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Monotonicity of the ratio of two Abelian integrals for a class of symmetric hyperelliptic Hamiltonian systems
Keywords:Hamiltonian system, Hyperelliptic Abelian integral, Monotonicity, Asymptotic expansion.
Abstract:
      In this paper we study the monotonicity of the ratio of two hyperelliptic Abelian integrals $I_0(h)=\oint_{\Gamma_h}ydx$ and $I_1(h)=\oint_{\Gamma_h}xydx$ for which $\Gamma_h$ is a continuous family of periodic orbits of a Newtonian system with Hamiltonian function of the form $H(x,y)=\frac{1}{2}{y^2}\pm \Psi(x)$, where $\Psi$ is an arbitrary even function of degree six.