Advertisement

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)

Abstract

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).

Keywords

Periodic Solution Closed Orbit Periodic Trajectory Multiplicity Result Newtonian Potential 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  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

Personalised recommendations