Solving the Pendulum Problem
a month ago
- #Differential Equations
- #Pendulum
- #Physics
- Introduction to a new LLM evaluation called Humanity’s Last Exam in Spring 2024, featuring a physics problem about a pendulum with a sliding pivot point.
- Problem statement involves a massless rod of length L with a pendulum displaced from the vertical, requiring understanding of calculus, trigonometry, and Newton’s Second Law.
- Derivation of the differential equation for the pendulum using Newton’s Second Law and trigonometry to find the tangential component of gravitational force.
- Explanation of the small angle approximation to simplify the differential equation, leading to a solvable form.
- Detailed steps to solve the simplified differential equation using integration, separation of variables, and trigonometric identities.
- Comparison between the exact and approximate solutions, highlighting the complexity introduced without the small angle approximation.
- Use of elliptic integrals (Legendre’s Form and Jacobi’s Form) to solve the non-linear differential equation, with detailed algebraic manipulation and substitution steps.
- Summary of the solution process for both the simplified and exact differential equations, emphasizing the periodic nature of the pendulum's motion.
- Mention of sources like James Crawford's work on pendulums and elliptic integrals, and Edmund Whittaker's Analytical Dynamics, for further reading.
- Acknowledgement of the complexity and educational value of the pendulum problem in classical mechanics.