标题： 哈密顿量生成速度无关力的一般表征 显示英文标题

标题： General characterisation of Hamiltonians generating velocity-independent forces

摘要： 从哈密顿量生成的动力学为量化提供了潜在途径，但标准哈密顿量只能生成保守力。 在Berry等人的研究中提出了能够生成非保守速度无关力的哈密顿量类别。 这类哈密顿量在过去已被分类，前提是它们在动量上是多项式的。 在这里，这一假设被放宽为解析性。 这样做发现了更广泛的哈密顿量类别。 通过将哈密顿量视为状态空间中的函数而不引入拉格朗日量并构造一个类似度量的张量，我们对生成速度无关力的哈密顿量发展出强有力的普遍约束，并展示了此类哈密顿量之间的意外二分法。 这些结果适用于任何维度的任何空间域，只要其具有定义良好的哈密顿动力学。 作为示例应用，我们将这些约束应用于对二维空间中所有哈密顿速度无关力以及不产生各向同性简谐运动的所有此类哈密顿量进行分类。 为了完整性，也讨论了一维空间的情况。 显示英文摘要

摘要： Dynamics generated from Hamiltonians enjoy potential pathways to quantisation, but standard Hamiltonians are only capable of generating conservative forces. Classes of Hamiltonians have been proposed in Berry et al. capable of generating non-conservative velocity-independent forces. Such Hamiltonians have been classified in the past, under the strict assumption that they are polynomial in momentum. This assumption is relaxed here to analyticity. In doing so, broader classes of Hamiltonians are discovered. By considering the Hamiltonian as a function of state space without introducing the Lagrangian and constructing a metric-like tensor, we develop strong general constraints on Hamiltonians generating velocity-independent forces and exhibit a surprising dichotomy between classes of such Hamiltonians. These results are applicable to any spatial domain of any dimension admitting well-defined Hamiltonian dynamics. As an example application, we apply these constraints to classify all Hamiltonian velocity-independent forces in two spatial dimensions, as well as all such Hamiltonians which do not generate an isotropic simple harmonic motion. The case of one spatial dimension is also discussed for the sake of completeness.