Beta 1.1 Release

Release $\beta$ 1.1!

The project does the following:

• Derives the Lagrangian, equations of motion, and effective potential for the Kapitza pendulum system, while inspecting the high-driving, low-amplitude regime.
• Implements numerical methods to solve the system and check that turning off the driving components recovers the simple pendulum.
• Dynamic widget analysis of the system with varying parameters.
• $U(\phi,t)$ and $U_{\rm eff}(\phi)$ comparisons for high-driving, low-amplitude regime, with separate PDF derivation.
• Manim animations of the system, with normalized state space axis as well.
• Chaos and bifurcation analysis: using Liapunov exponent to judge chaos. Manim animations of two pendula for chaos, with animated Liapunov exponent graph. Hopf bifurcation discrete analysis with checks on stability vs. instability of the vertical position using the bifurcation plot. Further analysis on the low-amplitude area, showing periodic motion but no chaos.

Included in this repository version:

• All project files: the project Jupyter notebook, the PDF derivation file, any .py numerical files, and the outputted images.
• The Conda environment used for the project, built for M-architecture OS.
• The presentation file given as a part of this project.

Included in the release:

• The videos.zip file containing all the animations made in the Project.ipynb file [6.34MB].

Changes between this and the last release:

• Update to plot titles in the repository and PendulumSolver.py file

Beta 1.0 Release

The first release of this project $\beta$ 1.0!

The project does the following:

• Derives the Lagrangian, equations of motion, and effective potential for the Kapitza pendulum system, while inspecting the high-driving, low-amplitude regime.
• Implements numerical methods to solve the system and check that turning off the driving components recovers the simple pendulum.
• Dynamic widget analysis of the system with varying parameters.
• $U(\phi,t)$ and $U_{\rm eff}(\phi)$ comparisons for high-driving, low-amplitude regime, with separate PDF derivation.
• Manim animations of the system, with normalized state space axis as well.
• Chaos and bifurcation analysis: using Liapunov exponent to judge chaos. Manim animations of two pendula for chaos, with animated Liapunov exponent graph. Hopf bifurcation discrete analysis with checks on stability vs. instability of the vertical position using the bifurcation plot. Further analysis on the low-amplitude area, showing periodic motion but no chaos.

Included in this repository version:

• All project files: the project Jupyter notebook, the PDF derivation file, any .py numerical files, and the outputted images.
• The Conda environment used for the project, built for M-architecture OS.
• The presentation file given as a part of this project.

Included in the release:

• The videos.zip file containing all the animations made in the Project.ipynb file [6.34MB].