One-dimensional conservative and dissipative models

(a) Data of a one-dimensional model. A one-dimensional model is declared by the following data: a fixed positive mass $m>0$, of dimension $\mu$ read in kilograms (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)); a time window $I_0\subseteq \mathbb{R}$, a nondegenerate open interval read in seconds (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length); an open configuration domain $Q\subseteq \mathbb{R}$, a nonempty open interval of the real line read in metres; and a force law $F:I_0\times Q\times \mathbb{R}⟶\mathbb{R},(t,x,v)⟼F(t,x,v),$ of dimension $\mu \mathrm{\ell }{\tau }^{-2}$, read in newtons, subject to the standing regularity of the framework: $F$ is continuous on $I_0\times Q\times \mathbb{R}$ and locally Lipschitz in the state variable $(x,v)$ locally uniformly in time (Local Lipschitz continuity in the state variable, locally uniform in time and parameters). The framework's three-dimensional force branch is Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance; the class declared here is a restricted model class adopted at the same primitive level, in which the physical position of a single label along one coordinate is the whole state. A model whose force law fails the standing regularity is not a one-dimensional model of this class. When a statement differentiates the force law or the potential, the additional smoothness it needs is stated in that statement.

(b) Motions. Let $J\subseteq I_0$ be an interval with nonempty interior (endpoints read with one-sided derivatives, as in First-order systems, initial value problems, and solutions on intervals). A motion of the model on $J$ is a curve $x:J\to Q$ of class $C^2$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral) with $m{x}^{\mathrm{\prime }\mathrm{\prime }}(t)=F(t,x(t),x^{\mathrm{\prime }}(t))(t\in J)\mathrm{.}$ In the notation of the three-dimensional branch this is the scalar restriction of its equation of motion clause (iv) to a single coordinate; the motions of the present class are declared to be these curves. A datum at time $t_0\in I_0$ is a triple $(t_0,x_0,v_0)$ with $x_0\in Q$ and $v_0\in \mathbb{R}$; a motion $x$ takes the datum when $x(t_0)=x_0$ and $x^{\mathrm{\prime }}(t_0)=v_0$.

(c) Phase system. The phase system of the model is the first-order system on the open set $I_0\times Q\times \mathbb{R}\subseteq {\mathbb{R}}^3$ with right-hand side $G_F(t,x,v):=(v,\frac{F(t,x,v)}{m}),$ read as a system of First-order systems, initial value problems, and solutions on intervals; its state variable is $z=(x,v)$. For an interval $J\subseteq I_0$ with nonempty interior, a phase solution on $J$ means a differentiable curve $z:J\to Q\times \mathbb{R}$ satisfying $z^{\mathrm{\prime }}(t)=G_F(t,z(t))$ for every $t\in J$, with domain-relative derivatives at included endpoints. Equivalently, for any $t_0\in J$ it is a solution in the cited IVP sense with datum $(t_0,z(t_0))$. For a $C^2$ curve $x:J\to Q$, being a motion is equivalent to $t\mapsto (x(t),x^{\mathrm{\prime }}(t))$ being a phase solution: the first component is $x^{\mathrm{\prime }}=v$ and the second is $x^{\mathrm{\prime }\mathrm{\prime }}=F(t,x,x^{\mathrm{\prime }})\mathrm{/}m$. Conversely a phase solution $(x,v)$ has $x^{\mathrm{\prime }}=v$ and $v^{\mathrm{\prime }}=F(t,x,v)\mathrm{/}m$; continuity of $F$ makes $v^{\mathrm{\prime }}$ continuous, so $x$ is $C^2$ and is a motion. This is the argument of clause (c) of The associated first-order system, motions, maximal motions and the Newtonian flow of a force model read in one coordinate. Here $x$ is the position or displacement coordinate and $v$ the velocity coordinate; both are read in the fixed inertial frame in which the model is declared. The phase system inherits the standing regularity: $G_F$ is continuous and locally Lipschitz in the state variable locally uniformly in time, so the Picard–Lindelöf hypotheses of Picard-Lindelöf local existence and uniqueness for first-order systems hold at every point of its domain.

(d) Potential presentation, residual forces and energy. The model presents its force law through a potential when it declares a function $U:Q\to \mathbb{R}$ of class $C^2$ in the sense of clause (a) of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$, read in joules, and a residual force $d:I_0\times Q\times \mathbb{R}\to \mathbb{R}$ of the standing regularity of (a), with $F(t,x,v)=-U^{\mathrm{\prime }}(x)+d(t,x,v)((t,x,v)\in I_0\times Q\times \mathbb{R});$ here $U^{\mathrm{\prime }}$ is the ordinary derivative. This is the one-coordinate case of the potential presentation of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (a) for a time- and velocity-independent potential, and the scalar $U$ is a potential energy of the model's potential part. The model is a global potential model (equivalently, in the framework's usage, a conservative model) when $d\equiv 0$; the term "conservative" is used only in this sense, as recorded in that definition. For a conservative model the mechanical energy is the function $E:Q\times \mathbb{R}⟶\mathbb{R},E(x,v):=\frac{1}{2}m{v}^2+U(x),$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$. For a model with declared residual forces, the mechanical energy along a motion $x:J\to Q$ is $E(t):=\frac{1}{2}m{x}^{\mathrm{\prime }}(t{)}^2+U(x(t))$; it is defined along curve families even though the force law is not presented through $U$ alone.

(e) Equilibria. A time- and velocity-independent force law on $Q$ is one of the form $F(t,x,v)=f(x)$ for a function $f:Q\to \mathbb{R}$; the model is then autonomous. For such a law, a point $x_{\ast }\in Q$ is an equilibrium (or rest position) when $f(x_{\ast })=0$, and the phase-plane point $(x_{\ast },0)$ is the corresponding rest point of the phase system; the constant curve $x\equiv x_{\ast }$ is then a motion. When the force law is presented as $f=-U^{\mathrm{\prime }}$ with $U$ of class $C^2$, the equilibria are exactly the critical points of $U$: $f(x_{\ast })=0⟺U^{\mathrm{\prime }}(x_{\ast })=0$. A point $x_0\in Q$ is a strict local minimum of $U$ when $U(x)>U(x_0)$ for every $x\in Q$ with $0<\mid x-x_0\mid <r_0$ for some $r_0>0$. The Lyapunov stability and asymptotic stability of a rest point of the autonomous phase system are the notions of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system applied to the field $G_F$; they are not defined here anew.

(f) Oscillator-type data. Let $m>0$, $c\ge 0$, $k\in \mathbb{R}$ and let $f:I_0\to \mathbb{R}$ be continuous. A model is of oscillator type with data $(m,c,k,f)$ when its force law has the form $F(t,x,v)=f(t)-cv-kx((t,x,v)\in I_0\times Q\times \mathbb{R})$ with $Q$ a nonempty open interval containing $0$ when the equation is read about the origin, and the parameters are read with dimensions $[c]=\mu {\tau }^{-1}$ (kilograms per second), $[k]=\mu {\tau }^{-2}$ (kilograms per second squared) and $[f]=\mu \mathrm{\ell }{\tau }^{-2}$ (newtons). For such a model the equation of motion is the second-order equation $m{x}^{\mathrm{\prime }\mathrm{\prime }}+c{x}^{\mathrm{\prime }}+kx=f(t)$, and the motions are exactly the restrictions of the solutions of the mathematical model of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data with the same data, by the identification of that definition's phase system with the phase system (c). The parameters $c$ and $k$ are model data: they enter the model's declaration and are not derived by this item.

(g) Status. This item introduces vocabulary, objects, regularity and dimensions; it contains no existence, uniqueness, conservation, stability or boundedness assertion, no force law of nature and no measured value. In particular, clause (a) declares the one-dimensional class rather than deriving it from the three-dimensional branch, and clauses (d)-(f) only fix which data a model of the class carries. Every later statement of this page names the model data, the potential representative and the initial datum it uses.

• Why a separate restricted class. The three-dimensional branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance is formulated for $N$ labels in ${\mathbb{R}}^{3N}$; the models of this page are declared in one scalar coordinate, as the framework's one-dimensional class, because every claim below is a statement about that scalar equation. Reading a model of this class as the straight-line motion of a three-dimensional label, or as the effective reduction of a constrained system (for example the angle coordinate of the pendulum), is a modelling identification that is stated and used only where a consumer item declares it, together with the extra assumptions it needs; it is not a premise of this definition.

• The declared representative of the potential. As in Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c), adding a constant to $U$ changes no force; for the scalar class no time-dependent normalization is admitted in clause (d), because the potential is declared time-independent, and every energy statement of this page names the representative it uses.

• Dimensions are read from the framework conventions. The length coordinate is read in metres, time in seconds and mass in kilograms, and the dimension map for products, quotients and derivatives is the one fixed by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework; the entries $[c]$, $[k]$ and $[f]$ of clause (f) follow from it. No numerical value of any parameter is asserted.

Adopt a one-dimensional model $(m,I_0,Q,F)$ of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria, with phase system $G_F(t,x,v)=(v,F(t,x,v)\mathrm{/}m)$.

(a) Phase plane and phase curves. The phase plane of the model is ${\mathbb{R}}^2$ with coordinates $(x,v)$, where $x$ is the position and $v$ the velocity coordinate of clause (b) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria. They are real numerical SI coordinates: if $X_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ and $V_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ are the corresponding physical position and velocity, then $x=X_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{m})$ and $v=V_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}(1\mathrm{m}\mathrm{/}\mathrm{s})$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The phase domain of the model is the open set $I_0\times Q\times \mathbb{R}$ of (time, position, velocity) triples. The phase plane is read with the Euclidean structure and norm $\parallel (x,v){\parallel }_2=\sqrt{x^2+v^2}$ on these dimensionless numerical coordinates (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$); physical displacement and velocity are not added as dimensionful quantities. Distances used below are those of this norm. For a curve family $x:J\to Q$ of class $C^1$ its phase curve is $t\mapsto (x(t),x^{\mathrm{\prime }}(t))$; when $x$ is a motion of the model the phase curve is the corresponding solution of the phase system.

(b) The energy function of a conservative model. Let the model be a global potential model of clause (d) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria with potential $U$ of class $C^2$, so that $F=-U^{\mathrm{\prime }}$. The energy function of the model is $E:Q\times \mathbb{R}⟶\mathbb{R},E(x,v):=\frac{1}{2}m{v}^2+U(x),$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ (joules). For $E_0\in \mathbb{R}$ the energy level set of $E_0$ and the energy sublevel set of $E_0$ are ${\mathrm{\Sigma }}_{E_0}:=\{(x,v)\in Q\times \mathbb{R}:E(x,v)=E_0\},S_{E_0}:=\{(x,v)\in Q\times \mathbb{R}:E(x,v)\le E_0\}\mathrm{.}$

(c) Turning positions and turning points. Let the model be as in (b). A point $x\in Q$ is a turning position of the energy $E_0$ when $U(x)=E_0$, and the phase-plane point $(x,0)$ is the corresponding turning point of $E_0$. A turning position at which $U^{\mathrm{\prime }}$ does not vanish is a simple turning position. In the one-coordinate reading of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart the potential difference $U(x)-E_0$ is the sign-controlling quantity for the velocity of a motion of energy $E_0$; the statements of that kind are the content of the conservation theorem of this page and are not made here.

(d) Rest points and stability vocabulary. For a model with a time- and velocity-independent force law $f$, the equilibria, rest points, strict local minima and the Lyapunov and asymptotic stability of a rest point are those of clause (e) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria, read for the phase system $G_F$; Lyapunov stability is the notion of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system and linear stability is that of Invariant sets, orbital stability of an invariant set, and linear stability of an equilibrium, applied to $G_F$ at its rest point. For an oscillator-type model with data $(m,c,k,f)$ of clause (f) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria the phase system is the phase-plane system of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data, and the phase-plane reading of its equilibria is the one recorded there.

(e) Status. This item fixes geometric objects and reads mathematical stability vocabulary for the scalar phase system; it asserts no conservation of energy, no containment of a phase curve in a level set, no boundedness of level sets and no measured value.

• The phase plane is two-dimensional, and the model is not the pendulum. The coordinates $(x,v)$ are the position and velocity of the declared scalar model; a model with an angle coordinate would carry a quotient of the line by the periodicity of the physical configuration, which is not part of this definition. The pendulum model of the companion page declares its own coordinate interval.

• Level sets are not orbits. ${\mathrm{\Sigma }}_{E_0}$ is a subset of the phase plane defined by an equation; that the phase curve of a motion with initial energy $E_0$ lies in it is a conservation statement, proved separately. A level set may be empty, a single rest point, a closed curve or a union of curves, and this item asserts nothing about which case occurs.

(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.

Adopt a conservative one-dimensional model: the class of clause (a) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria with a time- and velocity-independent force law presented through a potential, that is $F(x)=-U^{\mathrm{\prime }}(x)(x\in Q)$ for a function $U:Q\to \mathbb{R}$ of class $C^2$ (so the residual force of clause (d) of that definition vanishes identically and the model is a global potential model). Write $E(x,v)=\frac{1}{2}m{v}^2+U(x)$ for the energy function of The phase plane of a one-dimensional model: phase curves, the energy function and its level sets, and turning points clause (b). Then:

1. Existence and uniqueness. Every datum $(t_0,x_0,v_0)$ with $t_0\in I_0$, $x_0\in Q$, $v_0\in \mathbb{R}$ has exactly one maximal motion: a motion $x:J_{\max }\to Q$ with $x(t_0)=x_0$ and $x^{\mathrm{\prime }}(t_0)=v_0$ that has no extension to a motion on a strictly larger interval. Its interval $J_{\max }$ is open, contains $t_0$ and is contained in $I_0$, and every motion of the model with the same datum is the restriction of $x$ to a subinterval.
2. Energy conservation. For every motion $x:J\to Q$ the function $t\mapsto E(x(t),x^{\mathrm{\prime }}(t))$ is constant on $J$.
3. Level-set containment and velocity determination. Let $x:J\to Q$ be a motion with datum $(t_0,x_0,v_0)$ and let $E_0:=E(x_0,v_0)$. Then $U(x(t))\le E_0$ for every $t\in J$; if $U(x(t))<E_0$ then $x^{\mathrm{\prime }}(t)=\pm \sqrt{\frac{2(E_0-U(x(t)))}{m}},$ the two signs giving the only two possible values; and $x^{\mathrm{\prime }}(t)=0$ holds exactly when $U(x(t))=E_0$. In particular the phase curve of $x$ is contained in the energy level set ${\mathrm{\Sigma }}_{E_0}$ of clause (b) of The phase plane of a one-dimensional model: phase curves, the energy function and its level sets, and turning points.
4. Rest motions. A point $x_{\ast }\in Q$ is an equilibrium of the model if and only if $U^{\mathrm{\prime }}(x_{\ast })=0$; in that case the constant curve $x\equiv x_{\ast }$ is the maximal motion with datum $(t_0,x_{\ast },0)$, and conversely the maximal motion with datum $(t_0,x_{\ast },0)$ is constant exactly when $U^{\mathrm{\prime }}(x_{\ast })=0$.

The statement is conditional on the declared potential $U$: replacing $U$ by $U+c$ with a constant $c$ changes $E$ on the whole phase plane by $c$ and leaves clauses 1, 3 and 4 unchanged. No claim of periodicity, boundedness of the position or of the velocity, or approach to an equilibrium is made.

• What conservation does and does not give. Clause 3 confines the motion to the sublevel set $\{U\le E_0\}$ and fixes the speed at each position where the level is crossed, up to sign; it does not by itself show that the motion oscillates between turning positions, that a period exists, or that the motion stays bounded for all forward times. Those require additional hypotheses on $U$ and are not claimed here.

• The sample-independent sign. On an interval on which the velocity does not vanish, its sign is constant, because a continuous function with no zero has constant sign; this is why the phase curve travels along one branch of the level set between consecutive turning points. No turning points are asserted to exist for a general $U$.

• Normalization. Adding a constant to $U$ shifts $E$ by that constant on the whole phase plane; ${\mathrm{\Sigma }}_{E_0}$ moves to ${\mathrm{\Sigma }}_{E_0+c}$ and the velocity formula of clause 3 is unchanged, as it must be, since the force is unchanged. The energy conservation statement is a statement about the declared representative $U$.

Adopt a conservative one-dimensional model as in 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: the class of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria with force law $F(x)=-U^{\mathrm{\prime }}(x)$ for a potential $U:Q\to \mathbb{R}$ of class $C^2$, phase system $G_F(x,v)=(v,-U^{\mathrm{\prime }}(x)\mathrm{/}m)$ on $Q\times \mathbb{R}$, and energy $E=\frac{1}{2}m{v}^2+U$. Let $x_{\ast }\in Q$ be a strict local minimum of $U$: there is $r_0>0$ with $\bar{B}(x_{\ast },r_0)\subseteq Q$ and $U(x)>U(x_{\ast })$ for $0<\mid x-x_{\ast }\mid <r_0$. Then:

1. Lyapunov stability. $(x_{\ast },0)$ is a Lyapunov stable rest point of the phase system in the sense of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system: for every $\epsilon >0$ with $\bar{B}((x_{\ast },0),\epsilon )\subseteq Q\times \mathbb{R}$ there is $\delta >0$ such that the unique maximal solution of the autonomous phase equation through the datum of every motion $x$ of the model with $\parallel (x(t_0),x^{\mathrm{\prime }}(t_0))-(x_{\ast },0){\parallel }_2<\delta$ for some $t_0\in I_0$ is defined for all $t\ge t_0$ and satisfies $\parallel (x(t)-x_{\ast },x^{\mathrm{\prime }}(t)){\parallel }_2<\epsilon (t\ge t_0)\mathrm{.}$ Here $x$ in the displayed all-time bound denotes that maximal extension; on the original motion interval it agrees with the given motion. The autonomous equation is read on $\mathbb{R}\times Q\times \mathbb{R}$, even when the declared time window $I_0$ is smaller; no enlargement of that declared window is asserted. The norm is the Euclidean norm on ${\mathbb{R}}^2$ of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$; the balls are those of Open ball, closed ball and sphere in a metric space. No asymptotic stability and no attraction are asserted.

2. Linearisation. $U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })$ exists ($C^k$ Euclidean maps and diffeomorphisms), the phase field $G_F$ is totally differentiable at $(x_{\ast },0)$ with $D{G}_F(x_{\ast },0)(\delta x,\delta v)=(\delta v,-\frac{U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })}{m}\delta x),$ and the linearised equation is $m{\xi }^{\mathrm{\prime }\mathrm{\prime }}+U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })\xi =0,$ which is the equation of the oscillator-type model of clause (f) of The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria with data $(m,0,U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast }),0)$ (The damped driven oscillator model: second-order equation, phase-plane system and characteristic data). If in addition $U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })>0$, then that linearised model is the harmonic model of The harmonic (linear restoring) model: a declared linear force law with parameter $k>0$ with spring constant $U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })$, and its undamped frequency in the sense of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data is $\sqrt{U^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })\mathrm{/}m}$.

The statement is conditional on the potential $U$ and on the strict local minimum; no conclusion is drawn for a critical point that is only a weak minimum, a maximum, or a point where $U^{\mathrm{\prime }\mathrm{\prime }}$ vanishes identically on one side.