Numerical verification of the steepness of three and four degrees of freedom Hamiltonian systems

We describe a new algorithm for the numerical verification of steepness, a necessary property for the application of Nekhoroshev’s theorem, of functions of three and four variables. Specifically, by analyzing the Taylor expansion of order four, the algorithm analyzes the steepness of functions whose Taylor expansion of order three is not steep. In this way, we provide numerical evidence of steepness of the Birkhoff normal form around the Lagrangian equilibrium points L4–L5 of the spatial restricted three-body problem (for the only value of the reduced mass for which the Nekhoroshev stability was still unknown), and of the four-degrees- of-freedom Hamiltonian system obtained from the Fermi – Pasta – Ulam problem by setting the number of particles equal to four.