The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria

(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.