The phase plane of a one-dimensional model: phase curves, the energy function and its level sets, and turning points

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.