The monotonicity of the period function for planar Hamiltonian vector fields (Q1090829)

Classical Hamiltonian systems on the plane with Hamiltonians of the form \(H(x.y)=(1/2)y^ 2+V(x)\) are considered. Here V(x) is a smooth potential function with a nondegenerate relative minimum at the origin. The phase portrait of the Hamiltonian vector field \(X=y(\partial /\partial x)-V^ 1(\partial /\partial y)\) has a center at the origin surrounded by periodic orbits each of which is an energy curve of H with energy \(E\in (0,E_*)\). Periodic function T which assigns to each period orbit its minimum period is studied. Let \(K(E)=\{x\in R:V(x)\leq E\}\) and define \(N(x)=(V'(x))^ 4(V(x)/V'(x)^ 2)''.\) The following theorem is proved. If N(x)\(\geq 0\) for all \(x\in K(E)\), then T is monotone increasing on (0,E). If N(x)\(\leq 0\) for all \(x\in K(E)\) then T is monotone decreasing on (0,E). Several examples are shown.