Nonlinear Dynamics

, Volume 21, Issue 4, pp 307–315

Application of Symmetries to Central Force Problems

  • F. M. Mahomed
  • F. Vawda

DOI: 10.1023/A:1008317327402

Cite this article as:
Mahomed, F.M. & Vawda, F. Nonlinear Dynamics (2000) 21: 307. doi:10.1023/A:1008317327402


We determine the Noether point symmetries associated with theusual Lagrangian of the differential equation of the orbit of the twobody problem. This gives rise to three definite natural forms (apartfrom the linear form) when the Lagrangian admits three Noether pointsymmetries which enables the solution of the orbit equation in terms ofelementary functions. The other forms for which the orbit equation hastwo point symmetries that arise in the Lie classification do not occurfor the usual Lagrangian. For central force problems we obtain inaddition to the six general central forces found by Broucke [8], newforce laws which lead to integrability in terms of known functions. Thisis achieved by the two Noether cases and further cases by use ofequivalence transformations of the orbit differential equation.Moreover, we give an extension of what is sometimes referred to asNewton's theorem of revolving orbits. We also make use of the Lieclassification to provide two new cases of integrable (in terms of knownfunctions) orbit differential equations and new force laws.

central forceorbitNewtonLieNoethersymmetryfirst integrals

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • F. M. Mahomed
    • 1
  • F. Vawda
    • 2
  1. 1.Centre for Differential Equations, Continuum Mechanics and Applications, Department of Computational & Applied MathematicsUniversity of the WitwatersrandSouth Africa
  2. 2.Department of Computational & Applied MathematicsUniversity of the WitwatersrandSouth Africa