Advertisement

Annali di Matematica Pura ed Applicata

, Volume 117, Issue 1, pp 339–347 | Cite as

Analytical theory of non-linear oscillations (+)

VIII. — Second order conservative systems whose solutions are all oscillating with the least period 2π
  • Chike Obi
Article

Summary

This paper establishes what are essentially necessary and sufficient conditions under which every solution of the differential equation\(\ddot x + g\left( x \right) = 0\left( {\dot x = dx/dt} \right)\) in the real domain oscillates with the least period 2π independent of the initial conditions of the solution and in addition gives expressions for the Fourier coefficients of a solution with given initial conditions.

Keywords

Differential Equation Analytical Theory Fourier Coefficient Real Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [1]
    I. Koukles -N. Piskounov,Sur les vibrations tautochrones dans les systèmes conservatifs et non conservatifs, C. R. Acad. Sci. URSS,17 (1937), pp. 471–475.Google Scholar
  2. [2]
    J. J. Levin -S. S. Schatz,Non-linear oscillations of fixed period, J. Math. Anal. Appl.,7 (1973), pp. 284–288.CrossRefGoogle Scholar
  3. [3]
    C. Obi,Analytic theory of non-linear oscilations. VII: The periods of the periodic solutions of the equation\(\ddot x + g\left( x \right) = 0\), J. Math. Anal. Appl.,55 (1976), pp. 295–301.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    C. Obi,Periodic solutions of non-linear differential equations of the second order IV, Proc. Cambridge Philos. Soc.,47 (1951), pp. 741–751.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    C. Obi,Analytical theory of non-linear oscillations. IX: Limit oscillations and unbounded solutions of the equation\(\ddot x + \lambda ^2 g\left( x \right) = \} \) cos ωt, Ann. Mat. Pura Appl. (to appear).Google Scholar
  6. [6]
    C. Obi, Equations of the form\(\ddot x + g\left( x \right) = 0\) with periodic solutions whose Fourier series are finite, Nigerian Journal of Science (to appear).Google Scholar
  7. [7]
    M. Urabe,Potential forces which yield periodic motions of fixed period, J. Math. Mech.,10 (1961), pp. 569–578.zbMATHMathSciNetGoogle Scholar
  8. [8]
    M. Urabe, Relations between periodics and amplitudes of periodic solutions of\(\ddot x + g\left( x \right) = 0\), Funkcialaj Ekvacioj,6 (1964), pp. 63–88.zbMATHMathSciNetGoogle Scholar
  9. [9]
    E. T. Whittaker - G. N. Watson,Modern Analysis, Cambridge (1946).Google Scholar

Copyright information

© Nicola Zanichelli Editore 1978

Authors and Affiliations

  • Chike Obi
    • 1
  1. 1.LagosNigeria

Personalised recommendations