Continuous piecewise linearization method for approximate periodic solution of the relativistic oscillator

Abstract

This paper presents an approximate periodic solution to the vibration of the relativistic oscillator using a novel analytical method called continuous piecewise linearization method. First, an equivalent conservative equation for the vibration of the relativistic oscillator was derived in a simple straightforward manner that elucidates the physical meaning of the conservative equation. The continuous piecewise linearization method was then applied to derive periodic solutions for the displacement and velocity of the relativistic oscillator based on the conservative equation. The results of the present method were compared with results of published methods and exact numerical solution and the maximum error of the present method was less than 0.002%. The model derivations and the solutions presented in this paper are considerably simple and very accurate and can be used to introduce the relativistic oscillator in relevant undergraduate courses on dynamics. Essentially, knowledge of freshman calculus is sufficient to comprehend and implement the continuous piecewise linearization method for the relativistic oscillator.

Keywords

Relativistic oscillator continuous piecewise linearization method Duffing-type oscillator harmonic balance method homotopy perturbation method

Introduction

The linear harmonic oscillator is all too familiar in undergraduate courses on mechanics and physics. Its governing equation is a second-order linear differential equation with well-known analytical solutions in terms of circular harmonic functions. On the other hand, the relativistic oscillator is usually introduced as an advanced concept in post-graduate courses because the governing equation is a second-order nonlinear differential equation with complex nonlinearity and its analytical solution is quite difficult.

The relativistic oscillator is useful for accurate and realistic modeling and simulation of very high-frequency mechanical vibrations. These vibrations are characterized by high maximum velocities, which are bounded to the speed of light ( $c = 3 \times 10^{8} m / s$ ) by relativistic effect. For velocities much lower than the speed of light, the relativistic effect is negligible and the relativistic oscillator produces a simple harmonic response that is the same as the nonrelativistic (Newtonian) simple harmonic oscillator (SHO). Hence, the relativistic oscillator can be used for analysis of the simple harmonic response of mechanical systems. But this would be an unnecessary exercise since the nonrelativistic SHO can do this job perfectly and is much simpler to analyze.

During very high-frequency vibrations the relativistic effect becomes significant and limits the maximum velocity of the oscillator to the speed of light. The response in this case is anharmonic. The Newtonian SHO introduces significant errors when used to determine the response of very high frequency vibrations because it produces a simple harmonic response with unbounded maximum velocity. Therefore, the SHO can violate the velocity limit of the special theory of relativity during very high frequency vibrations of mechanical systems. The velocity limit of the motion of bodies is one of the key achievements of the special theory of relativity, which is undoubtedly the best known theory of motion today.

Molecular/atomic vibrations (typical frequency of $10^{13}$ – $10^{14}$ Hz) fall under the category of vibrations with significant relativistic effect but this is a physics problem. For mechanical engineering application, ultrasonic vibrations (typical frequency above 20 kHz) may possess significant relativistic effect and can be studied using the relativistic oscillator. Ultrasonic vibrations can be applied in a variety of ways and systems such as ultrasonic transducers, ultrasonic welding of plastics,¹ material property enhancement,² and medical imaging (sonography). Another important mechanical engineering application of the relativistic oscillator is the vibration of micro-electromechanical systems (MEMS) also known as micro-machines. MEMS are small mechanical devices that vibrate at very high frequencies (typically between 1 MHz and 100 MHz) and therefore exhibit relativistic effect. The resonistor is an example of a MEMS device that can be modeled as a cantilever beam with linear elastic stiffness. Its dynamic response can be analyzed using the relativistic oscillator. Many applications of MEMS devices exist and the vibrations of a good proportion of these devices can be studied as relativistic oscillators.

Very accurate numerical solutions of the governing equation for the relativistic oscillator can be obtained but such numerical solutions are not physically insightful and not very useful for parametric analysis. Consequently, approximate analytical solutions are usually sought. Several approximate analytical methods for solution of nonlinear oscillators exist and can be classified as perturbation^3–6 and nonperturbation methods.^7–15 The perturbation methods have the general limitations of requiring (a) a small-parameter that is typically valid for weak nonlinearity and (b) asymptotic expansions that are only valid for small displacements. Recent perturbations methods^4–6,16 attempt to provide modifications that enable solutions for strong nonlinear and large-amplitude oscillations. However, it has been demonstrated that the recent perturbations methods are only accurate in predicting the frequency–amplitude response and not the oscillation history.¹⁷ Actually, the higher order approximations of the recent perturbation methods produce large unbounded errors in the oscillation history just like their classical counterparts.¹⁷ In contrast, the nonperturbation methods do not have any special or small parameter and are generally not based on asymptotic expansions. Some approximate analytical methods that have been used to study the relativistic oscillator are harmonic balance method (HBM),^10,11,18 homotopy perturbation method (HPM),^12,13 and differential transform method (DTM).¹⁴ The solution of the relativistic oscillator obtained by these analytical methods are too complicated for an average senior year undergraduate. In this paper, a nonperturbation method called continuous piecewise linearization method was applied to derive simple periodic solutions for the relativistic oscillator. The CPLM algorithm is an iterative analytic algorithm for finding periodic solutions of Duffing-type conservative oscillators. It was formulated recently¹⁵ and has been shown to produce very accurate results for small- and large-amplitude oscillations as well as weak and strong nonlinear oscillations. The main advantage of the CPLM is that it remains simple and produces highly accuracy solutions notwithstanding the complexity of the nonlinear restoring force considered. The goal of this paper is to present CPLM solutions for the relativistic oscillator that can be taught in relevant undergraduate courses based on knowledge of freshman calculus.

Dynamic modeling of the relativistic oscillator

Mathematical description of the relativistic oscillator

The relativistic oscillator consists of a linear spring attached to a fixed support at one end and connected to a rest mass $m_{0}$ at the other end. The mass moves in one dimension and its momentum is given by the product of its mass and velocity. According to the special theory of relativity the oscillator has a relativistic mass while in motion given as $m = m_{0} / \sqrt{1 - {(\dot{x} / c)}^{2}}$ where $\dot{x}$ is the derivative of $x$ w.r.t. the dimensional time, $t$ . Therefore, the momentum in relativistic sense is

p = m_{0} \dot{x} / \sqrt{1 - {(\dot{x} / c)}^{2}}

(1)

From Newtonian mechanics, it is well known that the rate of change of momentum for a freely vibrating oscillator is equal and opposite to the restoring force. Hence, we can write that

\dot{p} + K_{L} x = 0

(2)

where

K_{L}

is a linear stiffness constant and

x

is the displacement. Using equation (1) the rate of change of momentum can be expressed as

\dot{p} = \frac{d}{d t} (\frac{m_{0} \dot{x}}{\sqrt{1 - {(\dot{x} / c)}^{2}}}) = \frac{m_{0}}{{[1 - {(\dot{x} / c)}^{2}]}^{3 / 2}} \ddot{x}

(3)

From equations (2) and (3) the equation of motion of the relativistic oscillator is derived thus

\ddot{x} + ω_{0}^{2} {[1 - {(\frac{\dot{x}}{c})}^{2}]}^{3 / 2} x = 0

(4)

where

ω_{0} = \sqrt{K_{L} / m_{0}}

is the natural frequency of the nonrelativistic linear oscillator and the initial conditions are

x (0) = 0

and

\dot{x} (0) = V_{0}

. The nonlinearity in equation (4) is due to the relativistic effect. At first glance, equation (4) seems to be the motion of a damped oscillator because of the presence of

{\dot{x}}^{2}

but the relativistic oscillator is actually a conservative one with an energy balance given by¹⁹

\frac{m_{0} c^{2}}{\sqrt{1 - {(\dot{x} / c)}^{2}}} + \frac{1}{2} K_{L} x^{2} = m_{0} c^{2} + \frac{1}{2} K_{L} a^{2}

(5)

where

a

is the amplitude of the dimensional displacement. The

{\dot{x}}^{2}

acts as a coordinate parameter rather than a parameter producing a damping effect.¹⁰ By introducing nondimensional variables

X = ω_{0} x / c

and

τ = ω_{0} t

, equation (4) can be written in nondimensional form:

X^{″} + {[1 - {(X^{′})}^{2}]}^{3 / 2} X = 0

(6)

with initial conditions as

X (0) = 0

and

X ′ (0) = β_{0} = V_{0} / c

, where

V_{0}

is the initial dimensional velocity. The prime in equation (6) represents differentiation w.r.t. the nondimensional time,

τ

, and

0 < X ′ < 1

. In the limit that

X ′ \to 0

equation (6) behaves like a typical nonrelativistic linear harmonic oscillator with a time period that is independent of the displacement amplitude. On the other hand, when

X ′ \to 1

the mass is approaching the speed of light and the time-period is strongly dependent on the displacement amplitude.

Conservative model of the relativistic oscillator

The general form of a conservative oscillator is $\ddot{x} + F (x) = 0$ and $F (x)$ is the restoring force, which is a function of the displacement. In the previous section, the equation of motion of the relativistic oscillator derived (see equation (4)) is in the form $\ddot{x} + F (x, \dot{x}) = 0$ , which gives the false impression that the relativistic oscillator is a dissipative system, but equation (5) shows clearly that it is a conservative system. In order to derive the conservative equation of motion for the relativistic oscillator, equation (1) can be rewritten as

\dot{x} = \frac{1}{m_{0}} \frac{p}{\sqrt{1 + p^{2} / m_{0}^{2} c^{2}}}

(7)

Differentiating equation (2) gives $\ddot{p} + K_{L} \dot{x} = 0$ and substituting equation (7) in this equation, we get

\ddot{p} + ω_{0}^{2} \frac{p}{\sqrt{1 + p^{2} / m_{0}^{2} c^{2}}} = 0

(8)

Equation (8) is the conservative equation of motion for the relativistic oscillator expressed in terms of momentum, and the momentum restoring force is $F (p) = p / \sqrt{1 + p^{2} / m_{0}^{2} c^{2}}$ . The initial conditions to equation (8) are $p (0) = p_{0}$ and $\dot{p} (0) = 0$ where $p_{0} = m_{0} V_{0} / \sqrt{1 - {(V_{0} / c)}^{2}}$ . By introducing the nondimensional variables $P = p / m_{0} c$ and $τ = ω_{0} t$ the nondimensional form of equation (8) is

P ″ + \frac{P}{\sqrt{1 + P^{2}}} = 0

(9)

Using equation (2), the nondimensional momentum can be expressed as $P = β / \sqrt{1 - β^{2}}$ where $β = \dot{x} / c$ is a nondimensional velocity representing the ratio of the particle velocity to the speed of light. Hence, the initial conditions to equation (9) are $P (0) = P_{0}$ and $P ′ (0) = 0$ where $P_{0} = β_{0} / \sqrt{1 - β_{0}^{2}}$ .

Equation (9) has been derived by Mickens¹⁸ and used in other studies to derive approximate periodic solutions to the relativistic oscillator.^10–12 Mickens¹⁸ derived equation (9) from equation (6) using a transformation variable and this was done from purely mathematical considerations. Consequently, the physical meaning of the transformation variable remained ambiguous. Here, equation (9) has been derived in a simple and straightforward manner by normalizing the conservative equation governing the momentum response. From the present derivation, it is clear that the transformation variable used by Mickens¹⁸ is the nondimensional momentum restoring force. The latter is a function of the nondimensional momentum, which is a ratio of the relativistic momentum to the momentum of the rest mass moving at the speed of light.

Periodic solution of the relativistic oscillator

Basic idea of the CPLM

The basic idea of the CPLM is the piecewise discretization and linearization of the nonlinear compliance curve, which is a plot of the nonlinear restoring force against the displacement. The linearized restoring force produces a standard nonhomogeneous linear differential equation. The solution to the latter gives an approximate solution to the original nonlinear differential equation over a time range that is automatically determined by the CPLM algorithm. As the algorithm moves from one discretization to the other, the constants in the approximate solution and the time range for which the solution is applicable are updated, thereby making it possible to obtain approximate solutions for a complete oscillation cycle. Further details on the piecewise discretization and linearization technique of the CPLM can be obtained from the literature.^15,20,21

Now, let us consider a general 1-D nonlinear conservative system represented as $\ddot{u} + F (u) = 0$ where $F (u)$ is the restoring force and the displacement occurs in the $u$ -direction. The linearized force for each $n$ discretization of the nonlinear compliance is given as¹⁵

F_{r s} (u) = K_{r s} (u - u_{r}) + F_{r}

(10)

where

K_{r s} = [F (u_{s}) - F (u_{r})] / (u_{s} - u_{r})

is the linearized stiffness for the discretization bounded by points

r

and

s

, and

F_{r} = F (u_{r})

. Also,

r = 0, 1, 2, 3, \dots, n - 1

and

s = r + 1

are the start and end points of each discretization. If equation (10) is substituted into the general equation of motion for a conservative system, we arrive at the standard linear differential equation for each discretization as follows

\ddot{u} + K_{r s} u = K_{r s} u_{r} - F_{r}

(11)

The displacement and velocity for equation (11) are respectively given as

u (t) = R_{r s} \sin (ω_{r s} t + φ_{r s}) + C_{r s}

(12)

and

\dot{u} (t) = ω_{r s} R_{r s} \cos (ω_{r s} t + φ_{r s})

(13)

where the constants in equations (12) and (13) are calculated as:

R_{r s} = {[{(u_{r} - C_{r s})}^{2} + {({\dot{u}}_{r} / ω_{r s})}^{2}]}^{1 / 2}

ω_{r s} = \sqrt{K_{r s}}

, and

C_{r s} = u_{r} - F (u_{r}) / K_{r s}

. The initial conditions and other parameters (

φ_{r s}

and

Δ t

) are determined based on the stage of the oscillation. For the negative velocity oscillation stage, the initial conditions for each discretization are

u_{r} (0) = A - r Δ u

and

{\dot{u}}_{r} (0) = - \sqrt{| 2 \int_{A}^{u_{r}} - F (u) d u |}

; where

Δ u = A / n

and the other parameters are calculated as

φ_{r s} = {\begin{array}{l} 0.5 π {\dot{u}}_{r} = 0 \\ π + \tan^{- 1} [ω_{r s} (u_{r} - C_{r s}) / {\dot{u}}_{r}] {\dot{u}}_{r} < 0 \end{array}; and

Δ t = {\begin{array}{l} (0.5 π - φ_{r s}) / ω_{r s} (u_{s} - C_{r s}) \geq R_{r s} \\ (0.5 π + \cos^{- 1} [(u_{s} - C_{r s}) / R_{r s}] - φ_{r s}) / ω_{r s} (u_{s} - C_{r s}) < R_{r s} \end{array}

For the positive velocity oscillation stage, the initial conditions for each discretization are $u_{r} (0) = - A + r Δ u$ and ${\dot{u}}_{r} (0) = \sqrt{| 2 \int_{A}^{u_{r}} - F (u) d u |}$ , whereas the other parameters are calculated as

φ_{r s} = {\begin{array}{l} - 0.5 π {\dot{u}}_{r} = 0 \\ \tan^{- 1} [ω_{r s} (u_{r} - C_{r s}) / {\dot{u}}_{r}] {\dot{u}}_{r} > 0 \end{array}; and

Δ t = {\begin{array}{l} (0.5 π - φ_{r s}) / ω_{r s} (u_{s} - C_{r s}) \geq R_{r s} \\ (0.5 π - \cos^{- 1} [(u_{s} - C_{r s}) / R_{r s}] - φ_{r s}) / ω_{r s} (u_{s} - C_{r s}) < R_{r s} \end{array}

The time at the end of each discretization can now be calculated as $t_{s} = t_{r} + Δ t$ and the end conditions $u_{s}$ and ${\dot{u}}_{s}$ are calculated by replacing $r$ with $s$ in the formulae for initial conditions.

CPLM solution for the relativistic oscillator

The CPLM solution in equations (12) and (13) can be applied to the dimensional momentum equation (see equation (8)) or the nondimensional momentum equation (see equation (9)), but the nondimensional equation will be used here. Hence, the nondimensional momentum restoring force is $F (P) = P / \sqrt{1 + P^{2}}$ and the CPLM solutions for the nondimensional momentum and momentum rate are given as

P (τ) = R_{r s} \sin (ω_{r s} τ + φ_{r s}) + C_{r s}

(14)

and

P ′ (τ) = ω_{r s} R_{r s} \cos (ω_{r s} τ + φ_{r s})

(15)

respectively. The constants in equations (14) and (15) are calculated using the expressions in the previous section by replacing

(u, t)

with

(P, τ)

. Having obtained the solution of the nondimensional momentum, the solutions for the displacement and velocity of the relativistic oscillator can now be determined. To achieve this equation (2) is rewritten as

x (t) = - \frac{1}{K_{L}} \dot{p} = - \frac{m_{0} c ω_{0}}{K_{L}} P ′ (τ)

(16)

Substituting equation (15) in equation (16) and simplifying, we get

x (t) = - \frac{c ω_{r s} R_{r s}}{ω_{0}} \cos (ω_{r s} τ + φ_{r s})

(17)

The nondimensional displacement can be derived from equation (17) as

X (τ) = - ω_{r s} R_{r s} \cos (ω_{r s} τ + φ_{r s})

(18)

Again, from equation (2) we can write

\dot{x} (t) = - \frac{1}{K_{L}} \ddot{p} = - \frac{m_{0} c ω_{0}^{2}}{K_{L}} P ″ (τ)

(19)

Differentiating equation (15) and substituting the result in equation (19), the corresponding CPLM solution for the velocity is obtained as shown

\dot{x} (t) = c K_{r s} R_{r s} \sin (ω_{r s} τ + φ_{r s})

(20)

The nondimensional velocity can be derived from equation (20) as

X ′ (τ) = K_{r s} R_{r s} \sin (ω_{r s} τ + φ_{r s})

(21)

From the above, it is obvious that the CPLM solution for the relativistic oscillator is much simpler than previously published approximate analytical solutions.^10–14,18 Also, we note that the nondimensional displacement is equal to the negative of the nondimensional momentum (see equations (15) and (18)), and the nondimensional velocity is equal to the nondimensional momentum restoring force (i.e. from equation (9), $F (P) = - P^{″} (τ),$ which can be obtained from equation (15)).

Results and discussions

Previous studies^15,17 have shown that complete validation of a solution method for nonlinear oscillators requires testing of the oscillation frequency (or time period) and the displacement and velocity profiles. This validation process is applied in respect of the CPLM results presented and discussed here.

Table 1 summarizes the CPLM time period estimates for different values of the nondimensional amplitude in comparison with exact numerical solutions and estimates of other published methods.^10,12 The initial conditions corresponding to the amplitudes are also given. The numerical solutions were obtained by direct numerical integration of equation (9) using the “StiffnessSwitching” algorithm of the NDSolve function in Mathematica™. The StiffnessSwitching algorithm is particularly useful for the solution of highly nonlinear and stiff systems for which other numerical schemes may not reach a convergent solution.

Table 1.

Comparison of CPLM time period estimates with results of exact and other approximate methods.

	Initial condition		Time period with percentage relative error
$A$	$P_{0}$	$β_{0}$	HBM¹⁰	HPM¹²	CPLM	Exact
0.1	0.1001249220	0.099626788032342	6.29496(0.001475899)	6.29496(0.001475935)	6.29496(0.001429695)	6.29505
1	1.1180339887	0.745355992499930	7.41149(0.417270865)	7.37263(0.10923942)	7.38082(0.001761353)	7.38069
2	2.8284271247	0.942809041582063	–	–	10.0368(0.000996343)	10.0367
5	13.4629120178	0.997252742061945	–	–	20.8303(0.000960149)	20.8301
10	50.9901951359	0.999807747763291	40.12507(0.697224711)	40.07185(0.828954563)	40.4071(0.000742449)	40.4068
20	200.9975124224	0.999987623992128	80.25015(0.06103102)	80.14369(0.071704693)	80.2023(0.001371551)	80.2012
30	450.9988913512	0.999997541798641	120.37522(0.200793627)	120.21554(0.067872512)	120.136(0.001664808)	120.134
40	800.9993757800	0.999999220699165	160.50030(0.250028246)	160.28738(0.117041818)	160.103(0.001873829)	160.1
50	1250.9996003197	0.999999680511335	200.62537(0.272575483)	200.35923(0.139559146)	200.083(0.0014994)	200.08
60	1800.9997223764	0.999999845850325	240.75044(0.284688371)	240.43108(0.151655965)	240.071(0.001666202)	240.067
70	2450.9997960016	0.999999916769328	280.87552(0.292267873)	280.50292(0.159225413)	280.062(0.001785351)	280.057
80	3200.9998437988	0.999999951202377	321.00059(0.297013105)	320.57477(0.16396435)	320.051(0.000312451)	320.05
90	4050.9998765737	0.999999969531889	361.12566(0.300425572)	360.64661(0.16737229)	360.051(0.001944207)	360.044
100	5000.9999000200	0.999999980007997	401.25074(0.302654248)	400.71846(0.169598009)	400.047(0.001749825)	400.04

HBM: harmonic balance method; HPM: homotopy perturbation method; CPLM: continuous piecewise linearization method.

The percentage relative errors shown in bracket were calculated as $100 \times | T - T_{e x} | / T_{e x}$ where $T_{e x}$ is the exact period.

The percentage relative error of the time period estimates are given in brackets. The CPLM results were generated for 50 discretization and the maximum error for the time period estimates is less than 0.002%. This is two orders more accurate than the maximum errors of the HBM and HPM results. The second-order approximations for the frequency–amplitude relations of the HBM¹⁰ and HPM¹² used in Table 1 are given in the appendix and were derived in terms of elliptic functions. The series expansion of the elliptic function solutions were only derived for small- ( $A \leq 1$ ) and large-amplitude ( $A \geq 10$ ) oscillations. Consequently, the HBM and HPM time period estimates for amplitudes in the range $1 < A < 10$ are not given in Table 1 because very large errors occur if the series solutions for small- or large-amplitude oscillations are applied in this range.

The oscillation histories of the relativistic oscillator for different nondimensional amplitudes are shown in Figures 1 to 5. The figures show a perfect match between the CPLM results (line plots) and the corresponding exact numerical results (marker plots). The amplitudes considered cover the nonrelativistic (Figure 1), relativistic (Figures 2 and 3) and ultra-relativistic (Figure 4 and 5) responses of the relativistic oscillator. In the nonrelativistic regime, the velocity of the mass is much less than the speed of light so that the nonlinear relativistic effect is negligible. This condition will generally hold true for $β_{0} \leq 0.1$ . Therefore, the oscillations are purely sinusoidal and the time period is independent of the displacement amplitude. For the relativistic regime ( $0.1 < β_{0} < 0.99$ ) the oscillations are no longer sinusoidal and the time period is determine by the displacement amplitude and the spring stiffness.²² As $β_{0}$ increases the displacement history looks more like a saw-tooth wave while the velocity history looks more like a square wave with curvature occurring only around the turning points.^10,12 In the ultra-relativistic regime ( $β_{0} > 0.99$ ) the displacement history is approximately a saw-tooth wave while the velocity history is a square wave. Hence, the displacement of the ultra-relativistic regime can be likened to the path of a light photon bouncing back and forth between two parallel mirrors.²² The time period–amplitude relation becomes a straight line (Figure 6) and this shows a strong dependence of the time period on the displacement amplitude. Previous studies²² have demonstrated that the time period in the ultra-relativistic regime is governed by the expression $T = 4 a / c$ , which is a linear relationship between the time period and the amplitude with a slope of $4 / c$ . Hence, the behavior predicted in Figure 6 is in order. The implication is that the effect of the spring stiffness on the time period is negligible during ultra-relativistic vibrations. It should be noted that the boundaries for the three regimes in terms of $β_{0}$ are based on observations from the simulations. No mathematical or theoretical justification is claimed and therefore, the boundaries are only approximate with possibility of overlaps.

Figure 1.

Periodic oscillations in the nonrelativistic regime: $A = 0.1$ and $β_{0} = 0.0996268$ .

Figure 2.

Periodic oscillations in the relativistic regime: $A = 1$ and $β_{0} = 0.74536$ .

Figure 3.

Periodic oscillations in the relativistic regime: $A = 2$ and $β_{0} = 0.94281$ .

Figure 4.

Periodic oscillations in the ultra-relativistic regime: $A = 10$ and $β_{0} = 0.99981$ .

Figure 5.

Periodic oscillations in the ultra-relativistic regime: $A = 100$ and $β_{0} = 0.99999998$ .

Figure 6.

Time period–amplitude response in the ultra-relativistic regime.

Conclusions

Much of the existing literature on the periodic solution of the relativistic oscillator is too advanced for senior year undergraduates. As a result the physical insight to be gained from this conservative oscillator is generally not accessible to most undergraduates in relevant discipline where the traditional linear harmonic oscillator is being taught. In this paper, the CPLM algorithm is used to present very simple and accurate solutions to the relativistic oscillator that can be taught alongside the simple harmonic oscillator in relevant undergraduate courses. Knowledge of freshman calculus is sufficient to comprehend and implement the CPLM solution for the relativistic oscillator. The CPLM solutions for each discretization (equations (14), (15), (17), (18), (20), and (21)) are in the form of simple harmonic response, which mechanical engineering undergraduates are already familiar with from freshman and sophomore courses. This justifies introducing the relativistic oscillator and the corresponding CPLM solutions to mechanical engineering undergraduates.

The results of the present study show that the CPLM solution is valid for all range of amplitudes (from nonrelativistic to ultra-relativistic) and that the errors in its predictions for $n = 50$ are less than 0.002%, which is remarkably excellent given its simplicity. The CPLM solution can be implemented by hand calculations for fewer discretization ( $n = 5$ ) with reasonable accuracy.¹⁵ Also, it can be implemented easily using MS Excel spreadsheet or simple Matlab or Mathematica™ codes. Nevertheless, the CPLM algorithm can only produce closed-form analytical solutions for conservative systems but not dissipative systems. Consequently, the CPLM algorithm can only be applied to the damped relativistic oscillator as a semi-analytic construct²¹ in which the time at the end of each discretization is calculated using numerical schemes for finding roots e.g. Newton–Raphson method.

An important deduction from the CPLM solution is the relationship between the nondimensional displacement ( $X (τ)$ ) and the nondimensional momentum rate ( $P ′ (τ)$ ) on the one hand, and the nondimensional velocity ( $X ′ (τ)$ ) and the nondimensional momentum restoring force ( $F (P)$ ) on the other hand. This explicit relationship simplifies the procedure for obtaining $X (τ)$ and $X ′ (τ)$ once $P (τ)$ has been determined from equation (9). Thereafter, the definitions of $X (τ)$ and $X ′ (τ)$ can be used to determine the history of the dimensional displacement ( $x (t)$ ) and velocity ( $\dot{x} (t)$ ) for given values of the rest mass and linear spring constant.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix

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