发表机构
Czech Technical University in Prague; University of La Laguna（布拉格捷克理工大学; 拉古纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在余切丛哈密顿力学框架下引入污垢映射概念，开发张量方法构造多项式污垢映射，刻画相关多项式丛映射并推导条件，通过欧氏平面和二维球面例子说明方法。
AI 中文摘要
在机械系统构形空间\(Q\)的余切丛\(T^*Q\)上与时间无关的哈密顿力学框架下，我们引入污垢映射的概念，它是所谓污垢变换（保持构形坐标的正则类变换）的非可逆推广。我们开发了一种构造多项式污垢映射的张量方法。我们表明，每个这样的映射都诱导出一个在哈密顿流下不变的\((1,1)\)张量场，其幂次的迹是多项式运动常数。对于机械哈密顿函数（半黎曼构形空间\((Q,g)\)上的动能加势能），我们完全刻画了由对称\((k + 1,0)\)张量场产生的多项式丛映射，并推导了确保其污垢性质所需的条件。欧几里得平面和二维球面上的几个具体例子说明了该方法。
英文摘要
Under the framework of time-independent Hamiltonian mechanics on the cotangent bundles $T^*Q$ of the configuration spaces $Q$ of mechanical systems, we introduce the notion of fouling map as a (not necessarily invertible) generalization of the so-called fouling transformations --canonoid transformations preserving configuration coordinates--. We develop a tensorial method for constructing polynomial fouling maps. We show that each such map induces a $(1,1)$-tensor field invariant under the Hamiltonian flow, whose traces of its powers are polynomial constants of motion. For mechanical Hamiltonian functions --the kinetic energy plus the potential energy on a semi-Riemannian configuration space $(Q,g)$--, we completely characterize polynomial bundle maps arising from symmetric $(k+1,0)$-tensor fields and derive the conditions ensuring their fouling nature. Several explicit examples on the Euclidean plane, on the 2-sphere and on a Riemannian manifold with a Liouville metric illustrate the method.
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