Apparent topologically forbidden interchange of energy surfaces under slow variation of a Hamiltonian

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 $H=E_A^{\prime }$ and $H=E_B^{\prime }$, where the $B^{\prime }$ curve now encircles the $A^{\prime }$ 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 $H=E_A^{\prime }$ and $H=E_B^{\prime }$ 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.

Figure 1

Apparent (not real) interchange of energy surfaces.

Figure 2

The parameter vector $\vec{\lambda }=({\lambda }_R,{\lambda }_L)$ starts from $(0,0)$ when $t=0$ and reaches $(1,0)$ when $t=\tau ,(1,1)$ when $t=2\tau ,(0,1)$ when $t=3\tau$, and returns back to $(0,0)$ when $t=4\tau$. Each of the four phases of the full cycle has duration $\tau$, and $\vec{\lambda }$ changes linearly with time during each of the four phases [Eq. (3)].

Figure 3

Snapshots at $t=0,t_0^B,\tau ,t_x^{A-},t_x^{A+},2\tau ,t_x^{B-},t_x^{B+},3\tau ,t_0^A$, and $4\tau$. The green (light gray) and red (dark gray) horizontal lines represent set $A$ and set $B$. Initially, set $A$ is above set $B$, where the height represent the energy of the ensemble. At $t=4\tau$, set $B$ is above set $A$.

Figure 4

Assumed evolution of curve $A$ and curve $B$ when $t=0,t_0^B,\tau ,t_x^{A-},t_x^{A+},2\tau ,t_x^{B-},t_x^{B+},3\tau ,t_0^A$, and $4\tau$.

Figure 5

Discontinuity of the adiabatic action invariant of $A$ particles at time $t_x^A$ in the hypothetical evolution.

Figure 6

Final result at $t=4\tau$ from regular simulation with different slowness parameters $\tau$.

Figure 7

Six figures are snapshots in phase space showing the evolution of sample points along a positive energy curve at $t=0$ to the resulting curve at $t=\tau =20$, from both the regular and the robust simulations. The regular simulation evolves only those 2000 initial phase points. Its result is shown by the thick red (light gray) “curve,” which is made up of 2000 red (light gray) points. The thin blue (dark gray) curve is the result from the robust simulation starting initially with 2000 points.

Figure 8

Snapshots for $0<t<\tau$ with a larger $\tau =200$. Red (light gray) points are results from regular simulation, and the blue (dark gray) thin curve is of robust simulation with $\epsilon =0.0531$. The inset in panel (f) shows a magnification of the small rectangular near the origin, verifying that the origin $(q,p)=(0,0)$ is still enclosed by the true time-evolved curve at time $t=\tau$.

Figure 9

Energy curve dynamics from a robust simulation illustrate the real dynamics of the separatrix “crossing” in the second phase. The insert shows a magnification about the origin for (b). The “X” and “I” symbols label regions that are exterior and interior to the closed curve; these designations are also used for Figs. 10–11.

Figure 10

Energy curve dynamics from robust simulation illustrates the real dynamics of the separatrix “crossing” in the third phase. A schematic illustration of a magnification around the origin is shown as the inset in panel (b).

Figure 11

Energy curve dynamics from robust simulation illustrates the real dynamics of the separatrix “crossing” in the fourth phase, starting with a Hamiltonian level curve of action $J_A\left(0\right)-J_x$ at time $t=3\tau$. The inset shows a schematic illustration of a magnification around the origin of the robust curve calculation result at time $t=4\tau$.

Figure 12

Snapshots for the full protocol with $\tau =20$. Panel (f) is a zoom magnification of a small region of panel (e).

Figure 13

Result of two energy curves after going through the whole protocol with (b) and without (a) friction, $4\tau =4000$ and $N=2000$.

Figure 14

Simulated histograms of action $J$ after evolving the system different numbers of cycles.

Figure 15

Simulated histogram in phase space after 4000 repeating cycles. The curve $J=2J_x$ is shown in blue (dark gray).

Figure 16

The potential at $t=2\tau$.

Figure 17

Simulated histograms of action $J$ after evolving the system different numbers of cycles with unequal left and right well depths are shown. We use irrational ratio between well depths $({\lambda }_L,{\lambda }_R)=(1,(\sqrt{5}-1)/2)$. $J_0$ is the initial action value and $J_{tot}=J_L+J_R$.

Figure 18

Billiard system with unchanged potential and changing potential.

Figure 19

Potential function for billiard system when $t=0,\tau ,2\tau ,3\tau$, and $4\tau$.