The relativistic feature of a non-dissipative classical harmonic oscillator restored by a relativistic linear Hooke force

The relativistic feature of a non-dissipative classical harmonic oscillator (CHO) is investigated due to its vast applications in science and technology [H. Nieto-Chaupis, in International Conference on Engineering and Emerging Technologies (ICEET) (IEEE, Dubai, 2024), pp. 1–4; J. Jahanpanah et al., Int. J. Quantum Chem. 122, e26911–e26918 (2022); J. Li et al., Sci. Rep. 15, 7534 (2025); G. Liu et al., Int. J. Non-Linear Mech. 100, 10–19 (2018); and N. Sharif et al., J. Low Freq. Noise, Vib. Act. Control 43, 250–262 (2023)]. The semi-relativistic aspect has already been studied by solving the nonlinear differential equation Frel=dprel/dt⌣=−kX using various approximate methods [M. P. Solon and J. P. H. Esguerra, Phys. Lett. A 372, 6608–6612 (2008); D. Clark et al., Eur. J. Phys. 33, 1041–1051 (2012); J. Biazar and M. Hosami, J. Egypt. Math. Soc. 22, 45–49 (2014); E. H. Hutten, Nature 205, 892 (1965); R. E. Mickens, J. Sound Vib. 212(5), 905–908 (1998); and A. Beléndez et al., J. Sound Vib. 311, 1447–1456 (2008)]. The main aim of this research is to extend the feature of a CHO from the present semi-relativistic state to a relativistic state by resolving the latter nonlinear differential equation in the presence of a relativistic linear Hooke force rather than the simple non-relativistic restoring force −kX. Consequently, the second-order nonlinear differential equation x¨+αrel2β1−x⋅232x=0 (x=ω0cX) is constructed and simplified to the semi-relativistic case when the relativistic coefficient αrelβ tends toward unity (0<αrelβ≤1). The solution is carried out up to second-order approximation by the harmonic balance method and first-order approximation by the Krylov–Bogoliubov–Mitropolsky method. The results indicate that the restitution coefficient krelβ, amplitude Brelβ, and angular frequency ωrelβ of a relativistic CHO are simultaneously changed by varying the normalized initial relativistic velocity β=x⋅0/c (0≤β<1). It is finally demonstrated that the sum of kinetic and potential energies of a relativistic CHO satisfies the energy conservation law.