Normal forms for the Laplace resonance

Abstract We describe a comprehensive model for systems locked in the Laplace resonance. The framework is based on simplest possible dynamical structure provided by Keplerian problem perturbed resonant coupling truncated at second order eccentricities. reduced Hamiltonian, constructed transformation to coordinates, then submitted suitable ordering of terms and study its equilibria. Henceforth, normal forms are computed. main result identification two different classes In first class, only one kind stable equilibrium present: paradigmatic case that Galilean system. three kinds equilibria least them characterised high value forced eccentricity ‘first planet’: here, exo-planetary system GJ-876, which combination libration centres admits triple conjunctions otherwise not case. form obtained averaging with respect free oscillations describes argument arbitrary amplitudes allows us determine width agreement analytic predictions numerical integration toy models very good.