Conditions on nonlinearity of oscillatory equations inducing the periapsidal precession

  • Emilija CelakoskaEmail author
  • Ana M. Lazarevska
Research Article


The equations of motion in the theory of general relativity obtained for the Schwarzschild metric yield an oscillatory differential equation with weak quadratic nonlinearity. This nonlinearity induces the well-known parametric expression for the relocation of the orbital periapsis, usually termed as periapsidal precession. It still represents a hard test for a gravitational theory viability. In the standard process of obtaining the precession, some approximation methods are employed, however it seems that the methods can provide better information then it is usually presented in the literature. We give an in-depth analysis of the oscillatory nonlinear differential equation as a dynamical system, also analyzing the conditions for obtaining the precession given other nonlinearities. Then, we outline a procedure for obtaining this precession for more general types of nonlinearities and the conditions which apply on them.


Perturbation methods Linearization Perihelion shift Lambert W function 

Mathematics Subject Classification

83C10 70F15 37C10 37N05 



  1. 1.
    Janssen, M., Norton, J.D., Renn, J., Sauer, T., Stachel, J.: Introduction to Volumes 1 and 2: The Zurich notebook and the genesis of general relativity. In: Janssen, M., Norton, J.D., Renn, J., Sauer, T., Stachel, J. (eds.) The genesis of general relativity, Boston studies in the philosophy of science, vol. 250. Springer, Dordrecht (2017)Google Scholar
  2. 2.
    Tolman, R.C.: Relativity, thermodynamics and cosmology. Clarendon Press, Oxford (1934)zbMATHGoogle Scholar
  3. 3.
    Schiff, L.I.: On experimental tests of the general theory of relativity. Am. J. Phys. 28(4), 340–343 (1960)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Polyanin, D.A., Zaitsev, V.F.: Handbook of exact solutions for ordinary differential equations. Chapman & Hall/CRC, New York (2002)zbMATHGoogle Scholar
  5. 5.
    Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie, Sitzungberichte der Königlich Preussichen Akademie der Wissenschaften, vol XLVII, no. 18 Nov., 1915, pp. 831–839Google Scholar
  6. 6.
    Moller, C.: The theory of relativity. Clarendon Press, Oxford (1955)zbMATHGoogle Scholar
  7. 7.
    D’Inverno, R.: Introducing Einstein relativity. Clarendon Press, Oxford (1998)zbMATHGoogle Scholar
  8. 8.
    Dean, B.: Phase-plane analysis of perihelion precession and Schwarzschild orbital dynamics. Am. J. Phys. 67(1), 78–86 (1999)ADSCrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Department of Mathematics and Informatics, Faculty of Mechanical EngineeringSS Cyril and Methodius University SkopjeSkopjeRepublic of Macedonia
  2. 2.Institute of Automatics and Fluid Engineering, Faculty of Mechanical EngineeringSS Cyril and Methodius University SkopjeSkopjeRepublic of Macedonia

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