Instability criterion for multidimensional nonlinear Hamiltonian systems

A differential-geometrical approach is proposed for the investigation of instability in multidimensional nonlinear conservative systems. The critical value $E_c$ of the total energy for onset of instability of the motion in the two-dimensional case is calculated as the smallest value of the potential $U(x,y)$ on the line of zero curvature $K(x,y)=0$ of the potential-energy surface: $E_c=\min U(x,y\mid K=0)$. The criterion is generalized to the multidimensional case and illustrated by definite examples of the Hènon–Heiles systems and the reduced three-dimensional Yang–Mills problem.