, Volume 47, Issue 4, pp 857–862 | Cite as

Closed form integration of a hyperelliptic, odd powers, undamped oscillator

  • Giovanni Mingari Scarpello
  • Daniele RitelliEmail author


A known one-dimensional, undamped, anharmonic, unforced oscillator whose restoring force is a displacement’s odd polynomial function, is exactly solved via the Gauss and Appell hypergeometric functions, revealing a new fully integrable nonlinear system. Our t=t(x) equation—and its correspondent x=x(t) obtained via the Lagrange reversion approach—can then added to the (not rich) collection of highly nonlinear oscillating systems integrable in closed form. Finally, the hypergeometric formula linking the period T to the initial motion amplitude a is then assumed as a benchmark for ranking the approximate values of the relevant literature.


Nonlinear oscillator Hypergeometric functions Duffing-type equation 


  1. 1.
    Agostinelli C, Pignedoli A (1978) Meccanica razionale, vol 1. Zanichelli, Bologna Google Scholar
  2. 2.
    Appell P (1880) Sur les series hypergeometriques de deux variables, et sur des equations differentielles lineaires aux derives partielles. CR Acad Sci Paris 90:296–298 Google Scholar
  3. 3.
    Atkinson CP (1962) On the superposition method for determining frequencies of nonlinear systems. In: ASME proceedings of the 4th national congress of applied mechanics, pp 57–62 Google Scholar
  4. 4.
    Burton TD, Hamdan MH (1983) Analysis of nonlinear autonomous conservative oscillators by a time transformation method. J Sound Vib 87(4):543–554 MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Citterio M, Talamo R (2006) On a Korteweg-de Vries-like equation with higher degree of non-linearity. Int J Non-Linear Mech 41(10):1235–1241 MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Citterio M, Talamo R (2009) The elliptic core of nonlinear oscillators. Meccanica 44:653–660 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gauss KF (1812) Disquisitiones generales circa seriem infinitam. Commun Soc Regia Sci Göttingen Rec, 2 Google Scholar
  8. 8.
    Lagrange JL (1770) Nouvelle méthode pour résoudre les équations littérales par le moyen des séries. Mém Acad R Sci Belles-Lett Berlin 24: 251–326 Google Scholar
  9. 9.
    Ke L-L, Yang J, Kitipornchai S (2010) An analytical study on the nonlinear vibration of functionally graded beams. Meccanica 45:743–752 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mingari Scarpello G, Ritelli D (2004) Higher order approximation of the period-energy function for single degree of freedom hamiltonian systems. Meccanica 39:357–364 MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Mingari Scarpello G, Ritelli D (2009) The hyperelliptic integrals and π. J Number Theory 129:3094–3108 MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Pirbodaghi T, Hoseini SH, Ahmadian MT, Farrahi GH (2009) Duffing equations with cubic and quintic nonlinearities. Comput Math Appl 57:500–506 MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Recktenwald G, Rand R (2007) Trigonometric simplification of a class of conservative nonlinear oscillators. Nonlinear Dyn 49:193–201 MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Roy L (1945) Cours de Mècanique Rationelle, vol I. Gauthier-Villars, Paris Google Scholar
  15. 15.
    Sinha SC, Srinivasan P (1971) Application of ultraspherical polynomials to non-linear autonomous systems. J Sound Vib 18:55–60 ADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Slater LJ (1966) Generalized hypergeometric functions. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  17. 17.
    Yildirim A (2010) Determination of periodic solutions for nonlinear oscillators with fractional powers by He’s modified Lindstedt-Poincaré method. Meccanica 45:1–6 MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Whittaker ET, Watson GN (1962) A course of modern analysis, 4th edn. Cambridge University Press, Cambridge zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.MilanoItaly
  2. 2.Dipartimento di Matematica, per le scienze economiche e socialiBolognaItaly

Personalised recommendations