Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting

Adopt, as this page's impact and open-mass extensions of the force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance and Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch, the following three model extensions. Each is an adopted model, not a consequence of the smooth-force postulates; each applies only where a statement explicitly invokes it.

(i) The isolated reciprocal impulse model. A two-label force model with masses $m_1,m_2>0$ (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)), time window $I_0$, configuration domain $Q$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator) and continuous external-plus-pair force maps $F_1,F_2$ is run in the isolated reciprocal impulse model at $t_{\ast }\in I_0$ when there is an interval $I\subseteq I_0$ with $t_{\ast }\in \mathrm{int}I$ and a position curve $x=(x_1,x_2):I\to Q$ such that the following conditions hold. This curve is called an impulse motion; only its restrictions away from $t_{\ast }$ are motions of the associated first-order system of The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, and no differentiability at $t_{\ast }$ is required:

1. $x_1,x_2$ are continuous on $I$ and twice continuously differentiable on each of $I\cap (-\mathrm{\infty },t_{\ast })$ and $I\cap (t_{\ast },\mathrm{\infty })$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, $C^k$ Euclidean maps and diffeomorphisms), and $m_i{x}_i^{\mathrm{\prime }\mathrm{\prime }}(t)=F_i(t,x(t),x^{\mathrm{\prime }}(t))$ for every $t\in I\setminus \{t_{\ast }\}$;
2. the one-sided limits $v_i^{\pm }:={\lim }_{t\to t_{\ast }\pm }{x}_i^{\mathrm{\prime }}(t)$ exist in ${\mathbb{R}}^3$ (Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions);
3. there is a vector $J\in {\mathbb{R}}^3$, the impulse, with $m_1(v_1^{+}-v_1^{-})=J,m_2(v_2^{+}-v_2^{-})=-J\mathrm{.}$

The two equations of clause 3 are the reciprocal finite-jump form of the isolated impulse: the two labels receive equal and opposite momentum changes. The regular force maps remain continuous on their state domain; their evaluations along the curve can have different one-sided limits because the velocity can jump. The dimension of the impulse is $[J]=\mu \mathrm{\ell }{\tau }^{-1}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

For the distributional rendering on $\mathrm{int}I$, assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Put $p_i(t)=m_i{x}_i^{\mathrm{\prime }}(t)$ and $f_i(t)=F_i(t,x(t),x^{\mathrm{\prime }}(t))$ off $t_{\ast }$, assigning both the value $0$ at $t_{\ast }$. Their one-sided limits are finite: those of $p_i$ are $m_i{v}_i^{\pm }$, and continuity of the state-domain force maps gives $f_i(t_{\ast }^{\pm })=F_i(t_{\ast },x(t_{\ast }),v^{\pm })$, with $v^{\pm }=(v_1^{\pm },v_2^{\pm })$. Also $p_i^{\mathrm{\prime }}=f_i$ off the impulse. Thus each component meets the piecewise-$C^1$ and one-sided derivative-limit hypotheses of The distributional derivative of a piecewise $C^1$ function with a jump: $(u_p{)}^{\mathrm{\prime }}=u_{p_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}}+[p]{\delta }_{t_{\ast }}$. Its identity, summed componentwise, gives for every real vector test function $\mathrm{\Phi }\in C_c^{\mathrm{\infty }}(\mathrm{int}I;{\mathbb{R}}^3)$ (each component is smooth and compactly supported, Test function space d of an open set) $-{\int }_{\mathrm{int}I}⟨p_i,{\mathrm{\Phi }}^{\mathrm{\prime }}⟩d\lambda ={\int }_{\mathrm{int}I}⟨f_i,\mathrm{\Phi }⟩d\lambda +s_i⟨J,\mathrm{\Phi }(t_{\ast })⟩,s_1=1,s_2=-1,$ where $\lambda$ is Lebesgue measure and the integral convention is that of Integral over a measurable subset. This is the pairing form of the adopted jumps. Unlike Isolated reciprocal impulse with full-vector restitution: momentum jumps $\pm J$, conservation of total momentum, the restitution solution and the kinetic-energy change $-\frac{1-e^2}{2}\mu \parallel v_2^{-}-v_1^{-}{\parallel }_2^2$, whose hypotheses require continuous time forces and assume the pairings to derive the jumps, this rendering allows distinct one-sided regular-force values.

(ii) The collinear restitution contact law. A model run in the isolated reciprocal impulse model at $t_{\ast }$ is collinear when there is a unit vector $n\in {\mathbb{R}}^3$ (the contact direction) with $v_i^{\pm }\parallel n$ for $i=1,2$ and $⟨v_2^{-}-v_1^{-},n⟩<0$ (the labels approach along $n$). For a collinear contact, adopt a coefficient of restitution $e\in [0,1]$ — a dimensionless model datum of the pair and the contact — and the restitution law $v_2^{+}-v_1^{+}=-e(v_2^{-}-v_1^{-})\mathrm{.}$ For $e=1$ the contact is called elastic, for $e=0$ perfectly inelastic and otherwise semi-elastic; the law is a constitutive assumption, not a consequence of the impulse equations of clause (i), and $e$ is not derived from the masses or the approach speed. The law is stated only for a single collinear contact; simultaneous contacts, repeated contacts and tangential (friction) impulses are outside the model.

(iii) The open-body material accounting. Let $I\subseteq \mathbb{R}$ be a nonempty open interval. Adopt for a body whose mass $m:I\to (0,\mathrm{\infty })$ and velocity $v:I\to {\mathbb{R}}^3$ are differentiable, and whose external force and exchanged matter are described by maps $F:I\to {\mathbb{R}}^3$, $r_{\mathrm{i}\mathrm{n}},r_{\mathrm{o}\mathrm{u}\mathrm{t}}:I\to [0,\mathrm{\infty })$ and $w_{\mathrm{i}\mathrm{n}},w_{\mathrm{o}\mathrm{u}\mathrm{t}}:I\to {\mathbb{R}}^3$ (mass rates and inertial velocities of the matter entering and leaving, measured in the same inertial frame as $v$), the material balance identities $m^{\mathrm{\prime }}=r_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}},(mv{)}^{\mathrm{\prime }}=F+r_{\mathrm{i}\mathrm{n}}{w}_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}}{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}\mathrm{.}$ They are the mathematical rendering of "gains $r_{\mathrm{i}\mathrm{n}}\mathrm{d}t$ of matter with velocity $w_{\mathrm{i}\mathrm{n}}$, loses $r_{\mathrm{o}\mathrm{u}\mathrm{t}}\mathrm{d}t$ with velocity $w_{\mathrm{o}\mathrm{u}\mathrm{t}}$, and feels the external force $F$", and they are adopted rather than derived from a closed-system law. Their product-rule consequence $m{v}^{\mathrm{\prime }}=F+r_{\mathrm{i}\mathrm{n}}(w_{\mathrm{i}\mathrm{n}}-v)-r_{\mathrm{o}\mathrm{u}\mathrm{t}}(w_{\mathrm{o}\mathrm{u}\mathrm{t}}-v)$ is proved in Variable-mass momentum balance: from $m^{\mathrm{\prime }}=r_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and $(mv{)}^{\mathrm{\prime }}=F+r_{\mathrm{i}\mathrm{n}}{w}_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}}{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ to $m{v}^{\mathrm{\prime }}=F+r_{\mathrm{i}\mathrm{n}}(w_{\mathrm{i}\mathrm{n}}-v)-r_{\mathrm{o}\mathrm{u}\mathrm{t}}(w_{\mathrm{o}\mathrm{u}\mathrm{t}}-v)$; in particular the closed-system law $(mv{)}^{\mathrm{\prime }}=F$ is recovered only when $r_{\mathrm{i}\mathrm{n}}{w}_{\mathrm{i}\mathrm{n}}=r_{\mathrm{o}\mathrm{u}\mathrm{t}}{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}$. The quantities have types and SI dimensions $[m]=\mu$, $[v]=[w_{\mathrm{i}\mathrm{n}}]=[w_{\mathrm{o}\mathrm{u}\mathrm{t}}]=\mathrm{\ell }{\tau }^{-1}$, $[F]=\mu \mathrm{\ell }{\tau }^{-2}$ and $[r_{\mathrm{i}\mathrm{n}}]=[r_{\mathrm{o}\mathrm{u}\mathrm{t}}]=\mu {\tau }^{-1}$; they are model data, not measured values.

• Why the extensions are adopted rather than derived. A motion of the smooth-force branch with a continuous force map has a $C^1$ momentum curve and no jumps; hence no choice of smooth force maps produces the momentum jump of clause 3 with $J\mathrm{\ne }0$ at an isolated time. The closed, fixed-label smooth-force model does not account for matter crossing a body boundary; clause (iii) adopts that additional accounting. Constant body mass alone does not exclude exchange: its mass balance permits $r_{\mathrm{i}\mathrm{n}}=r_{\mathrm{o}\mathrm{u}\mathrm{t}}>0$, and different entering and leaving inertial velocities can give a nonzero momentum flux. The extensions enlarge the model class; they are not consequences of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance or Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch, and the exact statement that a smooth-force motion cannot jump is proved on this page as part of Consequences of the adopted impact and open-mass models: continuity of total momentum, the smooth-force exclusion, the restitution solution with $\mathrm{\Delta }K=-\frac{1-e^2}{2}\mu \parallel v_2^{-}-v_1^{-}{\parallel }_2^2$, and the open-body and rocket laws.

• What the data are. $m_i$, $J$, $e$, $r_{\mathrm{i}\mathrm{n}}$, $r_{\mathrm{o}\mathrm{u}\mathrm{t}}$, $w_{\mathrm{i}\mathrm{n}}$, $w_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and the contact direction $n$ are model data, not measurements. The empirically informed reading of $e$ (a pair property inferred from bounce or collision experiments) requires a separately specified experiment and statistical analysis; this page asserts no measured value and no law of nature, exactly as for the potentials of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions.

• Exclusions. Coefficient values outside $[0,1]$; more than one contact in a motion; contact intervals rather than isolated times; tangential and rolling contacts; deformable contact geometry; rotational degrees of freedom; and any identification of the impulse with the time integral of a bounded smooth force are outside the adopted models. Clause (iii) excludes internal structure of the leaving mass, pressure forces on the open boundary and non-inertial frames; its spatially distributed refinement is the finite-regularity open-flow model of The finite-regularity open-flow model: moving-control-region mass and momentum balances, their material and fixed special cases, and the inlet/outlet decomposition reproducing the open-body and rocket laws.