The damped oscillator model: a harmonic restoring force with a phenomenological linear drag $-c\dot{x}$, $c\ge 0$

(i) Setting. Work in the one-dimensional model class of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria with time window $I_0=\mathbb{R}$, configuration domain $Q=\mathbb{R}$, a fixed mass $m>0$, the spring constant $k>0$ of The harmonic (linear restoring) model: a declared linear force law with parameter $k>0$, and a drag coefficient $c\ge 0$ of dimension $\mu {\tau }^{-1}$, read in kilograms per second (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The model's force law is declared through the additive composition of force contributions, in the sense of clause (v) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, read in one coordinate: $F(x,v)=-kx-cv\mathrm{.}$

The damped oscillator model is the one-dimensional model with this force law; its equation of motion is $m{x}^{\mathrm{\prime }\mathrm{\prime }}+c{x}^{\mathrm{\prime }}+kx=0,$ and its data $(m,c,k,0)$ are oscillator-type data in the sense of clause (f) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria. The model is autonomous and depends on the velocity; it is undamped when $c=0$, in which case it is exactly the harmonic model of The harmonic (linear restoring) model: a declared linear force law with parameter $k>0$, and strictly damped when $c>0$.

(ii) Potential presentation with a residual force. The model presents its force law through the harmonic potential $U(x)=\frac{1}{2}k{x}^2$ of The harmonic (linear restoring) model: a declared linear force law with parameter $k>0$ together with the residual force $d(t,x,v)=-cv,$ in the sense of clause (d) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria: $F(x,v)=-U^{\mathrm{\prime }}(x)+d(t,x,v)$. The mechanical energy of the model is therefore the function $E(x,v)=\frac{1}{2}m{v}^2+\frac{1}{2}k{x}^2$ of clause (d), defined along every motion; the drag force is not presented through a potential and the model is not a global potential model when $c>0$.

(iii) Regularity. Let $X:=x_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{m})$ and $V:=v_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{m}\mathrm{/}\mathrm{s})$ be the dimensionless numerical SI state coordinates (the real state coordinates in the displayed force-map signature), and let $K:=k_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{N}\mathrm{/}\mathrm{m})$, $C:=c_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{s})$ be the numerical coefficients. The numerical force map $\widehat{F}:=F_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{N})$ is $\widehat{F}(X,V)=-KX-CV$, so $\mid \widehat{F}(X,V)-\widehat{F}(X^{\mathrm{\prime }},V^{\mathrm{\prime }})\mid \le K\mid X-X^{\mathrm{\prime }}\mid +C\mid V-V^{\mathrm{\prime }}\mid \le (K+C)\parallel (X,V)-(X^{\mathrm{\prime }},V^{\mathrm{\prime }}){\parallel }_2\mathrm{.}$ The real number $K+C$ is a Lipschitz constant for this normalized numerical coordinate map; the dimensionful coefficients $k$ and $c$ are not added. The force map is continuous and locally Lipschitz in the state variable, uniformly in time (Local Lipschitz continuity in the state variable, locally uniform in time and parameters). It is an admissible force law of the class of clause (a) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria, and its equation is exactly the homogeneous model $m{x}^{\mathrm{\prime }\mathrm{\prime }}+c{x}^{\mathrm{\prime }}+kx=0$ of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data.

(iv) Physical status and regime. The drag term is a phenomenological constitutive model, adopted rather than derived: it represents the net force of a medium whose microscopic state is not part of the model. The linear velocity-proportional form is a restricted regime — slow motion through a viscous medium, of which the Stokes force $c=6\pi \eta a$ is the recorded example in the framework's drag postulate The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient — and the coefficient $c$ is a model parameter depending on body and medium. Other regimes are different models: drag growing quadratically with speed, or Coulomb sliding friction, whose force contains the factor $v\mathrm{/}\mid v\mid$ and is undefined at $v=0$ and is not of the standing regularity. The model also neglects the random component of the environment's force; the recorded source states that this neglect is admissible only in a stated energy regime, in which the oscillator's energy greatly exceeds the energy scale of the environment's fluctuations. No value of $c$, $m$ or $k$ is asserted, and a measured value used in an application is an empirical premise with its own conditions and uncertainty.

(v) Status. Adopted assumption of the framework when a model invokes it; the damped oscillator selects no law of nature on its own, and no fluctuation–dissipation relation is asserted. Its conditional consequences are the subject of the damped-motion theorems of this page.

• Why a postulate and not a theorem. The equation of motion determines no force law; the drag contribution is additional model input, exactly as in The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient. The linear form is the simplest velocity-proportional law of the standing regularity, and it is the form of the recorded viscous-regime treatments; neither reason makes it a derivation.

• $c=0$ is a parameter value, not a change of model class. The model is declared for all $c\ge 0$; at $c=0$ the residual force vanishes and the model is a global potential model, while for $c>0$ the mechanical energy is not conserved and the page's dissipation theorem applies.