Stability of Elliptic Periodic Points with an Application to Lagrangian Equilibrium Solutions

Abstract

We study the stability of elliptic rest points and periodic points of Hamiltonian systems of two degrees of freedom. We try to understand to what extent the linear stability would imply nonlinear stability. In case of one and one half degrees of freedom, linear stability, most of the times, do imply nonlinear stability provided that certain conditions on the eigenvalues are satisfied. This is a result of Herman. However, this is no longer the case for two degrees of freedom. We will analyse this case and it turns out that we can still say a lot about the nonlinear stability even in this case. As an example, we consider the Lagrange solutions ( $$L_4$$ and $$L_5$$ ) of circular restricted three body problem. For certain mass ratios of the primaries, the Lagrangian solution is elliptic and the high order term in the normal form is non-degenerate, and therefore the Lagrangian point is stable from the standard KAM theory. However, there is one particular value, famously named $$\mu _0$$ , where the fourth order term in the normal form happens to be degenerate. We will apply our result to this case.