For various values of n, d, and the phase space dimension, we construct simple examples of Hamiltonian and reversible systems possessing smooth d-parameter families of invariant n-tori carrying conditionally periodic motions. In the Hamiltonian case, these tori can be isotropic, coisotropic, or atropic (neither isotropic nor coisotropic). The cases of non-compact and compact phase spaces are considered. In particular, for any N no less than 3 and any vector omega in R^N, we present an example of an analytic Hamiltonian system with N degrees of freedom and with an isolated (and even unique) invariant N-torus carrying conditionally periodic motions with frequency vector omega (but this torus is atropic rather than Lagrangian and the symplectic form is not exact). Examples of isolated atropic invariant tori carrying conditionally periodic motions are given in the paper for the first time. The paper can also be used as an introduction to the problem of the isolatedness of invariant tori in Hamiltonian and reversible systems.

中文翻译：

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具有光滑不变环族的哈密顿和可逆系统

对于 n、d 和相空间维数的各种值，我们构造了哈密顿和可逆系统的简单示例，这些系统具有带有条件周期性运动的不变 n-tori 的平滑 d 参数族。在哈密顿量的情况下，这些环面可以是各向同性、各向同性或 atropic（既不是各向同性也不是各向同性的）。考虑了非紧致和紧致相空间的情况。特别地，对于任何不小于 3 的 N 和 R^N 中的任何向量 omega，我们给出了一个具有 N 个自由度和一个独立的（甚至是唯一的）不变 N 环面的解析哈密顿系统的例子，它携带有条件的周期性运动用频率向量 omega （但这个圆环是 atropic 而不是 Lagrangian 并且辛形式不精确）。论文中首次给出了携带条件周期运动的孤立的退变不变环面的例子。该论文还可以作为对哈密顿系统和可逆系统中不变环面的孤立性问题的介绍。