In this paper, we present algebraic tools to obtain normal forms of \(\omega \)-Hamiltonian vector fields under a semisymplectic action of a Lie group, by taking into account the symmetries and reversing symmetries of the vector field. The normal forms resulting from the process preserve the Hamiltonian condition and the types of symmetries of the original vector field. Our techniques combine the classical method of normal forms of Hamiltonian vector fields with the invariant theory of groups.

中文翻译：

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$$\omega $$ 对称的哈密尔顿向量场的范式

在本文中，我们提出了代数工具，通过考虑矢量场的对称性和反演对称性，在李群的半辛作用下获得\(\omega \) -哈密尔顿矢量场的范式。该过程产生的范式保留了哈密顿条件和原始矢量场的对称类型。我们的技术将哈密顿向量场范式的经典方法与群不变理论结合起来。