Scope and limits of the one-dimensional conservative and dissipative models: declared class, approximation status, empirical parameters and open extensions

The one-dimensional class is declared, not derived. The models of this page are restricted model classes of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria: a scalar coordinate, a mass and a scalar force law with the standing regularity. Their relation to motions of the three-dimensional force branch along a straight line, or to effective one-dimensional reductions of constrained systems such as the pendulum, is a modelling identification that a consumer item states and uses explicitly; this page contains no three-dimensional reduction theorem, and the pendulum example of the companion page declares its effective model equation with its source support.

Declared assumptions versus deductions. The harmonic, damped and driven models are adopted postulates (The harmonic (linear restoring) model: a declared linear force law with parameter $k>0$, The damped oscillator model: a harmonic restoring force with a phenomenological linear drag $-c\dot{x}$, $c\ge 0$, The sinusoidally driven oscillator model: a prescribed external force $f_0\cos (\mathrm{\Omega }t)$ added to the damped oscillator); every physics theorem on the page is a conditional deduction from those adopted models: existence and uniqueness, energy conservation and level-set containment (Existence, uniqueness and conservation of energy for a one-dimensional conservative model: phase curves lie in energy level sets and the velocity is determined by the position and energy up to sign), Lyapunov stability at a strict minimum (A strict local minimum of the potential is a Lyapunov stable equilibrium of the conservative one-dimensional model, and its linearisation is an oscillator-type model), the sinusoid classification, energy and phase ellipses (The motions of the harmonic model are the sinusoids $A\cos (\omega (t-t_0))+B\sin (\omega (t-t_0))$: amplitude and phase, the least period, the conserved energy $k{R}^2\mathrm{/}2$ and the elliptical phase curves), the damping classification and the dissipation identity (Classification of the motions of the damped oscillator by the discriminant, and nonincreasing energy $E^{\mathrm{\prime }}=-c{x}^{\mathrm{\prime }2}$), decay and asymptotic stability (Strictly damped oscillator: explicit decay bounds, convergence to rest and asymptotic stability of the origin), and the response, boundedness and resonance statements (The driven oscillator: steady response and amplitude profile, bounded damped and off-resonant motions, and unbounded undamped resonance). No deduction here establishes that a real system obeys a model.

The harmonic model is a local quadratic approximation. In the displacement coordinate $q=x-x_{\ast }$ centered at a strict minimum with $U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })>0$, the quadratic Taylor term is $\frac{1}{2}{U}^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })q^2$ after dropping the additive constant $U(x_{\ast })$. It gives the linearized equation about that minimum; the approximation is exact only for the declared harmonic model. For the pendulum, the difference between the nonlinear and the linearized motion is bounded only in the finite-time sense of The pendulum field near the downward equilibrium: bounded force derivative, cubic remainder and the finite-time small-angle error bound, with the explicit small-amplitude and time hypotheses of that statement; no uniform-in-time approximation is asserted anywhere on this page.

The drag is phenomenological and fluctuation-free. The linear drag model is adopted for a stated viscous regime; its coefficient depends on body and medium; the random component of the environment's force is neglected. Likharev's Section 5.1, footnote 4 (Chapter 5 p. 2; PDF p. 88), limits that omission to oscillator energy much larger than the fluctuation energy scale. In the source's thermal-equilibrium setting, write $T\ge 0$ for absolute temperature in kelvins, $k_B>0$ for the Boltzmann constant in joules per kelvin, $\mathrm{\hslash }>0$ for the reduced Planck constant in joule seconds, and ${\omega }_0=\sqrt{k\mathrm{/}m}$ in inverse seconds. The reported scale is $E_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{c}}:=\max \{k_{B}T,\mathrm{\hslash }{\omega }_0\mathrm{/}2\}$, compared with $E_{\mathrm{o}\mathrm{s}\mathrm{c}}:=m{v}^2\mathrm{/}2+k{x}^2\mathrm{/}2$ above the declared potential minimum. The condition is $E_{\mathrm{o}\mathrm{s}\mathrm{c}}\gg E_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{c}}$: the ratio $E_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{c}}\mathrm{/}{E}_{\mathrm{o}\mathrm{s}\mathrm{c}}$ must be small where this approximation is used. The source gives no universal numerical tolerance or error estimate. These thermal and quantum quantities report a source's applicability caveat; they are not premises of the page's deterministic proofs or constants derived within this mechanical framework. Since the adopted strictly damped equation predicts $E_{\mathrm{o}\mathrm{s}\mathrm{c}}\to 0$, its exact long-time decay theorem does not assert indefinite physical applicability when $E_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{c}}>0$. No fluctuation–dissipation relation, derivation of $c$ or validity outside the linear regime is claimed; regimes of quadratic drag or of Coulomb friction are different models outside this page.

The drive is prescribed and sinusoidal. The driving-force contribution is a function of time alone with an idealized sinusoidal waveform; the total force also includes the declared spring and drag terms. No back-reaction of the motion on the drive, no non-sinusoidal waveform and no apparatus characterization is part of the model. The boundedness and resonance clauses of The driven oscillator: steady response and amplitude profile, bounded damped and off-resonant motions, and unbounded undamped resonance are statements about this model's motions.

Measured parameters are empirical premises. The values of $m$, $k$, $c$, $f_0$ and $\mathrm{\Omega }$ are model data; when a value is obtained from a calibration or an apparatus specification, its uncertainty, the calibration method and the operating conditions are premises of any comparison with observation, and agreement of a model prediction with data is statistical evidence about the combined model and apparatus assumptions, never a proof of the model. This page asserts no measured value and no empirical agreement.

Stability conclusions and their boundaries. The conservative model has a Lyapunov stable rest point at a strict minimum; the strictly damped model has a globally asymptotically stable rest point; the strict-minimum stability conclusion also covers degenerate strict minima, while no conclusion is drawn at other degenerate critical points, at maxima, or for the undamped driven model. In particular the linearized repulsive frequency of a maximum is not used to claim nonlinear instability of the model.

Open extensions. Nonlinear oscillators with large amplitude (the exact pendulum and its secular drift), parametric resonance, nonlinear damping and self-oscillation, dissipative models with fluctuations, and numerical integration studies are outside this page's scope and belong to the later pages of the track; the recorded sources' treatments of those topics are not prerequisites of the claims proved here.