The Gaussian curvature of associated manifold of dynamical systems
It is well known that the trajectories of a particle moving in the x-y- plane under the influence of a potential V(x,y) are geodesics in the region \({\mathcal D}:=\{(x,y)\in {\mathbb{R}}^ 2:\) \(V(x,y)<E\}\) with respect to the Jacobi-metric \((ds)^ 2=\lambda_ E(x,y)[(dx)^ 2+(dy)^ 2]\), where \(\lambda_ E(x,y):=E-V(x,y)\). The associated Gauss curvature is \(K=\lambda_ E^{-3}[(\nabla V)^ 2+(E-V)\nabla^ 2V]\). In particular \(K>0\) if \(\nabla^ 2V>0\), which is the case, e.g. in the Henon-Heiles example. The authors of the paper under review, quoting a famous theorem of E. Hopf, want to find a connection between K and ''ergodic'' or ''non-ergodic'' behavior of the trajectories. In particular, they claim that the system is ergodic in subregions of \({\mathcal D}\) in which \(K<0\). (A counterexample to this claim is \(V(x,y)=-(x^ 2+y^ 2)\) in which \(K=-\lambda_ E^{- 3}\cdot E<0\) for \(E>0\), although the flow is simply \(\Phi_ t(x,y)=(e^ tx\cdot e^ ty)\), i.e. ''non-ergodic'' in every sense of the word.) Since Hopf's theorem pertains to a compact Riemannian surface of negative curvature and \({\mathcal D}\) is noncompact Hopf's theorem cannot be invoked in the situation at hand.
ergodic'' or ''non-ergodic'' behavior of the trajectories
scientific article; zbMATH DE number 3913312