Monotonicity of the period function for planar Hamiltonian vector fields: A Generalization of Chicone's Criterion

This paper investigates the monotonicity of the period function associated with planar Hamiltonian systems of the form $H(x,y) = F(x) + G(y)$. We establish sufficient conditions ensuring the monotonicity of the period function corresponding to a nondegenerate center, expressed explicitly in terms of the functions F and G. Our approach extends Chicone's classical criterion, originally formulated for systems of the type $x' = y$, $y' = -F'(x)$, to a broader Hamiltonian framework. As a main application, we analyze the monotonicity of the period function associated with the center at (0,0) of the polynomial Hamiltonian system $$H(x,y) = (1/2)x^2 + (a/3)x^3 + (b/4)x^4 + (1/2)y^2 + (c/4)y^4,$$ as a function of the parameters $a, b, c \in \mathbb{R}$.