Authors : Muhammad Umar Jabbar
DOI : 10.2139/ssrn.6846759
Volume : 1
Issue : 1
Year : 2026
Page No : 11
The classical virial theorem, 2⟨T⟩ = 2n⟨V ⟩, holds only for conservative systems with a pure <div> power-law potential V ∝ x </div> <div> 2n; it breaks down once damping and periodic forcing are intro- </div> <div> duced, as in most real driven oscillators. This paper derives an exact identity for the time- </div> <div> averaged energy partition of a periodically driven, linearly damped, non-harmonic oscillator, </div> <div> mx¨ + γx˙ + kx2n−1 = F0 cos(Ωt), in its non-equilibrium steady state (NESS). Multiplying the </div> <div> equation of motion by x and time-averaging over the NESS, where the boundary term ⟨ </div> <div> d </div> <div> dt (xx˙)⟩ </div> <div> vanishes, yields 2⟨T⟩ = 2n⟨V ⟩ + 2ζ⟨xx˙⟩ − ⟨x cos(Ωt)⟩, with ζ = γ/2m. The identity is exact </div> <div> and holds for all n ≥ 1, ζ > 0, k > 0, and Ω > 0, reducing to the classical virial theorem as </div> <div> ζ → 0 and F0 → 0. Numerical verification by adaptive eighth-order Runge–Kutta integration </div> <div> (SciPy DOP853) confirms the identity to within 0.13% across thirty independent parameter </div> <div> combinations spanning n ∈ {1, . . . , 6} and ζ ∈ {0.05, . . . , 0.25}, with residuals consistent with </div> <div> floating-point precision. The two correction terms — a damping cross-correlation and a forc- </div> <div> ing cross-correlation — define a measurable scalar, the virial defect ∆, shown to be negative </div> <div> throughout the tested damping range for the Duffing case (n = 2), indicating systematically less </div> <div> kinetic energy than the classical prediction. The virial defect thus provides a directly measur- </div> <div> able, parameter-free diagnostic of departure from virial equilibrium, with physical implications </div> <div> for nonlinear mechanical energy accounting and for driven RLC circuits with nonlinear capaci- </div> <div> tance. </div>