The motion of a particle on the surface of a general cone

Abstract

We use the formalism of Lagrange to find the equations of motion of a particle on the inner surface of a “general cone”. The equations of motion are challenging to solve, but we can evaluate them numerically with different software, to obtain the particle’s trajectory on the surface as a function of parameters such as angular momentum Lθ, cone shape and initial conditions, and then we find the total free-fall time of the particle. The results show a special cone in which the free fall time has a minimum for a fixed angular momentum and fall height. Differences in the free-fall times and the particle’s trajectory also analyzed for a two-coordinate (r, z) and three-coordinate (r, θ, z) system. This work shows the importance of learning to use software (Wolfram Mathematica, Python, POV-Ray) to help with some complex theoretical problems. Finally, the results can easily be generalized for other more complex surfaces.