Multiple periodic trajectories in a relativistic gravitational field

  • A. Ambrosetti
  • U. Bessi
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)


Let us consider the Newtonian potential:
$$U(x) = {\text{ }}\frac{1}{{\left| x \right|}} + \gamma w(x)$$
where γ ∈ R and W: R n R is smooth. If we want to take into account the relativistic correction to the motion of a particle under the potential U with given energy h < 0, we are led to consider a motion at the same energy governed by the potential:
$${U_\kappa }(x) = {\text{ }}\left( {1 + \frac{4}{3}hk} \right)U(x) - \kappa {U^2}(x)$$
where \(k = \frac{3}{{{c^2}}}\) , c being the speed of light (cfr. chap 2, §8 of 6).


Periodic Solution Closed Orbit Periodic Trajectory Multiplicity Result Newtonian Potential 
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  1. [1]
    A. Ambrosetti, V. Coti Zelati, “Closed orbits of fixed energy for singular hamiltonian systems”, Archive Rat. Mech. and Anal., to appear.Google Scholar
  2. [2]
    A. Ambrosetti, V. Coti Zelati, “Closed orbits of fixed energy for a class of N-body problems”, preprint S.N.S., April 1990Google Scholar
  3. [3]
    A. Ambrosetti, G. Mancini, “On a theorem by Ekeland and Lasry concerning the number of periodic hamiltonian trajectories”, J.Diff. Eq. 43, No 2 (1982) 249 – 256.Google Scholar
  4. [4]
    V.Benci “A geometrical index for the group S1 and some applications to the study of periodic solutions of ordinary differential equations” Comm. Pure and Applied Math., 34 (1981) 393-423Google Scholar
  5. [5]
    I. Ekeland, J.M.Lasry, “On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface”, Ann. of Math. 112 (1980), 283 – 319.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    T. Levi Civita, “Fondamenti di meccanica relativistica”, Bologna, 1928.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • A. Ambrosetti
    • 1
  • U. Bessi
    • 1
  1. 1.Scuola Normale SuperiorePisaItaly

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