Status audit and model boundaries of the force page: open systems, frame selection, breakdown and empirical scope

This remark records the status of the notions of the force page. It is bookkeeping: it asserts no law and no empirical claim, and every mathematical statement it repeats is proved in the item cited.

Adopted assumptions. The force branch is the postulate Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: the force map with its standing continuity and state-local-Lipschitz hypotheses, the equation of motion $m_i{\ddot{x}}_i=F_i(t,x,\dot{x})$, the additive composition of declared force contributions, and the transformation rule under a change of inertial frame. The restricted interaction models — external-plus-pair decompositions, the optional antisymmetry and centrality assumptions, potential forces and the gravitational pair-potential branch with its coupling parameters — are the postulate Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch. The linear drag model, with its declared coefficient and its regime and fluctuation qualifications, is the postulate The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient.

Defined objects. The associated first-order system, motions, initial data, maximal motions and the Newtonian flow are the definition The associated first-order system, motions, maximal motions and the Newtonian flow of a force model; the configuration domain, the mass operator and the collision exclusion are Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, and the units and dimensions read throughout are Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.

Conditional deductions. Local existence, uniqueness and the existence of the maximal motion are Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder; the maximal-extension and breakdown criterion, including the autonomous trapping statement, is Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain. Both are conditional on the adopted force branch and the cited mathematical ODE theorems; neither asserts that any physical system realizes the model.

Computed examples. The uniform-force model with its explicit quadratic motion, its uniqueness and its frame covariance is The uniform-force model: explicit global motion of a label under a constant declared field, with uniqueness, dimensions and frame covariance. The companion page carries the worked model checks, the projectile computation, the terminal-velocity computation and the two counterexamples (nonuniqueness under a merely continuous force, and a singular collision with finite breakdown time).

Deliberately not claimed on this page. No conservation law (momentum, angular momentum, energy) is asserted: such statements require their own hypotheses on the interaction structure and belong to later pages. No constraint reactions and no constrained dynamics are adopted. No global existence theorem for all force maps is claimed; the maximal interval may end at a finite time. No measured value of any mass, field strength, drag coefficient or gravitational coupling is asserted, and no comparison with reported data is performed.

• The declared labels are a modelling choice. A model of the force branch names a finite set of labels; every force exerted by a body outside that set is an external force of the model, and the environment's state is not modelled. Splitting a physical situation into "the system" and "the environment" is part of the formulation, and different splits give different models with different external forces.

• Pair structure does not survive the boundary. The antisymmetry and centrality options of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch are properties of the pair forces between declared labels. For a proper subsystem, the forces to the omitted labels appear as external forces that need satisfy no such antisymmetry; the subsystem's mechanical momentum is exchanged with the environment and need not be conserved. This is the open-system warning: the balance laws of later pages are statements about stated closed or specified systems, not automatic properties of an arbitrary collection of labels.

• External data may select a useful frame or location. The equation of motion is kinematically covariant under changes of inertial frame (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (vi)). The uniform-force model of The uniform-force model: explicit global motion of a label under a constant declared field, with uniqueness, dimensions and frame covariance remains a uniform-force model in every inertial frame, with its parameter changing by ${\mathbf{g}}^{\mathrm{\prime }}=R\mathbf{g}$; it does not select a unique inertial frame. A medium at rest, as in The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient, supplies a preferred rest frame for that drag description, while a stationary external inverse-square center also supplies a preferred rest velocity. Explicitly, in the coordinates of one inertial frame let the prescribed source worldline be ${\gamma }_c(t)=(t,0)$ for $t\in I_0$ (a smooth curve), and let $F(t,x,w)=-kx\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^3$ for $x\mathrm{\ne }0$, with $k>0$ of dimension $\mu {\mathrm{\ell }}^3{\tau }^{-2}$. Under the transfer $t^{\mathrm{\prime }}=t+b$, $y=Rx+\upsilon t+a_0$ of clause (vi), its spatial coordinate curve becomes $c_{\ast }(t^{\mathrm{\prime }})=\upsilon (t^{\mathrm{\prime }}-b)+a_0$, and the force becomes $F_{\ast }(t^{\mathrm{\prime }},y,w)=-k(y-c_{\ast }(t^{\mathrm{\prime }}))\mathrm{/}\mathrm{\mid }y-c_{\ast }(t^{\mathrm{\prime }}){\mathrm{\mid }}^3$ on $y\mathrm{\ne }{c}_{\ast }(t^{\mathrm{\prime }})$. Thus a nonzero boost makes the center move and the field time-dependent. The source selects the class of inertial frames in which it is at rest, with origins and spatial axes still free; it does not select a unique inertial chart or violate covariance of the transferred model. The Galilean frames of Inertial frames, stationary observers and the adopted Galilean transformations between frames remain the frames in which the force branch is formulated; model-specific external data can make a particular frame useful for describing an application.

• The particle idealization is per application. The worldlines of the model are those of labels with no size, spin or internal structure; a concrete body is described by such a label only when those omissions are immaterial to the question asked, and the model must state that regime where it matters.

• Hypotheses are used, not decorative. The standing regularity of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iii) is exactly the hypothesis of the local existence and uniqueness theorem Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder; the companion counterexample on the examples page exhibits a continuous force map with two distinct motions through one datum, so continuity alone cannot replace the Lipschitz clause.

• Finite-time breakdown is not a flaw of the formulation. By Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain, at a finite right endpoint the full time-state graph leaves every compact subset of $D=I_0\times U$. If that endpoint is the finite right boundary of the prescribed open time window, the time coordinate alone can leave every such compact set while the state remains bounded and away from the excluded configurations. When $I_0=\mathbb{R}$, the state leaves every compact subset of $U=Q\times {\mathbb{R}}^{3N}$; it is unbounded on every terminal interval or, if bounded on a terminal interval and $Q$ is proper, positions approach $Q^c$ in the theorem's stated sense. The companion singular-collision counterexample exhibits both state-space alternatives at once for an inverse-square external model.

• Empirical status. The force maps, potentials, coefficients and constants of this page are model data. A drag coefficient or a field strength taken from measurement is an empirical premise carrying its conditions and uncertainty. The drag postulate omits fluctuations as an idealization; Likharev's energy criterion is specific to the damped harmonic oscillator in Section 5.1 and does not impose an energy condition on the free-particle drag model, which defines no corresponding $E$, $T$ or ${\omega }_0$ (The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient). A realistic field is uniform only as an approximation. Agreement between a computed motion and a reported observation tests the combined model, apparatus and calibration assumptions and is evidence about them, never a proof of the model. This page records no observation at all.