Hamilton系统的解析不变量
Configurational invariants of Hamiltonian systems
摘要：
在本文中，我们探讨的一般条件，以使一个2维的自然哈密顿系统具有第二个不变量，这是一个多项式的动量，因此是刘维尔可积。我们研究的可能性，不变量是由一个给定的能量超曲面上的哈密顿流保存（弱可积性），并推导出额外的必要条件，以保持在任意能量（强可积性）。利用零复坐标，我们证明了在弱可积情形下，多项式的首阶系数是任意全纯函数，而在强可积情形下，首阶系数在坐标系中是多项式.我们回顾了迄今为止得到的结果与强不变量的程度四，并提供了一些新的例子弱可积系统与线性和二次不变量。
英文摘要：
In this paper we explore the general conditions in order that a 2-dimensional natural Hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore Liouville integrable. We examine the possibility that the invariant is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have conservation at arbitrary energy (strong integrability). Using null complex coordinates, we show that the leading order coefficient of the polynomial is an arbitrary holomorphic function in the case of weak integrability and a polynomial in the coordinates in the strongly integrable one. We review the results obtained so far with strong invariants up to degree four and provide some new examples of weakly integrable systems with linear and quadratic invariants.
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