接触卡农映射及其守恒量与耗散量
Contact canonoid maps and their conserved and dissipated quantities
- Czech Technical University in Prague（布拉格捷克理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究接触卡农映射的守恒与耗散量，通过多项式构造和归一化方法，在奇异映射中建立守恒不变量，并给出维度相关的上界及实例。
AI中文摘要：
我们研究了与接触卡农变换相关的守恒量和耗散量，并将其推广到可能奇异的光滑映射。定义条件施加在拉回接触一次形式上，而不要求其保持接触性质。一种多项式构造从具有与原始接触形式相同耗散率的一次形式产生首次积分。两种归一化将该构造推广到其相关哈密顿量具有不同耗散率的映射。在相关哈密顿量非零的集合上，我们构造了一个光滑的不变自同态，其迹是守恒的。该构造包含了原始哈密顿量的零点，此时自同态消失。我们将张量迹与哈密顿量归一化多项式的系数联系起来，并在2n+1维中获得n个函数无关的迹不变量的上界。一个五维奇异映射达到了该上界。例子包括阻尼自由粒子的奇异映射、具有不等耗散率的变换、阻尼谐振子以及在线性摩擦下的重力运动。
英文摘要：
We study conserved and dissipated quantities associated with contact canonoid transformations and their extension to possibly singular smooth maps. The defining condition is imposed on the pulled-back contact one-form, without requiring it to remain contact. A polynomial construction produces first integrals from one-forms having the same dissipation rate as the original contact form. Two normalizations extend this construction to maps whose associated Hamiltonians have different dissipation rates. On the set where the associated Hamiltonian is nonzero, we construct a smooth invariant endomorphism whose traces are conserved. This construction includes zeros of the original Hamiltonian, where the endomorphism vanishes. We relate the tensor traces to the coefficients of the Hamiltonian-normalized polynomial and obtain an upper bound of n functionally independent trace invariants in dimension 2n + 1. A five-dimensional singular map attains this bound. Examples include singular maps for the damped free particle, a transformation with unequal dissipation rates, the damped harmonic oscillator, and motion under gravity with linear friction.
