Solving the Pendulum Problem (ethansteere.net)

In Spring 2024, a new evaluation framework for large language models (LLMs) named Humanity’s Last Exam was introduced, featuring a complex physics challenge centered on the pendulum problem. The problem involves deriving a differential equation that describes the motion of a pendulum with a sliding pivot point, which requires a solid grasp of calculus, Newtonian mechanics, and trigonometry for its solution. The article outlines the stepwise approach to both linear and non-linear forms of the pendulum’s motion, culminating in an understanding of how to solve for the position of the pendulum over time, employing techniques such as integration and variable separation. This exploration is significant for the AI/ML community as it highlights the intersection of advanced programming and classical mechanics, illustrating how LLMs can tackle intricate real-world problems. By detailing the mathematical derivations and leveraging known functions like the Incomplete Elliptic Integral of the First Kind, the findings not only demonstrate the capabilities of AI in solving complex physics problems but also serve as a practical case study for developing algorithms that manage similar challenges across various domains. Such advancements enhance the potential for LLMs in academic research and educational tools, emphasizing their role in bridging theoretical knowledge with practical application.