Apparent topologically forbidden interchange of energy surfaces under slow variation of a Hamiltonian.
Apparent topologically forbidden interchange of energy surfaces under slow variation of a Hamiltonian.
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哈密顿量缓慢变化下能量面的明显拓扑禁止交换。
DOI：
10.1103/physreve.91.052913
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
E. Ott
中科院分区：
文献类型：
--
作者：
Zhixin Lu;C. Jarzynski;E. Ott
在本文中，我们考虑点粒子在一个特定类型的单自由度，缓慢变化，时间周期哈密顿运动。在大部分时间周期中，粒子保持其作用，但当接近并穿过分界线时，作用的守恒就被打破了，正如以前的理论研究所示。这些交叉的影响是数值解显示出明显的矛盾。具体地，我们考虑两个初始恒定能量相空间曲线H=E（A）和H=E（B）在时间t=0，其中H是哈密顿量，E（A）和E（B）是两个初始能量。曲线H=E（A）环绕曲线H=E（B）。然后，我们在这些曲线上散布许多初始条件（粒子），并在数值上从t=0向前跟踪它们的轨道一个周期。在循环结束时，最初在曲线H=E（A）和H=E（B）上的绝大多数点现在看起来位于两条新的恒定能量曲线H=E（A）'和H=E（B）'上，其中B'曲线现在环绕A'曲线（与A曲线环绕B曲线的初始情况相反）。由于汉密尔顿动力学的唯一性，在动力学下演化的曲线不能相互交叉。因此，视曲线H=E（A）'和H=E（B）'必须只是尊重曲线交叉的拓扑排除的真实情况的近似表示。在本文中，我们解决这个明显的悖论，并研究其后果。为此，我们引入了一个“强大的”数值模拟技术研究复杂的时间演化的哈密顿系统的相空间曲线。我们还考虑了非常微小的摩擦如何产生重大后果，以及当遵循非常大量的循环时会发生什么。我们还讨论了这种现象如何可能扩展到混沌运动在高维哈密顿系统。
In this paper we consider the motion of point particles in a particular type of one-degree-of-freedom, slowly changing, temporally periodic Hamiltonian. Through most of the time cycle, the particles conserve their action, but when a separatrix is approached and crossed, the conservation of action breaks down, as shown in previous theoretical studies. These crossings have the effect that the numerical solution shows an apparent contradiction. Specifically we consider two initial constant energy phase space curves H=E(A) and H=E(B) at time t=0, where H is the Hamiltonian and E(A) and E(B) are the two initial energies. The curve H=E(A) encircles the curve H=E(B). We then sprinkle many initial conditions (particles) on these curves and numerically follow their orbits from t=0 forward in time by one cycle period. At the end of the cycle the vast majority of points initially on the curves H=E(A) and H=E(B) now appear to lie on two new constant energy curves H=E(A)' and H=E(B)', where the B' curve now encircles the A' curve (as opposed to the initial case where the A curve encircles the B curve). Due to the uniqueness of Hamilton dynamics, curves evolved under the dynamics cannot cross each other. Thus the apparent curves H=E(A)' and H=E(B)' must be only approximate representations of the true situation that respects the topological exclusion of curve crossing. In this paper we resolve this apparent paradox and study its consequences. For this purpose we introduce a "robust" numerical simulation technique for studying the complex time evolution of a phase space curve in a Hamiltonian system. We also consider how a very tiny amount of friction can have a major consequence, as well as what happens when a very large number of cycles is followed. We also discuss how this phenomenon might extend to chaotic motion in higher dimensional Hamiltonian systems.