Unlike many energy-conserving oscillatory systems, the relativistic harmonic oscillator (RHO) represents a unique case. In many of these systems, the nonlinearity is a nonlinear function of the amplitude, x(t). Usually, the occurrence in the nonlinearity of the velocity, x ˙ ( t ) , represents damping and energy non-conservation. The RHO is an energy-conserving system, in which the nonlinearity involves the velocity. Furthermore, in popular systems, as the amplitude of oscillation, a, tends to infinity, the period of oscillations, T, tends to zero and the maximal velocity tends to infinity. In the case of the RHO, the period tends to infinity and the velocity tends to a constant, the speed of light, c.

The study of relativistic oscillations has progressed in several avenues: The exploitation of relativistic oscillators for the purpose of the description of bound states of quarks in High-Energy Physics [1] [2] [3] [4]; The study of the Dirac oscillator theoretically [5] - [12] and experimentally [13] [14], and the Quantum-Mechanical RHO [15] - [23]. Oscillations in the weak-relativistic limit have been of interest in Plasma Physics [24] - [36]. The emergence of chaotic behavior in a parametrically forced relativistic van der Pol oscillator is discussed in Ref. [37]. The behavior of an ensemble of RHO’s is studied in Ref. [38]. Interest in the RHO, in both weak- and extreme-relativistic limits has won renewed attention in the study of media, through which light propagates at velocities that are much lower than its speed in vacuum. A recent example is that of the dynamics of an oscillator composed of ultra-cold Bose-condensed atoms [39] [40] [41], where the effective speed of light is lower than in vacuum by orders of magnitude, e.g., 143 mm/s [41]. The energy-momentum relation of the atoms is nearly identical to that of a massive relativistic particle, with an effective mass.

Over many years, the major effort in the analysis has been dedicated to either generating numerical solutions or to various approximation schemes in the weak relativistic limit by expansions in powers of β² ( β = ( max | x ˙ | ) / c ) for β ≪ 1 [42] - [50]. Owing to the emergence of secular terns in asymptotic expansion methods, the validity in time of approximate solutions is limited. Typically, the lowest-order approximation (O(β²)) in such analyses has a 1% - 2% error relative to the numerical solution up to only β ≈ 0.2. One important result of Ref. [43] is the presentation the period of oscillations in the frame of reference of an outside observer in terms of complete elliptic integrals.

Qualitative progress has been attained through the exploitation of the transformation from the time variable in the observer’s frame of reference to that of the oscillating mass (proper time) [51] - [57]. The oscillation period in proper time was derived in terms of a complete elliptic integral in Ref. [51]. The equivalence of the equation of motion of the RHO with a specific form of the equation of motion of a Duffing equation was found in Refs. [52] [53] [55] [57]. This has led to the derivation of a closed-from solution for the equation of the RHO in terms of Jacobi elliptic functions [51] [56].

The purpose of his paper is to expose the detailed characteristics of the solution of the equation of motion of the RHO in the weak- and extreme-relativistic limits.

Preliminary information is presented in Section 2. Exploiting the known closed-form expressions of the period, T, and the oscillation amplitude, x_max, the time and position variables, t and x, are transformed in Section 3 into dimensionless variables, θ and η, respectively. η is (2π)-periodic in θ and bounded: |η| ≤ 1. This transformation relieves the analysis in the weak-relativistic limit of the restriction on the validity in time of approximate solutions, encountered in standard asymptotic expansion perturbative analyses of the weak-relativistic limit. In addition, it naturally leads to a novel approach to the analysis of the extreme-relativistic limit.

The weak-relativistic limit is discussed in Section 4 through a Normal Form expansion [58] - [66]. As the solution is (2π)-periodic in θ, there is no need to worry about the growth of secular errors in the approximate solution for θ ≫ 1 . A direct consequence is that the error in the approximate solution turns out to remain very small up to rather high values of β. For example, in an expansion through O(β²), the approximations for the dimensionless position, η, and for the velocity, η', agree with the full numerical solution within a fraction of 1% up to β ≈ 0.6 and β ≈ 0.45, respectively.

The expansion in powers of β² fails in the extreme relativistic limit (β  1). In this limit, over most of the period of oscillations, the velocity x ˙ ,oscillates between two constant values, ± v_max, (v_max = c·β) and is reduced to zero rapidly over a short time interval near the turning point. The analysis is presented in Section 5 through the introduction of a small parameter, μ, which measures the proximity to the extreme relativistic limit.

Near the turning point (θ = 0), the velocity varies rapidly over a range of O(μ²) in θ. Hence, for finding an approximate solution for η over this narrow range in the extreme-relativistic limit, a variable transformation is required, θ = μ 2 z , so that z is of O(1). The resulting expansion of the solution in powers of μ², expected to provide a good approximation to the solution near the turning point (θ = O(μ²) (z = O(1))), generates an excellent approximation to the solution for 0 ≤ θ ≤ 2π, even in the lowest order in μ², for small μ. Unique features of the Fourier expansion of the solution are discussed in Section 6. Some concluding comments are presented in Section 7.

The dynamical equation of the RHO is:

m ω 0 2 x + m x ¨ ( 1 − ( x ˙ / c ) 2 ) 3 / 2 = 0 , (1)

where m is the mass of the oscillation particle, ω₀ is the period of the oscillator in the non-relativistic limit and c is the speed of light.

The total energy is given by:

E = 1 2 m ω 0 2 x 2 + m c 2 1 − ( x ˙ / c ) 2 = m c 2 1 − β 2 ,   ( β = max ( | x ˙ | / c ) ) . (2)

Solving Equation (2) for x ˙ yields the period of oscillations, T [43]:

T = 1 ω 0 4 2 ( 1 + 1 1 − β 2 E ( k ) − ( 1 / 1 + 1 1 − β 2 ) K ( k ) ) ,   k 2 = ( 1 − 1 − β 2 1 + 1 − β 2 ) . (3)

In Equation (3), E(k) and K(k) are the complete elliptic integrals [67]:

E ( k ) = ∫ 0 π / 2 1 − k 2 sin φ 2 d φ ,   K ( k ) = ∫ 0 π / 2 ( 1 / 1 − k 2 sin φ 2 ) d φ . (4)

The maximal values of the amplitude of oscillations and of the velocity are obtained by setting x ˙ = 0 and x = 0, respectively, in Equation (2), yielding:

x max = 2 c ω 0 1 − 1 − β 2 ( 1 − β 2 ) 1 / 4 ,   x ˙ max = β c . (5)

In the limit of small relativistic effects ( β ≪ 1 ), the expansion of E, T (using expansions of elliptic functions presented in [67]) and x_max in powers of β² is given by:

E = m c 2 ( 1 + 1 2 β 2 + 3 8 β 4 + 5 16 β 6 + O ( β 8 ) ) T = 2 π ω 0 ( 1 + 3 16 β 2 + 129 1024 β 4 + 1595 16384 β 6 + O ( β 8 ) ) x max = β c ω 0 ( 1 + 3 8 β 2 + 31 128 β 4 + 187 1024 β 6 + O ( β 8 ) ) x max T = β c 2 π ( 1 + 3 16 β 2 + 83 1024 β 4 + 761 16384 β 6 + O ( β 8 ) ) (6)

In the far relativistic limit ( β ≈ 1 ) it turns out to be convenient to define a new small parameter:

β 2 = 1 − μ 4 ,   ( μ ≪ 1 ) . (7)

Equation (6) is then replaced by (the expansion of the elliptic integrals can be found in Ref. [67]):

E = m c 2 μ 2 T = 4 2 ω 0 1 μ ( 1 − μ 4 3 − 6 log 2 + 4 log μ 16 + O ( μ 8 ) ) x max = 2 c ω 0 1 μ ( 1 − 1 2 μ 2 − 1 8 μ 4 − 1 16 μ 6 + O ( μ 8 ) ) x max T = c 4 ( 1 − 1 2 μ 2 + 1 16 μ 4 ( 1 − 6 log 2 + 4 log μ )                     − 1 32 μ 6 ( 5 − 6 log 2 + 4 log μ ) + O ( μ 8 ) ) (8)

Note that both T and x_max tend to infinity as E^1/2.

Figure 1 shows the phase-space plot of the numerical solution of Equation (1) for a sequence of values of β. x and t are expressed in terms of dimensionless entities:

τ = ω 0 t ,   X ( τ ) = ( ω 0 / c ) x ( τ / ω 0 ) . (9)

At high values of β, the phase-space plot tends to a rectangle. This has been observed in the many numerical analyse cited above.

Rescaling x and t as follows:

x ( t ) = x max η ( θ ) ,   t = ( T / 2 π ω 0 ) θ , (10)

Equation (1) becomes

η ( θ ) + ( 2 π ω 0 T ) 2 η ″ ( θ ) ( 1 − ( 2 π x max c T ) 2 η ′ ( θ ) 2 ) 3 / 2 = 0 . (11)

η(θ) is (2π)-periodic in θ and bounded: |η(θ)| ≤ 1. Exploiting Equations (10) and (6), yields that the maximal values of η and η' are both finite for all 0 ≤ β² ≤ 1:

η max ≡ max ( | η | ) = 1 η ′ max ≡ max | ( η ′ ( θ ) ) | = ( β c T 2 π x max ) → { 1 − 3 16 β 2 + O ( β 4 )         β → 0 2 π + μ 2 π + O ( μ 4 log μ )         β → 1 (12)

Figures 2-4 show the dependence of the solution on β. For β ≪ 1 , the phase-space plot is very close to a circle. As β is increased, it approaches a rectangle. Concurrently, η' evolves from a sine-like function of θ to a periodic step function and that of η – to a saw-tooth pattern.

For β ≪ 1 , Equation (11) is reduced to the equation of a classical harmonic oscillator perturbed by a small nonlinear term:

η ( θ ) + ( 1 + O ( β 2 ) ) η ″ ( θ ) ( 1 − β 2 ( 1 + O ( β 2 ) ) η ′ ( θ ) 2 ) 3 / 2 = 0 . (13)

Hence, this calls for an expansion in powers of β².

In the extreme-relativistic limit, Equation (11), using Equations (7) and (8), and expanding in powers of μ ≪ 1 :

η ( θ ) + π 2 8 μ 2 ( 1 + O ( μ 4 log μ ) ) η ″ ( θ ) ( 1 − ( π 2 4 ) ( 1 − μ 2 + O ( μ 4 ) ) η ′ ( θ ) 2 ) 3 / 2 = 0 . (14)

A convenient way to analyze Equation (11) is to transform it to a first-order equation for a complex entity:

z ( θ ) = η ( θ ) + i η ′ ( θ ) . (15)

Expressing η and η' in terms of z and z^*, Equation (11) is transformed into:

z ′ ( θ ) = z ( θ ) − z * ( θ ) 2 i   + i { − ( ω 0 T 2 π ) 2 z ( θ ) + z * ( θ ) 2 ( 1 − ( 2 π x max c T ) 2 ( z ( θ ) − z * ( θ ) 2 i ) 2 ) 3 / 2 } (16)

The boundary conditions at the turning point are:

η ( 0 ) = z ( 0 ) + z * ( 0 ) 2 = 1 ,   η ′ ( 0 ) = z ( 0 ) − z * ( 0 ) 2 i = 0 . (17)

One now writes z(θ) as:

z ( θ ) = z 0 ( θ ) + β 2 z 2 ( θ ) + β 4 z 4 ( θ ) + β 6 z 6 ( θ ) + ⋯ (18)

and exploits the expansion of T and x_max (Equation (6)) to obtain an expansion of Equation (16) in powers of β².

The zero-order term of this expansion yields:

z ′ 0 ( θ ) = − i z 0 ( θ ) ⇒ z 0 ( θ ) = ρ 0 e − i ( θ + θ 0 ) . (19)

Thus, as expected, the zero-order term is (2π)-periodic in θ. The O(β²) term in the expansion of Equation (16) is:

z ′ 2 ( θ ) = − i z 2 ( θ ) − 3 8 i { ρ 0 ( 1 − ρ 0 2 ) cos ( θ + θ 0 ) + ρ 0 3 cos ( 3 ( θ + θ 0 ) ) } . (20)

For the solution to be periodic in θ, the coefficient of the resonant term cos(θ + θ₀) must vanish, yielding

ρ 0 = 1 . (21)

One now proceeds in the same fashion through higher orders, requiring that the higher-order corrections, z₄ and z₆ of Equation (18) are periodic in θ (i.e., the coefficients of resonant terms in the dynamical equations for z₄ and z₆ vanish) and that the initial conditions are obeyed by the zero-order term:

z 0 ( 0 ) + z 0 ∗ ( 0 ) 2 = 1 ,   z 0 ( 0 ) − z 0 ∗ ( 0 ) 2 i = 0 . (22)

The result is the following expression for η:

η ( θ ) = cos θ + β 2 3 64 ( − cos θ + cos 3 θ ) + β 4 ( − 97 cos θ + 72 cos 3 θ + 25 cos 5 θ 4096 )     + β 6 ( − 11619 cos θ + 6786 cos 3 θ + 4000 cos 5 θ + 833 cos 7 θ 786432 ) + O ( β 8 ) (23)

The fact that the period, T, is known explicitly means that the restriction of the validity of the approximation that emerges in well-known asymptotic expansion methods is avoided. In these methods, the period is approximated by a truncated expansion in powers of the small parameter, limiting the validity of the approximation to times of O(1) or O(1/β²), depending on the analysis method used. This limitation is avoided here. The approximation to the full numerical solution up the rather high values of β is excellent. For β = 0.1, the maximal relative error generated by the approximate solution of Equation (23) is, at most, O(.01%). In expansions through O(β²), O(β⁴) and O(β⁶), the approximation for η agrees with the full numerical solution of Equation (11) within a fraction of 1% up to β ≈ 0.6, 0.7 and 0.8, respectively, for all θ. The approximation for η', agrees with the full numerical solution within a fraction of 1% up to β ≈ 0.4, 0.55 and 0.65, respectively, for all θ. Even for β ≈ 0.8, the relative error of the approximate expression for the amplitude (through O(β⁶), Equation (23)) varies between 0 and close to 1% (see Figure 5) and for the velocity—between 1% and 5% (see Figure 6).

In the extreme relativistic limit, the expansion in powers of β² fails. Figures 7(a)-(c) show how the approximate solution (Equation (23)) begins to deviate

from the numerical solution of Equation (11) as β grows. The errors in η and η' are shown in Figure 8 and Figure 9, respectively, in a computation through O(β⁶), for β = 0.99. The error in η is, at most about 10%. In η', it exceeds 60% near the turning point.

In this limit, the velocity, x ˙ , is constant at (±c β) most of the time. It vanishes rapidly over a short time gap near the turning point (See Figure 1). In the scaled coordinates, the dependence of η on θ is a straight line, except very close to the turning point (see Figure 3).

To guide oneself towards a choice of a scaled variable that is appropriate for the analysis of Equation (11) near the turning point, one uses μ ≪ 1 of Equation (7), which is a measures of the proximity to the extreme-relativistic limit. Observe that η' is now very close to zero. Using Equation (8) for T and x_max and replacing η' by zero in Equation (14), the O(μ²) approximation to the equation becomes:

η ( θ ) + π 2 8 μ 2 η ″ ( θ ) = 0 . (24)

For the boundary conditions at the turning point (θ = 0), η(0) = 1 and η'(0) = 0, Equation (24) is solved by

η ( θ ) = cos ( 2 2 π θ μ ) . (25)

Exploiting the leading-order term in η'_max (Equation (12)), one finds an approximation for θ_S, the value of θ, at which the slope η' from Equation (25) becomes equal to the constant (maximal) slope that prevails throughout most of the range 0 ≤ θ ≤ (π/2). Equation (25) yields θ_S = O(μ²):

η ′ ( θ S ) ≈ − η ′ max ⇒ θ S ≈ π 4 μ 2 ( 1 + 1 2 μ 2 ) . (26)

Thus, for a valid expansion in powers of μ² near the turning point, one should rescale θ as:

θ = μ 2 z , (27)

so that z is of O(1). One now expands η(θ) in powers of μ²:

η ( θ ) = 1 + μ 2 F 2 ( z ) + μ 4 F 4 ( z ) + μ 6 F 6 ( z ) + O ( μ 8 ) ,   ( z = ( θ / μ 2 ) ) . (28)

Using Equations (28) and (7), one expands Equation (11) in powers of μ², and solves for the higher-order corrections in Equation (28). The boundary conditions for all F_2n are:

F 2 n ( z = 0 ) = F ′ 2 n ( z = 0 ) = 0 . (29)

Through O(μ⁶), F_2n are found to be given by:

F 2 ( z ) = 1 2 − π 2 + 16 z 2 2 π , (30)

F 4 ( z ) = 1 4 − π 2 + 16 z 2 4 π + z arcsin ( 4 z / π ) 2 π 2 + 16 z 2 , (31)

F 6 ( z ) = 1 4 + ( − π 128 − π z 8 ( π 2 + 16 z 2 ) + z 4 π 2 + 16 z 2 ) arcsin ( 4 z / π )     − π 3 arcsin ( 4 z / π ) 2 64 ( π 2 + 16 z 2 ) 3 / 2 + π 128 log ( 4 z + π 2 + 16 z 2 π )     + π 2 + 16 z 2 ( − 3 64 π − 3 log ( 2 ) 16 π + log ( μ ) 8 π )     + 1 π 2 + 16 z 2 ( − 13 π 64 + 13 π 16 log ( 2 ) − π 8 log ( μ ) ) (32)

Detailed inspection of F_2n for 2 n = 2, 4, 6, reveals that they all have the same qualitative dependence on θ. Except very near the turning point (θ = 0) they all tend to a linear dependence on θ. They vanish at roughly the same rate as θ varies from O(μ²) down to zero.

The surprising result is that an expansion that was meant to be valid around the turning point (θ ≈ O(μ²)) yields an excellent approximation to the solution throughout 0 ≤ θ ≤ 2 π for small μ. For sufficiently small μ, an excellent approximation is obtained with just the O(μ²) term:

η ( θ ) = 1 + μ 2 F 2 ( z ) . (33)

Figure 10 and Figure 11 show a comparison between the approximation, Equation (33), and the numerical solution of Equation (11) for μ = 0.1 (β = 0.99995). θ_S is then ≈ 0.0078. To stress the quality of the approximation, Figure 12 shows

how small the deviation of the approximate solution from the numerical solution remains even for θ ≫ θ S . Figure 13 shows the relative error in η'. It remains small for all 0 ≤ θ ≤ 2 π.

Finally, for θ =O(1), namely, away from the turning point, F4[z = θ/μ²] adds a μ² contribution. An excellent approximation is obtained by:

η ( θ ) = 1 + μ 2 2 − ( 2 + μ 2 π ) θ + O ( μ 4 log μ ) . (34)

The evolution of the velocity profile, η'(θ), in the extreme relativistic limit (β  1, μ  0) into a periodic step function (see Figure 4) offers unique characteristics for the Fourier spectrum of η'(θ). The Fourier coefficients tend to the limiting spectrum of the step function, the latter oscillating between ±(2/π) with a cycle length of (2π). The Fourier series for this step function is:

∑ n = 0 ∞ a 2 n + 1 sin ( 2 n + 1 ) θ ,   a 2 n + 1 = − 8 ( 2 n + 1 ) π 2 . (35)

The purpose of this Section is to discuss two aspects. The first is a demonstration of the qualitative difference between the Fourier spectra in the weak- and extremely-relativistic limits. The second is an additional test of the quality of the approximate solution obtained in Section 5.

The dynamics of the Relativistic Harmonic Oscillator (RHO) have been analyzed in the weak- and extreme-relativistic limits. Exploiting the fact that the period of oscillations can be computed in closed form and to any desired level of numerical accuracy, and using the scaled variables of Equation (10), for β < 1, a Normal Forms expansion yields a high quality approximation to the solution that is by far better than the standard application of asymptotic expansion methods.

The same formulation allows for the analysis of the extreme-relativistic limit as a boundary-layer problem, leading to a high-quality approximation for the solution for all 0 ≤ θ ≤ (2π) for (1 − β) << 1.

The author declares that no funds, grants, or other support were received during the preparation of this manuscript.

The author has no relevant financial or non-financial interests to disclose.

1) The author made all contributions to the conception or design of the work. No data acquisition was required. No new software was created;

2) The author drafted the work or revised it critically for important intellectual content;

3) The author approved the version to be published; and

4) The author agrees to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

Data sharing is not applicable to this article, as no datasets were generated or analyzed during the current study.

The author declares no conflicts of interest regarding the publication of this paper.

Zarmi, Y. (2023) On the Relativistic Harmonic Oscillator. Applied Mathematics, 14, 1-20. https://doi.org/10.4236/am.2023.141001