The harmonic (linear restoring) model: a declared linear force law with parameter $k>0$

(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 $m>0$ and a spring constant $k>0$ of dimension $\mu {\tau }^{-2}$, read in newtons per metre (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The harmonic model is the model whose force law is the linear restoring force $F(x,v):=-kx\mathrm{.}$ Its equation of motion is $m{x}^{\mathrm{\prime }\mathrm{\prime }}=-kx$, equivalently $m{x}^{\mathrm{\prime }\mathrm{\prime }}+kx=0$; the model is autonomous and velocity-independent, and its potential is $U(x):=\frac{1}{2}k{x}^2,$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ (joules), which presents the force law in the sense of clause (d) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria: $U$ is of class $C^2$ in that clause's sense, and $-U^{\mathrm{\prime }}(x)=-kx=F(x)$ (For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set). The harmonic model is therefore a conservative model of the class, with the potential representative $U$ declared above and no residual force.

(ii) Regularity and framework status. 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 $K:=k_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{N}\mathrm{/}\mathrm{m})$ be the numerical spring constant. In this representation the numerical force map $\widehat{F}:=F_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{N})$ is $\widehat{F}(X,V)=-KX$, so $\mid \widehat{F}(X,V)-\widehat{F}(X^{\mathrm{\prime }},V^{\mathrm{\prime }})\mid \le K\parallel (X,V)-(X^{\mathrm{\prime }},V^{\mathrm{\prime }}){\parallel }_2$. It is therefore locally Lipschitz in the state variable with global numerical-coordinate constant $K$, uniformly in time (Local Lipschitz continuity in the state variable, locally uniform in time and parameters). The harmonic model is thus an admissible model of the class of clause (a) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria, and the existence, uniqueness and conservation statements of this page apply to it. It is adopted as a restricted model class of the framework, at the same primitive level as the force maps of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: the force law is model input.

(iii) Physical reading and regime. The model idealizes a restoring force linear in the displacement: an ideal spring, or the quadratic approximation of a potential $U$ near a strict local minimum $x_{\ast }$ with $U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })>0$, where the Taylor expansion of the effective potential begins with the quadratic term $\frac{1}{2}{U}^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })(x-x_{\ast }{)}^2$ and the linearized equation carries the effective spring constant $U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })$ (the sources' linearization procedure). The model is adopted for a stated regime — displacements so small that the neglected remainder of the potential is adequate for the purpose at hand — and it is not claimed to describe large-amplitude oscillations, nonlinear springs, plasticity, or any given real system. The coefficient $k$ is a model parameter, not a constant of the framework; when its value is taken from a calibration experiment, that value is an empirical premise carrying the apparatus conditions and the measurement uncertainty, and the framework asserts none of it here. The linear form is the simplest restoring law satisfying the standing regularity; that is a modelling convenience, not a derivation.

(iv) Status. Adopted assumption of the framework when a model invokes it; it selects no law of nature on its own. Every consequence below is a conditional statement about the declared equation of motion.

• Zero, negative and nonlinear alternatives are different models. For $k=0$ the model is the free model $m{x}^{\mathrm{\prime }\mathrm{\prime }}=0$; for $k<0$ the force is repulsive near the origin; a force law with a cubic term is a different model. For the ideal planar point pendulum, declare a bob mass $M>0$ (kilograms), length $l>0$ (metres), fixed pivot, and constant downward gravitational acceleration $g>0$ (metres per second squared), with no friction. Its angular coordinate is a $C^2$ curve $\theta :J\to \mathbb{R}$ on an open time interval $J$ read in seconds; $\theta$ is the angle from the downward vertical, read in dimensionless radians. The angular model adopts inertia $\mathcal{I}:=M{l}^2$ (kilogram metres squared) and the scalar torque map $\mathcal{T}:\mathbb{R}\to \mathbb{R}$, $\mathcal{T}(\theta ):=-Mgl\sin \theta$ (newton metres), with equation $\mathcal{I}{\theta }^{\mathrm{\prime }\mathrm{\prime }}=\mathcal{T}(\theta )$, equivalently ${\theta }^{\mathrm{\prime }\mathrm{\prime }}=-(g\mathrm{/}l)\sin \theta$. Here sine is the everywhere-convergent real power series of Sine and cosine defined by their real power series. This is the recorded source's point-pendulum equation (Likharev, Section 1.2, Eq. (1.18), multiplied by $l$), adopted as an angular model, rather than a translational force law on the length coordinate $x$. The companion page's small-angle check quantifies its difference from the linearized angular equation; the linear theory does not describe its exact motions.

• The potential is declared, not measured. Only the value of $k$ enters the physics; the additive constant of $U$ is fixed by the convention $U(0)=0$ recorded above, which is part of the model's declaration, exactly as permitted by Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c); that clause requires a declared representative, not this particular zero.