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


  • Diego Gerardo Gómez Pérez Facultad de Ciencias Físico Matemáticas, Universidad Autónoma De Nuevo León.
  • Omar Gonzalez Amezcua Universidad Autónoma de Nuevo León Facultad de Ciencias Físico Matemática.



Angular momentum; cone; Wolfram mathematica


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.

Author Biography

Omar Gonzalez Amezcua, Universidad Autónoma de Nuevo León Facultad de Ciencias Físico Matemática.

 Departamento de Fisica


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Some codes to plot the particle trajectory are available in this link

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How to Cite

D. G. Gómez Pérez and O. G. Amezcua, “ The motion of a particle on the surface of a general cone”, Rev. Mex. Fis. E, vol. 21, no. 1 Jan-Jun, pp. 010206 1–, Jan. 2024.