A class of Newtonian forces, determining the acceleration F ( x , y ) of particles in the plane, is F =(Re F ( z ), Im F ( z )), where z is the complex variable x + i y . Curl F is non-zero, so these forces are nonconservative. These complex curl forces correspond to completely integrable Hamiltonians that are anisotropic in the momenta, separable in z and z * but not in x and y if the curl is nonzero. The Hamiltonians can be quantised, leading to unfamiliar wavefunctions, even for the (non-curl) isotropic harmonic oscillator. The formalism provides an alternative interpretation of the analytic continuation of one-dimensional real Hamiltonian particle dynamics, where trajectories are known to exhibit intricate structure (though not chaos), and is a Hermitian alternative to non-Hermitian quantisation.

中文翻译：

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经典和量子复杂的哈密顿卷曲力

确定平面中粒子的加速度F（x，y）的一类牛顿力是F =（Re F（z），Im F（z）），其中z是复变量x + iy。卷毛F不为零，因此这些力是非保守的。这些复杂的卷曲力对应于在矩量上各向异性的完全可积分的哈密顿量，如果卷曲为非零，则可在z和z *中分离，但在x和y中不可分离。可以对哈密顿量进行量化，即使对于（非卷曲）各向同性谐波振荡器，也会导致不熟悉的波函数。形式主义为一维真实哈密顿粒子动力学的解析连续性提供了另一种解释，其中已知轨道显示出复杂的结构（尽管不是混沌），并且是非赫米特量化的厄米特替代方法。