Comparison of the Newtonian and relativistic predicted trajectories for a low-speed periodically-delta-kicked system

Boon Leong Lan School of Engineering, Monash University, 46150 Petaling Jaya, Selangor, Malaysia

The dynamics of a periodically-delta-kicked Hamiltonian system moving at low speed (i.e., at speed much less than the speed of light) is studied numerically. In particular, the trajectory of the system predicted by Newtonian mechanics is compared with the trajectory predicted by special relativistic mechanics for the same parameters and initial conditions. We find that the Newtonian trajectory, although close to the relativistic trajectory for some time, eventually disagrees completely with the relativistic trajectory, regardless of the nature (chaotic, non-chaotic) of each trajectory. However, the agreement breaks down very fast if either the Newtonian or relativistic trajectory is chaotic, but very much slower if both the Newtonian and relativistic trajectories are non-chaotic. In the former chaotic case, the difference between the Newtonian and relativistic values for both position and momentum grows, on average, exponentially. In the latter non-chaotic case, the difference grows much slower, for example, linearly on average.

PACS: 05.45.-a

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It is expected that if the speed of a particle remains low, i.e., much less than the speed of light, the trajectory predicted by special relativistic mechanics remains very well approximated by the trajectory predicted by Newtonian mechanics for the same parameter(s) and initial conditions. However, it is shown here with a prototypical Hamiltonian system that the Newtonian predicted trajectory can eventually disagree completely with the relativistic predicted trajectory even though the particle speed is low. It is also shown that how fast the agreement breaks down depends on the nature (chaotic, non-chaotic) of the Newtonian and relativistic trajectories. The eventual complete breakdown of the agreement of the trajectories predicted by the two theories should occur also for other Hamiltonian systems moving at low speed and therefore it is possible to test the two different predicted trajectories against the measured trajectory of a slowmoving Hamiltonian system. In a recent paper [1], the trajectory predicted by Newtonian mechanics was compared with the trajectory predicted by Bohmian quantum mechanics for a periodically-delta-kicked system for the same parameters and initial conditions. The periodically-delta-kicked system is a one-dimensional Hamiltonian system in a sinusoidal potential that is periodically turned on for an instant; it could be, for example, a pendulum in a time-varying gravitational field [2,3] or an electron in a time-varying electric field in a plasma [4,5]. It was found [1] that the Newtonian trajectory does not agree with the Bohmian trajectory for all times. The agreement breaks down very fast if the Newtonian trajectory or one of its classical neighbors is chaotic; in contrast, the break down occurs much slower if neither the Newtonian

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trajectory nor its classical neighbors are chaotic. The difference between the Newtonian and Bohmian values for both position and momentum grows much faster, exponentially on average, in the chaotic case compared to the non-chaotic case. Earlier [6-8], similar results were found when comparing the Newtonian trajectory with the quantum mean trajectory for the same system. Extending the comparisons in [1,6-8], here I numerically compare the trajectory predicted by Newtonian mechanics for the periodically-delta-kicked system moving at low speed, i.e., at speed much less than the speed of light, with the trajectory predicted by special relativistic mechanics for the same parameters and initial conditions. Newton’s equation of motion for the periodically-delta-kicked system is easily integrated exactly [2,3] to produce a mapping, called the standard map, of the dimensionless scaled position X and dimensionless scaled momentum P from

( X n−1, Pn−1 ) , the values just before the nth kick, to ( X n , Pn ) , the values just before the (n+1)th kick:

Pn = Pn−1 −

K sin ( 2π X n−1 ) 2π

1(a)

X n = ( X n−1 + Pn ) mod1

1(b)

where n = 1, 2,... , and K is a dimensionless positive parameter. The map above has also served as an important model in the field of nonlinear dynamics because a number of dynamical problems are approximately reduced to it [2,9]. The relativistic equation of motion is also easily integrated exactly [4,5] to produce a mapping of the dimensionless scaled position X and dimensionless scaled

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momentum P from ( X n−1 , Pn−1 ) , the values just before the nth kick, to ( X n , Pn ) , the values just before the (n+1)th kick: Pn = Pn−1 −

K sin ( 2π X n−1 ) 2π

⎛ Pn X n = ⎜ X n−1 + ⎜ 1 + β 2 Pn2 ⎝

2(a)

⎞ ⎟ mod1 ⎟ ⎠

2(b)

where n = 1, 2,... , and in addition to K , there is another dimensionless positive parameter β . In general, according to special relativistic mechanics,

v = c

p m0c ⎛ p ⎞ 1+ ⎜ ⎟ ⎝ m0c ⎠

2

,

(3)

where v is the speed, p is the momentum and m0 is the rest mass. For the relativistic standard map [Eqs. 2(a) and 2(b)], since β P (P is the dimensionless scaled momentum) is [5] equal to

p , Eq. (3) becomes m0c

βP v = . 2 c 1+ ( β P)

(4)

We can easily see from Eq. (4) that β P  1 implies v  c . Detailed properties of the Newtonian standard map [Eqs. 1(a) and 1(b)] and the relativistic standard map [Eqs. 2(a) and 2(b)] can be found in references [2,9] and [4,5,10] respectively. Here, it suffices to note that the phase-space trajectories generated by the two maps can be either chaotic, or non-chaotic (periodic or quasiperiodic).

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I will now present and discuss the typical result of comparing the trajectories predicted by Newtonian mechanics and special relativistic mechanics for the periodically-delta-kicked system moving at low speed. In this example, the dimensionless scaled momentum P of the relativistic trajectory, with initial conditions X 0 = 0.5 and P0 = 99.9 , generated by the relativistic standard map [Eqs. 2(a) and

2(b)], with parameters K = 0.9 and β = 10−7 , is always ≈ 100 (in particular, P is always bounded between 99.6 and 100.4). Therefore, β P is always ≈ 10−5 . This implies, according to Eq. (4), that v is always ≈ 10−5 c , i.e., v is always merely 0.001% of c (for comparison, the orbital speed of the earth is about 10 times faster). In other words, the speed is always low. The Newtonian trajectory, which is generated by the Newtonian standard map [Eqs. 1(a) and 1(b)] with the same parameter K and initial conditions, is compared with the relativistic trajectory in Fig.1 for the first 140 iterations. The Newtonian and relativistic positions are plotted versus iteration in Fig. 1(a) while the Newtonian and relativistic momentums are plotted versus iteration in Fig. 1(b). In each figure, successive values of the dynamical quantity predicted by each theory are connected by a straight line to aid the eyes. The following behavior for both the position and momentum can clearly be seen in Fig. 1. The Newtonian values agree to certain extent with the relativistic values for the first 113 iterations. However, the Newtonian values no longer agree with the relativistic ones at all from iteration 114 onwards. The plotted Newtonian and relativistic values in Fig. 1 are values which are accurate to some degree, where the degree of accuracy decreases with increasing number of iterations. (The degree of accuracy was established using the standard method [9] of varying the numerical precision: here, by comparing the double 5

precision values (14 significant figures) to the corresponding values computed in quadruple precision (35 significant figures). For example, at iteration 114, the Newtonian position is 0.182857928… in double precision and 0.182857624… in quadruple precision and so the Newtonian position is accurate to 6 significant figures, i.e., 0.182857. By similar comparison, the relativistic position is also accurate to 6 significant figures but the value is 0.234650 which is entirely different compared to the Newtonian position.) Hence, the complete breakdown of the agreement of the Newtonian values with the relativistic values, for both position and momentum, after iteration 113 is not a numerical artifact. In all other cases studied (different initial conditions, different parameters, different very-small values of v/c), the Newtonian trajectory also eventually deviates completely from the relativistic trajectory, regardless of the nature (chaotic, nonchaotic) of each trajectory, i.e., even if both trajectories are non-chaotic. In other words, the eventual complete disagreement between the predictions of the two theories is not due to chaos. Furthermore, the accuracy check also confirmed that the eventual complete breakdown of the agreement of the Newtonian trajectory with the relativistic trajectory at low speed is not a numerical artifact. However, the speed of the breakdown depends on the nature of the Newtonian and relativistic trajectories. If either the Newtonian or relativistic trajectory is chaotic, the breakdown occurs very fast because the difference between the Newtonian and relativistic values for both position and momentum grows, on average, exponentially. In sharp contrast, if both the Newtonian and relativistic trajectories are non-chaotic, the breakdown occurs very much slower because the difference between the

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Newtonian and relativistic values grows, on average, significantly slower. For example, in the example above where both the Newtonian and relativistic trajectories are chaotic, the breakdown occurs at iteration 114. Fig. 2 shows that the difference between the Newtonian and relativistic values for both position and momentum grows, on average, exponentially up to at least iteration 114 at a rate of 0.13 which is close to the classical Liapunov exponent of 0.12 (calculation details of the classical Liapunov exponent is given in [8]). Soon after iteration 114, the difference saturates because both the chaotic Newtonian and relativistic trajectories are bounded above and below by impenetrable quasi-periodic trajectories, known [2-5,9,10] as KAM (Kolmogorov-Arnold-Moser) tori, in their respective phase space. However, for the same parameter values of the two maps, but different initial conditions: X 0 = 0.7 and P0 = 99.9 , where both the Newtonian and relativistic trajectories are non-chaotic

quasi-periodic ones, the breakdown occurs only after 187 million iterations. The breakdown occurs only after a very large number of iterations in this case because the difference between the Newtonian and relativistic values for both position and momentum only grows linearly on average. Fig. 3 shows the linear, on average, growth of the differences for the first 1,000 iterations, where the growth rate is 1.5 x10-10 for the position difference and 1.3 x10-10 for the momentum difference. For the chaotic case above ( X 0 = 0.5 , P0 = 99.9 , K = 0.9 , β = 10−7 ), Fig. 4 shows the relativistic trajectory and Fig. 5 shows the Newtonian trajectory in phase space for the first 300 thousand iterations, both calculated in quadruple precision. Each of the chaotic ‘sea’ is [11] a ‘fat’ fractal (fractal dimension is 2). The two chaotic seas, although similar, have some tiny differences: for example, the island located at 7

about (0.45, 99.9) in the Newtonian sea does not have smaller islands around it. For the non-chaotic case above ( X 0 = 0.7 , P0 = 99.9 , K = 0.9 , β = 10−7 ), Fig. 6 shows

the Newtonian and relativistic trajectories (calculated in quadruple precision) in phase space for 100 thousand iterations after the breakdown of the iteration-by-iteration agreement of the two trajectories. There is no discernible difference between the two quasi-periodic islands on the scale of the phase-space plot but fine differences appear upon magnification. Thus, in general, after the breakdown of the iteration-by-iteration agreement between the Newtonian and relativistic trajectories, there are only tiny differences between plots of the trajectories in phase space. Similar results of (1) eventual complete breakdown of agreement and (2) dependence of the speed of the breakdown on the nature of the trajectories were also obtained in the comparisons of a Newtonian trajectory generated by the Newtonian standard map [Eqs. 1(a) and 1(b)] with a relativistic trajectory generated by an approximate version of the relativistic standard map [Eqs. 2(a) and 2(b)]: Pn = Pn−1 −

K sin ( 2π X n−1 ) 2π

5(a)

⎛ ⎛ 1 ⎞⎞ X n = ⎜ X n −1 + Pn ⎜1 − β 2 Pn2 ⎟ ⎟ mod1 . ⎝ 2 ⎠⎠ ⎝

5(b)

The mapping above results from a binomial expansion of the square-root in Eq. 2(b) up to first order in β 2 Pn2 .The main results (1) and (2) in this paper parallel the results [1,6-8] of comparing a Newtonian trajectory with a quantum trajectory, either Bohmian [1] or mean [6-8], for the same system.

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The periodically-delta-kicked system studied in this paper is [2,4,5,9,11] a prototypical Hamiltonian system. For other Hamiltonian systems moving at low speed, it is therefore reasonable to expect the iteration-by-iteration agreement of the Newtonian predicted trajectory with the relativistic predicted trajectory to also break down and, after the breakdown, the differences between plots of the trajectories in phase space to also remain small.

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References

[1] B. L. Lan, Phys. Rev. A 61, 032105 (2000). [2] B. V. Chirikov, Phys. Rep. 52, 265 (1979). [3] G. Casati, B. V. Chirikov, F. M. Izrailev, J. Ford, in Stochastic Behavior in Classical and Quantum Hamiltonian System, G. Casati, J. Ford, Eds. (Springer, Berlin, 1979), pp. 334-351. [4] A. A. Chernikov, T. Tél, G. Vattay, G. M. Zaslavsky, Phys. Rev. A 40, 4072 (1989). [5] Y. Nomura, Y. H. Ichikawa, W. Horton, Phys. Rev. A 45, 1103 (1992). [6] B. L. Lan and R. F. Fox, Phys. Rev. A 43, 646 (1991). [7] B. L. Lan, Phys. Rev. E 50, 764 (1994). [8] R. F. Fox and T. Elston, Phys. Rev. E 49, 3683 (1994). [9] A. J. Lichtenberg, M. A. Lieberman, Regular and Stochastic Motion (Springer, New York, 1983), chaps. 4 and 5. [10]

H. C. Tseng, H. J. Chen, C. K. Hu, Phys. Lett. A 175, 187 (1993).

[11]

D. K. Campbell, Los Alamos Science, No. 15, 218 (1987).

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Figure Captions

Figure 1 Comparison of the Newtonian (squares) and relativistic (diamonds) (a) positions, (b) momentums, for a case where both the Newtonian and relativistic trajectories are chaotic. Figure 2 Natural-log of the absolute value of the difference between the Newtonian and relativistic (a) positions, (b) momentums, versus iteration for the chaotic case in Fig. 1. Straight-line fits up to iteration 120 are also plotted. Figure 3 Absolute value of the difference between the Newtonian and relativistic (a) positions, (b) momentums, versus iteration for a case where both the Newtonian and relativistic trajectories are non-chaotic quasi-periodic ones. Straight-line fits are also plotted. Figure 4 Plot of the relativistic trajectory (for the chaotic case in Fig. 1) in phase space for the first 300 thousand iterations. Figure 5 Plot of the Newtonian trajectory (for the chaotic case in Fig. 1) in phase space for the first 300 thousand iterations. Figure 6 Plot of the Newtonian and relativistic trajectories (for the non-chaotic case in Fig. 3) in phase space for 100 thousand iterations after the breakdown of the iteration-byiteration agreement of the two trajectories.

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