Abstract
In a central force system the angle between two successive passages of a body through pericenters is called the apsidal angle. In this paper we prove that for central forces of the form \(f(r)\sim \lambda r^{-\,(\alpha +1)}\) with \(\alpha <2\) the apsidal angle is a monotonous function of the energy, or equivalently of the orbital eccentricity.
Similar content being viewed by others
References
Bertrand, J.: Théoréme relatif au mouvement d’un point attiré vers un centre fixe. Comptes Rendus Acad. Sci. 77, 849–853 (1873)
Castelli, R.: The monotonicity of the apsidal angle in power-law potential systems. J. Math. Anal. Appl. 428, 653–676 (2015)
Danby, J.M.: Fundamentals of Celestial Mechanics, 2nd edn. Willmann-Bell Inc., Richmond (1988)
Grant, A.K., Rosner, J.L.: Classical orbits in power-law potentials. Am. J. Phys. 62(4), 310–315 (1994)
Mañosas, F., Rojas, D., Villadelprat, J.: Study of the period function of a two-parameter family of centers. J. Math. Anal. Appl. 452, 188–208 (2017)
Ortega, R., Rojas, D.: A short proof of Bertrand’s theorem using the theory of isochronous potentials (2017) (preprint)
Schaaf, R.: A class of Hamiltonian systems with increasing periods. J. Reine Angew. Math. 363, 96–109 (1985)
Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th edn. Cambridge University Press, Cambridge (1959)
Acknowledgements
The author want to thank Prof. Rafael Ortega for the fruitful discussions that led to the interpretation of the apsidal angle as the period function of an abstract oscillator. The author is partially supported by the MINECO Grant MTM2014-52209-C2-1-P and MEC/FEDER Grant MTM2014-52232-P.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declare that he have no conflict of interest.
Rights and permissions
About this article
Cite this article
Rojas, D. The Monotonicity of the Apsidal Angle Using the Theory of Potential Oscillators. Qual. Theory Dyn. Syst. 17, 631–635 (2018). https://doi.org/10.1007/s12346-017-0265-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12346-017-0265-9