具有对合函数族的泊松括号
Poisson brackets with prescribed family of functions in involution
摘要：
众所周知，关于泊松括号的对合函数在完全可积系统理论中具有特殊的作用。在Poisson流形上寻找与给定函数$h$对合的函数无关的函数是该理论的一个基本问题，并且对于由$h$定义的运动方程的显式积分非常有用.在本文中，我们提出了我们的研究结果的逆，可以这么说，问题。通过发展一种类似于P. Damianou和F. Petalidou，具有指定Casimirs的泊松括号，Canad. J. Math.，2012，第64卷，991-1018，为了建立具有规定的Casimir不变量的泊松括号，我们构造了一种算法，该算法产生具有给定的对合函数族的泊松括号。我们的方法使我们能够建设性地处理双汉密尔顿结构，因此也使我们能够处理在这样的框架中出现的完全可积系统。
英文摘要：
It is well known that functions in involution with respect to Poisson brackets have a privileged role in the theory of completely integrable systems. Finding functionally independent functions in involution with a given function $h$ on a Poisson manifold is a fundamental problem of this theory and is very useful for the explicit integration of the equations of motion defined by $h$. In this paper, we present our results on the study of the inverse, so to speak, problem. By developing a technique analogous to that presented in P. Damianou and F. Petalidou, Poisson brackets with prescribed Casimirs, Canad. J. Math., 2012, vol. 64, 991-1018, for the establishment of Poisson brackets with prescribed Casimir invariants, we construct an algorithm which yields Poisson brackets having a given family of functions in involution. Our approach allows us to deal with bi-Hamiltonian structures constructively and therefore allows us to also deal with the completely integrable systems that arise in such a framework.
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