Monotonicity criteria for an energy-period function in planar Hamiltonian systems (Q2758118)

The authors consider the Hamiltonian system NEWLINE\[NEWLINEJ\dot x=H_x(x),\;J=\left[ \begin{matrix} 0 & -1\\ 1 & 0\end{matrix} \right], \tag{1}NEWLINE\]NEWLINE where \(x=(x_1,x_2)\). The Hamiltonian \(H(x)\) is supposed to be twice differentiable for \(x\neq 0\). They study monotonicity of the energy-period function \(T(h)\). Here a new approach (based on an analysis of the associated variational equation) is developed. Moreover, monotonicity criteria for the eigenvalue-norm functions in two-point boundary value problems are found.