A cycloidal pendulum - one suspended from the cusp of an inverted cycloid - is isochronous, meaning its period is constant regardless of the amplitude of the swing. Please find the proof using energy methods: Lagrange's equations (in the images attached to the reply).

Background:
The standard pendulum period of \(2\pi\sqrt{L/g}\) or frequency \(\sqrt{g/L}\) holds only for small oscillations. The frequency becomes smaller as the amplitude grows. If you want to build a pendulum whose frequency is independent of the amplitude, you should hang it from the cusp of a cycloid of a certain size, as shown in the gif. As the string wraps partially around the cycloid, the effect decreases the length of the string in the air, increasing the frequency back up to a constant value.

In more detail:
A cycloid is the path taken by a point on the rim of a rolling wheel. The upside-down cycloid in the gif can be parameterized by \((x, y)=R(\theta-\sin\theta, -1+\cos\theta)\), where \(\theta=0\) corresponds to the cusp. Consider a pendulum of length \(L=4R\) hanging from the cusp, and let \(\alpha\) be the angle the string makes with the vertical, as shown (in the proof).

#Pendulum #Cycloid #Period #Frequency #SHM #TimePeriod #CycloidalPendulum #Lagrange #Cusp #Energy #KineticEnergy #PotentialEnergy #Lagrangian #Length #Math #Maths #Physics #Mechanics #ClassicalMechanics #Amplitude #CircularFrequency #Motion #Vibration #HarmonicMotion #Parameter #ParemeterizedEquation #GoverningEquations #Equation #Equations #DifferentialEquations #Calculus

Simulating harmonic motion with Leapfrog method and GeoGebra Python

I wrote an article about Leapfrog integration in Finnish last week. For the article I used Colab Python, the graphs were made with matplotlib-library. I decided to play with the code using GeoGebraPython from https://geogebra.org/python. I wanted to learn how to animate the objects in my code.

The idea for the Python code is to compare two simulation methods with the theoretical values. The first is using formulas of constant acceleration during the time step dt. In the code below the acceleration is calculated from harmonic force equation ma = -kx. In row 41 the constant H = -k/m.

The second method is Leapfrog integration, where the acceleration is calculated also at the new x-value, the velocity is calculated using average acceleration.

The animation is created using GeoGebra’s Point command. In row 45 the first variable 0 is x coordinate, second variable x is y-coordinate and the rest are properties for the point named paikka. The teorpaikka is calculated value for the theoretical value and lpaikka is the value from leapfrog integration.

The teorpaikka is calculated with equation

Later in code we change the visibility of the points.

Link to the GeoGebra/Python code.

references

My original article in Finnish
https://mikkorahikka.blog/2023/08/21/varahtelevan-jousen-liikkeen-simulointi-pythonilla-leapfrog-integrointi/

Original Python code in Colab
https://colab.research.google.com/drive/1JIcqE5mECFpF3naFlH2CVHwfXQfs9blp?usp=sharing

Leapfrog integration in Wikipedia
https://en.wikipedia.org/wiki/Leapfrog_integration

Peter Young’s article ”The leapfrog method and other “symplectic” algorithms for integrating Newton’s laws of motion”
https://young.physics.ucsc.edu/115/leapfrog.pdf

Rhett Allain in Wired. “The Leapfrog Numerical Method”
https://www.wired.com/2011/02/the-leapfrog-numerical-method/