The concept of extended Hamiltonian systems allows the geometrical interpretation of several integrable and superintegrable systems with polynomial first integrals of degree depending on a rational parameter. Until now, the procedure of extension has been applied only in the case of natural Hamiltonians. In this article, we give several examples of application to non-natural Hamiltonians, such as the two point-vortices, the Lotka-Volterra and some quartic in the momenta Hamiltonians, obtaining effectively extended Hamiltonians in some cases and failing in others. We briefly discuss the reasons of these results.

中文翻译：

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非自然哈密顿量的扩展

扩展哈密顿系统的概念允许对几个可积和超可积系统进行几何解释，这些系统具有取决于有理参数的多项式第一次积分。到目前为止，扩展程序仅适用于自然哈密顿量的情况。在本文中，我们给出了几个应用于非自然哈密顿量的例子，例如两个点涡、Lotka-Volterra 和动量哈密顿量中的一些四次，在某些情况下获得有效扩展的哈密顿量，而在其他情况下则失败。我们简要讨论这些结果的原因。