Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance

Adopt as physical assumptions of the framework the following statements about forces and motion, in the spacetime, frames and worldlines of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure, Inertial frames, stationary observers and the adopted Galilean transformations between frames and Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses.

(i) Setting and configuration domain. A model of the framework has $N\ge 1$ particle labels with fixed positive masses $m_1,\dots ,m_N$ (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses, clause (v)) and a configuration domain $Q$ in one fixed inertial frame, in the unconstrained branch of Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator clause (a): $Q\subseteq {\mathbb{R}}^{3N}$ is open, possibly with the collision set $\mathrm{\Delta }$ removed when the force law of the model is singular at coincident positions. The constrained branch of that definition, with its reaction forces, is not covered by this postulate; the dynamics of constrained systems is a separate development with its own assumptions. A state of the model is a pair $z=(x,v)\in Q\times {\mathbb{R}}^{3N}$ of positions and coordinate velocities of the labels in the chosen frame (Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors).

(ii) Time window. A model carries a nondegenerate open interval $I_0\subseteq \mathbb{R}$, its time window, read in seconds (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

(iii) Force map and standing regularity. A force map on this data is a map

$F:I_0\times Q\times ({\mathbb{R}}^3{)}^N⟶({\mathbb{R}}^3{)}^N,(t,x,v)⟼F(t,x,v)=(F_1(t,x,v),\dots ,F_N(t,x,v)),$

with components of dimension $\mu \mathrm{\ell }{\tau }^{-2}$, read in newtons (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The standing hypotheses of this branch, which every force map of a model is required to satisfy, are:

1. Continuity. $F$ is continuous on its domain $D:=I_0\times Q\times ({\mathbb{R}}^3{)}^N$.
2. State-local Lipschitz regularity, locally uniform in time. $F$ is locally Lipschitz in the state variable $z=(x,v)$, locally uniformly in time, in the sense of Local Lipschitz continuity in the state variable, locally uniform in time and parameters: every compact cylinder $C\subseteq D$ carries a finite $L\ge 0$ with $\parallel F(t,z)-F(t,z^{\mathrm{\prime }}){\parallel }_2\le L\parallel z-z^{\mathrm{\prime }}{\parallel }_2$ whenever $(t,z),(t,z^{\mathrm{\prime }})\in C$ share their time coordinate.

A map that fails clause 1 or clause 2 is not a force map of this branch, whatever its other merits. When a later statement differentiates the force map (for instance in a linearization or a smooth-dependence claim), it adds the required differentiability hypothesis of the state variable explicitly; clause 2 alone licenses only the existence, uniqueness and continuation statements proved on this page.

(iv) Equation of motion. Adopt: the motion of the labels of a model with force map $F$ in an inertial frame is governed by

$m_i{\ddot{x}}_i(t)=F_i(t,x(t),\dot{x}(t))(i=1,\dots ,N),$

imposed along families of $C^2$ coordinate curves of the labels $x=(x_1,\dots ,x_N):J\to Q$, where $J\subseteq I_0$ is an interval with nonempty interior and the derivatives are those of Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors. The precise notion of a motion of the model, its associated first-order system and its maximal motions is The associated first-order system, motions, maximal motions and the Newtonian flow of a force model. The force map is primitive data of the model: the equations above are the model's law, not an operational definition of $F$ from a measured acceleration, and $F$ is not an extra observable measured independently of the motion. The mass $m_i$ in (iv) is the fixed inertial mass of clause (v) of Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses; no other mass occurs in this clause.

(v) Additive composition of force contributions. When a model presents its force map as composed of contributions, the composition is the pointwise sum: for force maps $F^{(1)},\dots ,F^{(k)}$ on a common domain satisfying (iii),

$F(t,z)={\sum }_{a=1}^k{F}^{(a)}(t,z),$

and the sum again satisfies (iii), because a finite sum of continuous maps is continuous and a finite sum of maps that are locally Lipschitz in the state variable is locally Lipschitz with a locally uniform constant. This superposition of forces is an adopted modelling rule, not a theorem about nature: it says how the model's total force is composed from the contributions it declares. Two decompositions with the same sum define the same force map and the same model; the framework does not measure an individual contribution in isolation, and no observation of motions alone determines a decomposition. Nor does (v) assert that sums of motions are motions: if $F$ is not affine in the state, the sum of two solutions of (iv) generally does not satisfy (iv), and superposing solutions is a separate question from superposing forces.

(vi) Description in another inertial frame. Let $F_{\ast }=(o^{\mathrm{\prime }},u^{\mathrm{\prime }},(e_1^{\mathrm{\prime }},e_2^{\mathrm{\prime }},e_3^{\mathrm{\prime }}))$ be a second inertial frame related to the frame of (i) by the Galilean transformation

$t^{\mathrm{\prime }}=t+b,x_i^{\mathrm{\prime }}=R{x}_i+\upsilon t+a_0(i=1,\dots ,N)$

with $b\in \mathbb{R}$, $a_0,\upsilon \in {\mathbb{R}}^3$ and $R\in SO(3)$ (Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes clause 1; The orthogonal group $O(3)$ and the rotation group $SO(3)$); $\upsilon$ is the boost velocity denoted by $v$ in that theorem. Define the block-diagonal map $R_N:=\mathrm{diag}(R,\dots ,R):{\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N}$ and the repeated tuples ${\bar{a}}_0:=(a_0,\dots ,a_0)$ and $\bar{\upsilon }:=(\upsilon ,\dots ,\upsilon )$ in $({\mathbb{R}}^3{)}^N$. Thus $R_{N}x=(R{x}_1,\dots ,R{x}_N)$, and the same componentwise rule applies to velocities and forces; since $R^{\mathsf{T}}R=I_3$, $R_N^{\mathsf{T}}{R}_N=I_{3N}$. If $x:J\to Q$ is a family of motions of the labels in the first frame, then the description of the same worldlines in the second frame is $\eta (t^{\mathrm{\prime }}):=R_{N}x(t^{\mathrm{\prime }}-b)+\bar{\upsilon }(t^{\mathrm{\prime }}-b)+{\bar{a}}_0(t^{\mathrm{\prime }}\in J+b)\mathrm{.}$ By clause 2 of the cited theorem, each ${\eta }_i$ is $C^2$ and ${\eta }_i^{\mathrm{\prime }\mathrm{\prime }}(t^{\mathrm{\prime }})=R{\ddot{x}}_i(t^{\mathrm{\prime }}-b)$, so along the motion $m_i{\eta }_i^{\mathrm{\prime }\mathrm{\prime }}(t^{\mathrm{\prime }})=R{F}_i(t^{\mathrm{\prime }}-b,x(t^{\mathrm{\prime }}-b),\dot{x}(t^{\mathrm{\prime }}-b))\mathrm{.}$

Adopt: the same interaction, read in the second frame, is represented on the transferred data by the force map below. Let $P$ be the inverse affine transfer from new-frame data to old-frame data, $P(t^{\mathrm{\prime }},y,w):=(t^{\mathrm{\prime }}-b,R_N^{\mathsf{T}}(y-\bar{\upsilon }(t^{\mathrm{\prime }}-b)-{\bar{a}}_0),R_N^{\mathsf{T}}(w-\bar{\upsilon }))\mathrm{.}$ Its preimage of the original domain is $D_{\ast }:=P^{-1}(D)=\{(t^{\mathrm{\prime }},y,w):P(t^{\mathrm{\prime }},y,w)\in D\};$ it is open because $P$ is a homeomorphism and $D$ is open. Define $F_{\ast }:D_{\ast }\to ({\mathbb{R}}^3{)}^N,F_{\ast }(t^{\mathrm{\prime }},y,w):=R_{N}F(t^{\mathrm{\prime }}-b,R_N^{\mathsf{T}}(y-\bar{\upsilon }(t^{\mathrm{\prime }}-b)-{\bar{a}}_0),R_N^{\mathsf{T}}(w-\bar{\upsilon }))\mathrm{.}$ The map $F_{\ast }$ is continuous and locally Lipschitz in its state variable: the transfer and its inverse are affine, and at fixed time they act linearly on state differences; on compact time intervals their state Lipschitz constants are uniform, so precomposition preserves the stated regularity of $F$. The framework formulates the equation of motion in every inertial frame; a rule that does not transform by this law is a description tied to the frame in which it was written, and the framework attaches no kinematic covariance to it. When $Q={\mathbb{R}}^{3N}$ — the case of the models of the companion page — the transferred domain is again a product $I_0^{\mathrm{\prime }}\times {\mathbb{R}}^{3N}\times {\mathbb{R}}^{3N}$ with $I_0^{\mathrm{\prime }}=I_0+b$, and $F_{\ast }$ is a force map in the sense of (iii); for a proper open $Q$ the transferred position set $R_{N}Q+\bar{\upsilon }(t^{\mathrm{\prime }}-b)+{\bar{a}}_0$ may depend on $t^{\mathrm{\prime }}$, and the equation is then read on that transferred set.

(vii) What this branch excludes. Impulsive forcing (a finite momentum transfer at an instant is not a force map); Coulomb sliding friction, whose expression contains the factor $v\mathrm{/}\mid v\mid$ and is undefined at $v=0$; static friction, which requires an inequality or a set-valued reaction model; stochastic forcing; and forces that exist only as distributions. Each of these requires its own stated model and solution notion, and none is covered by (iii)–(iv). A singular force law is admissible on a configuration domain with the singular configurations removed — for instance an inverse-square law on the collision-excluded $Q$ of clause (i) — and what the removal of those configurations implies for existence in time is stated by the continuation theorem of this page.

(viii) Status. Clause (iii) states the hypotheses under which the theorems of this page are proved; clause (iv) is the adopted law of motion; clauses (v) and (vi) are the adopted composition and frame rules. Nothing here is deduced from experiment, from a symmetry principle or from the uniqueness theory of differential equations, and nothing here asserts that any physical interaction is exactly of this form: the theorems of this page are conditional consequences of the adopted model. The empirical significance of a concrete model lies in fully specified model parameters and reported observations compared with its predictions, and this postulate supplies neither.

• Adopted, not deduced. The Newtonian tradition and Galilean relativity motivate clauses (iv)–(vi), but they are assumptions of the model, as the sources themselves record; the postulate selects no numerical value of any parameter and no particular force law.

• The regularity hypotheses are used, not decorative. Clause 2 is exactly the local-Lipschitz hypothesis of the local existence and uniqueness theorem Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder and of the continuation theorem Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain. Continuity alone does not give uniqueness, as the companion counterexample on the examples page records.

• Open systems. The configuration domain and the force map are declared for a stated set of labels: forces exerted by bodies outside that set are external data of the model, and the pair structure of interactions is a further restricted assumption (Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch). The scope warning for such model boundaries is Status audit and model boundaries of the force page: open systems, frame selection, breakdown and empirical scope.

• No dynamics of constraints. Reactions that maintain a constrained configuration are not force maps of clause (iii) in general; their ideal model and its hypotheses belong to the constraint development, not here.