Bender C M

Bender C M Darg D W

Carl M Bender

Daniel W Hook

Darryl D Holm

Dorey P

Guckenheimer J

Heck A

Holm D D

Kato K

Nanayakkara A

Strogatz S H

Journal of Physics A Mathematical and Theoretical

arXiv

Crossref

Complex trajectories of a simple pendulum

Bender, CM

Holm, DD

Hook, DW

Spiral - Imperial College Digital Repository

arXiv:math-ph/0609068v1  25 Sep 2006preprint LA-UR-06-6814Complex Trajectories of a Simple PendulumCarl M Bender∗‡, Darryl D Holm†, and Daniel W Hook‡∗Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM87545, USAemail: cmb@wustl.edu†Department of Mathematics, Imperial College, London SW7 2AZ, UKemail: d.holm@ic.ac.ukandComputer and Computational Science, Los Alamos National Laboratory,MS D413 Los Alamos, NM 87545, USAemail: dholm@lanl.gov‡Blackett Laboratory, Imperial College, London SW7 2BZ, UKemail: d.hook@imperial.ac.ukAbstract. The motion of a classical pendulum in a gravitational field of strengthg is explored. The complex trajectories as well as the real ones are determined. If gis taken to be imaginary, the Hamiltonian that describes the pendulum becomes PT -symmetric. The classical motion for this PT -symmetric Hamiltonian is examined indetail. The complex motion of this pendulum in the presence of an external periodicforcing term is also studied.PACS numbers: 05.45.-a, 45.20.Jj, 11.30.ErSubmitted to: J. Phys. A: Math. Gen.1. IntroductionMany papers have been written on PT -symmetric quantum-mechanical Hamiltonians.In nearly all cases these Hamiltonians have had rising potentials and energies that areall discrete and real. An example of such a class of Hamiltonians is [1, 2, 3]H = 12p2 + x2(ix)ǫ (ǫ ≥ 0). (1)In a few instances quantum-mechanical Hamiltonians having real continuous spectrahave been examined [4].For the case of PT -symmetric quantum-mechanical Hamiltonians having discretespectra, the associated classical-mechanical systems have also been studied [2, 5, 6, 7].It has been found that the classical trajectories associated with real energies are PT -symmetric (they are symmetric under reflection about the imaginary axis) and exhibit‡ Permanent address: Department of Physics, Washington University, St. Louis MO 63130, USAComplex Trajectories of a Simple Pendulum 2remarkably interesting topological properties. These trajectories can visit multiplesheets of a Riemann surface. However, the classical mechanics associated with quantumsystems having continuous spectra have not yet been examined. Thus, in this paper weexplore the classical trajectories of the periodic potential analyzed in Ref. [4].Consider a classical-mechanical system consisting of a simple pendulum with a bobof mass m and a string of length L in a uniform gravitational field of magnitude g (seeFig. 1). The gravitational potential energy of the system is defined to be zero at theheight of the pivot point of the string. The pendulum bob swings through an angle θ.Therefore, the horizontal and vertical cartesian coordinates X and Y are given byX = L sin θ, Y = −L cos θ, (2)which gives velocitiesẊ = Lθ̇ sin θ, Ẏ = −Lθ̇ cos θ. (3)We define the potential and kinetic energies V and T asV = −mgL cos θ, T = 12m(Ẋ2 + Ẏ 2) = 12mL2θ̇2. (4)The Hamiltonian H = T + V for the pendulum is thereforeH = 12mL2θ̇2 − mgL cos θ. (5)Without loss of generality we set m = 1 and L = 1, and for consistency with the notationused in earlier papers on PT -symmetric Hamiltonians we relabel θ → x to getH = 12p2 − g cos x, (6)where p = ẋ. The classical equations of motion for this Hamiltonian readẋ =∂H∂p= p, ṗ = −∂H∂x= −g sin x. (7)The Hamiltonian H for this system is a constant of the motion and thus the energy Eis a time-independent quantity. For most of this paper we take E to be real because theenergy levels of the corresponding quantum-mechanical system are real.This paper is organized as follows: In Sec. 2 we study the complex trajectories forH in (6) when E and g are real. In Sec. 3 we study the complex trajectories for H whenE is real but g is imaginary. In Sec. 4 we examine briefly the complex chaotic motionthat results when the energy E of the pendulum is allowed to be complex and also whenthe pendulum is subject to a periodic external driving force.2. Pendulum in a Real Gravitational FieldIn this section we examine the solutions to Hamilton’s equations (7) for the case inwhich the gravitational field strength g is real. Without loss of generality we take g = 1.There are three cases to consider: (1) the complex closed trajectories that result when|E| < 1, (2) the complex open trajectories that occur when E > 1, and (3) the complexclosed trajectories that we obtain when E < −1.Complex Trajectories of a Simple Pendulum 3Figure 1. Configuration of a simple pendulum of mass m in a uniform gravitationalfield of strength g. The length of the string is L. The pendulum swings through anangle θ. We define the potential energy to be 0 at the height of the pivot.2.1. Closed periodic trajectories: |E| < 1.When the energy E of a simple pendulum lies in the range −1 ≤ E ≤ 1, the physicallyobserved motion is that of a swinging pendulum. For the example considered in thissubsection we choose E = 0. Since a classical turning point x0 is the solution to theequation V (x0) = E, the classical turning points for this case satisfy cosx0 = 0. Thus,x0 = π/2 + nπ (n ∈ Z). (8)The observed classical trajectory for this pendulum is represented by x(t) on the real-xaxis. This trajectory oscillates on the real axis between the turning points at x0 = ±12π.However, the full array of complex solutions to the classical equations of motion ismuch richer (see Fig. 2). In this figure we see that all classical trajectories are closedand periodic and lie in cells of width 2π that fill the entire complex-x plane. As aconsequence of Cauchy’s theorem, all of the closed orbits have exactly the same period;namely, the period of a real physical pendulum that undergoes periodic motion on thereal axis. This is because the period T for a particle of energy E for a Hamiltonianof the form H = 12p2 + V (x) is given in terms of a closed complex contour integral:T =∮Cdx/√2[E − V (x)], where C is the complex closed path of the particle. Sincethe only singularity that C encloses is the square-root branch cut joining the turningpoints, the value of T is independent of C.Note that the orbits in Fig. 2 that lie near the turning points are approximatelyelliptical and resemble the exactly elliptic complex trajectories of the simple harmonicoscillator, whose Hamiltonian is H = 12p2 + 12x2. These orbits are displayed in Fig. 3.Complex Trajectories of a Simple Pendulum 4Figure 2. Complex classical trajectories for the simple pendulum in (6). Thesetrajectories satisfy the equations of motion in (7) with g = 1 and E = 0. Note thatall trajectories are closed periodic orbits and lie in cells of width 2π that entirely fillthe complex plane. The usual physically observed motion of a swinging pendulum isrepresented by the line segments joining pairs of turning points on the real axis.Figure 3. Complex trajectories of the simple harmonic oscillator, whose Hamiltonianis H = 12p2 + 12x2 with E = 1. These orbits are exact ellipses. These ellipses closelyapproximate the complex orbits in the vicinity of the turning points in Fig. 2.2.2. Open trajectories: E > 1.When the energy E of a simple pendulum is sufficiently large (here, E > 1), thephysically observed motion is that of a rotating pendulum. The classical turning pointsfor this case are no longer on the real axis because there are no real solutions tocos x0 = E with E > 1. The rotating pendulum has no real turning points becauseits real trajectory never turns back. To find the complex turning points, we replace x0by a + ib and get two equations:sin a sinh b = 0, cos a cosh b = −E. (9)Complex Trajectories of a Simple Pendulum 5For the example considered in this subsection we choose E = cosh 1, and thus theturning points are situated at a = (2k + 1)π, where k ∈ Z, and b = ±1.The complex trajectories for this case are shown in Fig. 4. Note that the trajectoriesfill the entire complex plane and that there are no closed orbits. The turning pointshave a strong influence on the path of the complex pendulum bob. The trajectories inthe upper-half complex-x plane, for example, go just below and veer around the turningpoints above the real axis. The turning points seem to attract the trajectories in muchthe same way as was found in Ref. [6].Figure 4. Trajectories in the complex-x plane for the rotating pendulum. For thiscase the pendulum has energy E = cosh 1 and the gravitational field strength hasthe value g = 1. Note that the trajectory on the real axis corresponds to a physicalpendulum undergoing continuous circular motion. All of the complex trajectories areopen. The trajectories that terminate at turning points run vertically off to complexinfinity. One such trajectory that starts at x0 = π + i is shown.Note that the trajectories that terminate at turning points run vertically off tocomplex infinity. This feature is also a property of the Hamiltonian H = 12p2 + ix3,which was studied in detail in Ref. [2]. The complex trajectories for this Hamiltonianare shown in Fig. 5. There are three turning points, and the trajectory beginning at theturning point on the imaginary axis goes up the imaginary axis to infinity.One may ask how long it takes for a particle beginning at a turning point, say, theturning point at π + i, to reach infinity. The time T for this motion is represented bythe following integral:T =1√2∫ i∞+πx=i+πdx√E + cos x. (10)A numerical evaluation of this integral gives T = 1.97536 · · ·. Thus, the particle reachesthe point at complex infinity in a finite time.Complex Trajectories of a Simple Pendulum 6Figure 5. Complex trajectories for the Hamiltonian H = 12p2 + ix3 for E = 1. Alltrajectories are closed periodic orbits except for the trajectory beginning at x0 = i. Aparticle released from this turning point runs off to infinity in a finite time.2.3. Closed trajectories: E < −1There are no physically observable trajectories for this choice of the energy. However,the complex trajectories in this case can be calculated numerically (see Fig. 6) and wefind that they are all closed orbits that lie in periodic cells of width 2π. The turningpoints come in complex-conjugate pairs. In Fig. 6 we have chosen E = − cosh 1 and theturning points are situated at x0 = a + ib, where a = 2kπ (k ∈ Z) and b = ±1.3. Pendulum in an Imaginary Gravitational FieldIf we take the gravitational field strength g in (6) to be imaginary, the Hamiltonianbecomes PT symmetric, where P reflection consists of x → 12nπ − x. Setting g = i, theHamiltonian becomesH = p2/2 − i cos x (11)and the turning points are located at x0 = a + ib, wherecos a cosh b = 0, − sin a sinh b = E. (12)There are now two cases to consider, E > 0 and E < 0. In Fig. 7 we takeE = sinh 1 and find turning points at x0 =(n + 12)π + i(−1)n (n ∈ Z) and in Fig. 8 wetake E = − sinh 1 and find turning points at x0 =(n + 12)π + i(−1)n+1 (n ∈ Z). LikeComplex Trajectories of a Simple Pendulum 7Figure 6. Complex trajectories for the pendulum in (6) with g = 1 and E = − cosh 1.The turning points come in complex-conjugate pairs and the trajectories are all closedorbits.the complex trajectories in Fig. 4, the trajectories in these two figures are not closedand trajectories beginning at turning points run vertically off to infinity. In Fig. 7 onesuch trajectory is shown. This trajectory begins at 32π + i. The time T for a particle totravel from this turning point to infinity is given byT =1√2∫ i∞+3π/2x=i+3π/2dx√E + i cos x. (13)After a simple change of variables this integral becomesT =1√2∫ ∞s=1ds√sinh s − sinh 1=2√eK(−1/e2) = 1.84549 · · · , (14)where K is the complete elliptic function. Thus, the classical particle reaches complexinfinity in finite time.4. Further ConsiderationsThere are a number of interesting ways to extend and generalize the investigations inthe previous two sections.In Figs. 7 and 8 the energy for the pendulum in an imaginary gravitational field wastaken to be real. This is because a quantum-mechanical PT -symmetric Hamiltonianhas the virtue of possessing real energy levels. However, one might ask what happensif one takes the energy to be complex. In this case the complex trajectories are notspatially or temporally periodic, even though the turning points are periodic and areseparated horizontally by 2π. The behavior of the trajectories is illustrated in Fig. 9,where we have taken the energy to be E = i.Another way to extend these investigations is to study what happens when thereis an external periodic driving force of the form ǫ sin(ωt) [8, 9, 10, 11]. We have takenComplex Trajectories of a Simple Pendulum 8Figure 7. Complex trajectories for the pendulum in an imaginary gravitational fieldof strength g = i and positive real energy E = sinh 1.Figure 8. Complex trajectories for the pendulum in an imaginary gravitational fieldof strength g = i and negative real energy E = − sinh 1. Trajectories that begin atthe turning points run off to complex infinity in the positive and negative imaginarydirections as in Figs. 4, 5, and 7.ǫ = 0.2 and ω = 0.1 and in Figs. 10, 11, and 12 we have examined the complextrajectories up to times t = 100, t = 650, and t = 1000. The starting energy forthese figures is taken to be 0 and the value of g is 1. The initial value of x is takento be 12π + 0.1,which would lead to a complex trajectory if there were no drivingterm. We see that for short times the trajectory is noisy, but seems to be confinedto the central cell in Fig. 2. However, after a sufficiently long time the trajectorywanders into adjacent cells, as can be seen in Figs. 11 and 12. This result suggests itmight be productive to investigate complex trajectories of dynamical systems whose realtrajectories exhibit chaos, especially in the PT -symmetric case. One could begin withthe complex trajectories associated with real homoclinic chaos, as in the present case.In addition, the rich behavior seen in the complex spatial trajectories of PT -symmetricComplex Trajectories of a Simple Pendulum 9Figure 9. Complex trajectories for a pendulum in an imaginary gravitational field ofstrength g = i and complex energy E = i. In this case the trajectories are not closedand not spatially periodic.systems may be expected to be the projection onto the two-dimensional complex planeof interesting behavior in four-dimensional complex phase space. For example, thephase-space trajectories of PT -symmetric complex systems whose real parts are chaoticmight exhibit a complex Hamiltonian analog of Arnold diffusion, or some other kind ofHamiltonian diffusion.Figure 10. Complex trajectories for a pendulum in a gravitational field of strengthg = 1 and subject to an external periodic driving force. Here, the driving force hasthe form 0.2 sin(0.1t) and the initial condition is taken to be x = 12π +0.1. Ordinarily,such an initial condition would lead to a complex trajectory like that in Fig. 2. Forshort time (up to t = 100 in this figure) the trajectory remains confined to the centralcell in Fig. 2.We thank D. C. Brody for many useful discussions. CMB thanks the Theoretical Physics Group atComplex Trajectories of a Simple Pendulum 10Figure 11. Same as in Fig. 10 but now the trajectory is allowed to continue untilt = 650. Eventually, the trajectory excapes from its cell and begins to wander intonearby cells. It gains complex energy and it begins to resemble the trajectory in Fig. 9.Figure 12. Same as in Fig. 11 but now the time runs up to t = 1000. Note that thecomplex trajectory undergoes extreme fluctuations.Imperial College, London, for its hospitality. As an Ulam Scholar, CMB receives financial support fromthe Center for Nonlinear Studies at the Los Alamos National Laboratory and he is supported in partby a grant from the U.S. Department of Energy. The work of DDH was supported by the Royal Societyof London and by the U.S. Department of Energy Office of Science Applied Mathematical Research.[1] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).[2] C. M. Bender, S. Boettcher, and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).[3] P. Dorey, C. Dunning, and R. Tateo, J. Phys. A: Math. Gen. 34, L391 (2001); ibid. 34, 5679(2001).[4] C. M. Bender, G. V. Dunne, and P. N. Meisinger, Phys. Lett. A 252, 272-276 (1999).[5] A. Nanayakkara, Czech. J. Phys. 54, 101 (2004) and J. Phys. A: Math. Gen. 37, 4321 (2004).[6] C. M. Bender, J.-H. Chen, D. W. Darg, and K. A. Milton, J. Phys. A: Math. Gen. 39, 4219-4238Complex Trajectories of a Simple Pendulum 11(2006).[7] C. M. Bender and D. W. Darg (in preparation).[8] J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcationsof Vector Fields, Applied Mathematical Sciences, Vol. 42 (Springer, New York, 1983).[9] A. Heck and S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Textsin Applied Mathematics, Vol. 2 (Springer, New York, 2003), second edition.[10] P. J. Holmes and J. E. Marsden, J. Math. Phys. 23, 669 (1982).[11] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistryand Engineering (Perseus, Cambridge, Massachusetts, 1994).

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