Point mass dynamics on spherical hypersurfaces

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Abstract

The equations of motion are derived for a system of point masses on the (hyper-)surface Sn of a sphere embedded in ℝn+1 for any dimension n > 1. Due to the symmetry of the surface, the equations take a particularly simple form when using the Cartesian coordinates of ℝn+1. The constraint that the distance of the jth mass‖rj‖ from the origin remains constant (i.e. each mass remains on the surface) is automatically satisfied by the equations of motion. Moreover, the equations are a Hamiltonian system with a conserved energy as well as a host of conserved angular momenta. Several examples are illustrated in dimensions n = 2 (the sphere) and n= 3 (the glome).
Original languageEnglish
Article number20180349
Pages (from-to)1-11
Number of pages11
JournalPhilosophical Transactions of the Royal Society. A, Mathematical, Physical and Engineering Sciences
Volume377
Issue number2158
Early online date30 Sept 2019
DOIs
Publication statusPublished - Nov 2019

Keywords

  • Hamiltonian dynamics
  • Surfaces
  • Gravity

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