Monotonicity of the Period Function in Planar Separable Hamiltonian Systems: An Extension of Chicone’s Criterion

We investigate the monotonicity of the period function associated with planar Hamiltonian systems of the form $$H(x,y)=F(x)+G(y),$$ H ( x , y ) = F ( x ) + G ( y ) , where G G is an even function. We derive explicit sufficient conditions, expressed solely in terms of the functions F F , G G , and their derivatives, ensuring the monotonicity of the period function associated with a nondegenerate center. Our results extend Chicone’s classical criterion, originally established for potential systems, to a broader class of separable Hamiltonian systems. The effectiveness of the proposed criterion is illustrated through applications including polynomial, trigonometric, hyperbolic and relativistic Hamiltonian systems.