We consider the stability of the equilibrium position of a periodic Hamiltonian system with one degree of freedom. It is supposed that the series expansion of the Hamiltonian function, in a small neighborhood of the equilibrium position, does not include terms of second and third degree. Moreover, we focus on a degenerate case, when fourth-degree terms in the Hamiltonian function are not enough to obtain rigorous conclusions on stability or instability. A complete study of the equilibrium stability in the above degenerate case is performed, giving sufficient conditions for instability and stability in the sense of Lyapunov. The above conditions are expressed in the form of inequalities with respect to the coefficients of the Hamiltonian function, normalized up to sixth-degree terms inclusive.

中文翻译：

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哈密​​顿量的二次方和三次方零的情况下的一自由度典范系统的稳定性

我们考虑具有一个自由度的周期哈密顿系统的平衡位置的稳定性。假设在平衡位置的一小部分附近，哈密顿函数的级数展开不包括二阶和三阶项。此外，当哈密顿函数中的四阶项不足以获得关于稳定性或不稳定性的严格结论时，我们关注退化的情况。对上述退化情况下的平衡稳定性进行了完整的研究，从而为Lyapunov上的不稳定性和稳定性提供了充分的条件。上述条件以关于汉密尔顿函数系数的不等式形式表示，归一化至包括六阶项在内。