specific solution.
Specific cases are solved by applying the relevant boundry conditions to simplify it, but some of those cases may still have no solution.

[Holmes]

20 minutes ago, MigL said:

The differential equation discribes the general case ( all possible cases ), and as such, has no specific solution.
Specific cases are solved by applying the relevant boundry conditions to simplify it, but some of those cases may still have no solution.

I see, that's noteworthy. So by making (presumed) simplifying assumptions we may actually be making the problem harder to solve, if we remove assumptions about the string being unstretchable, there being no friction and so on - would we get to a more complex looking model but one that has an "easier" solution...

[Roscoe G. Little]

Can differential equations be used to solve such a problem? I understand that this topic is more related to physics, but I am sure that mathematics is indispensable here. I often use https://plainmath.net/post-secondary/calculus-and-analysis/differential-equations when I need help with various tasks. I know I can always get free solutions to bivariate data questions, but I'm wondering how you would represent the solution to this problem with an equation.

Edited April 12, 2022 by Roscoe G. Little