On the Influence of the Kinetic Energy¶on the Stability of Equilibria¶of Natural Lagrangian Systems

Abstract:

Natural Lagrangian systems (T,Π) on R ² described by the equation $\frac{d}{{dt}}\frac{{\partial T}}{{\partial \dot q}} - \frac{{\partial T}}{{\partial q}} = - \frac{{\partial \Pi }}{{\partial q}}$ are considered, where T(q, q̇) is a positive definite quadratic form in q̇ and Π(q) has a critical point at 0. It is constructively proved that there exist a C ^∞ potential energy Π and two C ^∞ kinetic energies T and T̃ such that the equilibrium q(t)≡ 0 is stable for the system (T,Π) and unstable for the system (T̃, Π). Equivalently, it is established that for C ^∞ natural systems the kinetic energy can influence the stability. In the analytic category this is not true.