IN THIS WORK, WE CONSIDER A GENERIC 2-DOF AUTONOMOUS HAMILTONIAN SYSTEM, WHICH LINEAR PART HAS ALL EIGENVALUES EQUAL TO ZERO. FOR THESE TYPE OF HAMILTONIAN, WE PROVIDE, IN AN EXPLICIT AND CONSTRUCTIVE WAY, THE NORMAL FORM UP TO THE TERM OF THE FOURTH ORDER. ADDITIONALLY, WE ANALYZE THE STABILITY OF THE NORMALIZED HAMILTONIAN IN SEVERAL CASES. PRECISELY, WE OBTAIN INSTABILITY OF THE TRUNCATED HAMILTONIAN BY EXPLICITLY BUILDING AN INVARIANT RAY SOLUTION.

WE PRESENT A GEOMETRICAL DESCRIPTION OF THE SYMMETRIES AND REDUCTION OF THE FULL GRAVITATIONAL 2-BODY PROBLEM AFTER COMPLETE AVERAGING OVER FAST ANGLES. OUR VARIABLES ALLOW FOR A WELL-SUITED FORMULATION IN ACTION-ANGLE TYPE COORDINATES ASSOCIATED WITH THE AVERAGED ANGLES, WHICH PROVIDE GEOMETRIC INSIGHT INTO THE PROBLEM. AFTER INTRODUCING EXTRA FICTITIOUS VARIABLES AND THROUGH A SYMPLECTIC TRANSFORMATION, WE MOVE TO A SINGULARITY-FREE QUATERNIONIC TRIPLE-CHART. THIS CHOICE ALLOWS FOR A GLOBAL CHART TO AVOID THE CLASSICAL SINGULARITIES ASSOCIATED WITH ANGLES AND RENDERS ALL THE INVARIANTS AS HOMOGENEOUS QUADRATIC POLYNOMIALS. ADDITIONALLY, IT PERMITS ONE TO QUICKLY WRITE THE HAMILTONIAN OF THE SYSTEM IN TERMS OF THE INVARIANTS AND THE POISSON STRUCTURE AT EACH STAGE OF THE REDUCTION PROCESS. IN CONTRAST WITH EXISTING LITERATURE, THE GEOMETRICAL APPROACH OF THIS RESEARCH COMPLETELY DESCRIBES ALL THE DYNAMICAL ASPECTS OF THE FULL REDUCED SPACE SINCE IT INVOLVES THE RELATIVE POSITION OF THE ROTATIONAL AND ORBITAL ANGULAR MOMENTA AND THEIR ORIENTATION, WHICH HAS YET TO BE CONSIDERED IN PREVIOUS STUDIES. OUR PROGRAM INCLUDES A PRELIMINARY PARAMETRIC ANALYSIS OF RELATIVE EQUILIBRIA AND A COMPLETE DESCRIPTION OF THE FIBERS IN THE RECONSTRUCTION OF THE REDUCED SYSTEM.