This Letter deals with the stability of nonlinear Hamiltonian dynamics. The Jacobi-Levi-Civita equation for the geodesic spread is shown to be a powerful tool for the characterization of the so called Hamiltonian chaos. The special case of two degrees of freedom is analyzed and used to study the origin of the instability properties of the Ne...I-2 molecule. Results are compared with those of the conventional methodology, resulting in complete agreement. Advantages of the geometrical framework are shown. It is demonstrated how the instability of geodesics is only determined by the projections of the curvature tensor on the transverse directions of the geodesic tangent vector. The relevant role of the phenomenon of parametric resonance in the explanation of the origin of instability in Hamiltonian systems was confirmed. (C) 2003 Published by Elsevier Science B.V.