Forces, initial data and Newtonian flow

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.

Adopt, whenever a model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance invokes them, the following restricted interaction assumptions. Throughout, the data $I_0$, $Q$, $m_1,\dots ,m_N$ and the force map $F$ are those of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, and a state writes $z=(x,v)$ with $x=(x_1,\dots ,x_N)\in Q$ and $v=(v_1,\dots ,v_N)\in {\mathbb{R}}^{3N}$.

(i) External-plus-pair decomposition. The model's force map is presented as

$F_i(t,z)=F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}(t,x_i,v_i)+{\sum }_{j\mathrm{\ne }i}{F}_{ij}(t,x_i,x_j,v_i,v_j)(i=1,\dots ,N),$

where each $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}$ is a single-label external force map and each $F_{ij}$ (for $i\mathrm{\ne }j$, an ordered pair) is a pair force map, all defined on their stated open domains and satisfying the standing continuity and state-local-Lipschitz hypotheses of clause (iii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with respect to the arguments they actually depend on. This is a restriction of the model class: an arbitrary force map $F$ of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance need not admit such a decomposition, and the decomposition is part of the model's declaration, not a theorem. It is relative to the declared set of labels: for a subsystem, a pair force to a label outside the declared set appears as an external force, and no antisymmetry or pairing property of the full system restricts that external force.

(ii) Optional pair assumptions. In addition, a model may adopt, separately or together, the following assumptions about a pair force $F_{ij}$:

1. Antisymmetry (weak form of the third law). $F_{ij}(t,x_i,x_j,v_i,v_j)=-F_{ji}(t,x_j,x_i,v_j,v_i)$ for all arguments in the domain.
2. Centrality. $F_{ij}(t,x_i,x_j,v_i,v_j)$ is parallel to $x_i-x_j$, i.e. lies in the real line spanned by $x_i-x_j$, whenever $x_i\mathrm{\ne }{x}_j$.

The strong form of the third law means antisymmetry together with centrality; centrality alone does not imply antisymmetry.

Each assumption is a further restriction of the model class and is adopted only where invoked. Both are sufficient hypotheses used by later balance statements, and neither is a consequence of the equation of motion: a velocity-dependent pair force, or an interaction mediated by a field omitted from the declared labels, need satisfy neither.

(iii) Potential model. A model may present its force map through a scalar potential $U:I_0\times Q\to \mathbb{R}$ of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms), of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$, read in joules (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework), through the rule

$F_i(t,z)=-{\mathrm{\nabla }}_{x_i}U(t,x)(i=1,\dots ,N),$

the gradient being the Euclidean gradient in the position variable of each label (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). This is a restriction of the force class: only force maps that are the negative gradient of such a potential belong to it. The same force map arises from every potential of the form $U(t,x)+c(t)$ with a differentiable function $c$ of $t$ alone whenever the declared $C^2$ regularity is retained. More generally, for each connected component $C$ of $Q$ (a maximal connected subset), two $C^2$ potentials for the same force differ by an independently chosen $C^2$ function $c_C(t)$ on $I_0\times C$. A single time function suffices when $Q$ is connected; openness alone does not require the constants on different components to agree. The chosen normalization is part of any later energy statement and is recorded with it, and no energy statement is made on this page.

(iv) Pair and external potentials are examples, not the general form. When $U$ has the form

$U(t,x)={\sum }_i{u}_i(t,x_i)+{\sum }_{i<j}{u}_{ij}(t,x_i,x_j)$

with $C^2$ scalar functions $u_i$, $u_{ij}$ on the respective open time-position domains, each with energy dimension, the decomposition (i) holds with $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}(t,x_i,v_i)=-{\mathrm{\nabla }}_{x_i}{u}_i(t,x_i)$ and, for every $i<j$, both ordered-pair maps given by

$F_{ij}(t,x_i,x_j,v_i,v_j)=-{\mathrm{\nabla }}_{x_i}{u}_{ij}(t,x_i,x_j),F_{ji}(t,x_j,x_i,v_j,v_i)=-{\mathrm{\nabla }}_{x_j}{u}_{ij}(t,x_i,x_j)\mathrm{.}$

Here the two gradients differentiate the first and second position arguments, respectively, of the same declared $u_{ij}$; no $u_{ji}$ is required. These maps are independent of velocities. Differentiating the finite sum for $U$ with respect to $x_i$ collects exactly the external term and one term for each $j\mathrm{\ne }i$, giving (i). Antisymmetry is not asserted for a general pair potential. Potentials of the still more special form $u_{ij}=u_{ij}(t,\mid x_i-x_j\mid )$ are the central pair potentials; these are examples, and no generality is claimed for any of these subclasses. Forces with nontrivial velocity dependence, such as the drag model The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient with $c>0$, lie outside every potential model of this clause; when $c=0$ that model is a constant force and has a potential.

(v) Newtonian gravitational branch. The model may adopt the pair potential

$U_{ij}(x_i,x_j)=-\frac{G_{\mathrm{N}}{m}_i^g{m}_j^g}{\mid x_i-x_j\mid }$

on the collision-excluded configuration domain ($x_i\mathrm{\ne }{x}_j$ for all $i\mathrm{\ne }j$), where $G_{\mathrm{N}}>0$ is a constant of dimension ${\mathrm{\ell }}^3{\mu }^{-1}{\tau }^{-2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework) and the numbers $m_1^g,\dots ,m_N^g>0$ are gravitational coupling parameters of the labels, primitive model data distinct from the inertial masses $m_i$ of the equation of motion. By clause (iii) this potential supplies the attractive inverse-square pair force $F_{ij}=-G_{\mathrm{N}}{m}_i^g{m}_j^g(x_i-x_j)\mathrm{/}\mid x_i-x_j{\mid }^3$, the content of the elementary differentiation ${\mathrm{\nabla }}_x(1\mathrm{/}\mid x-y\mid )=-(x-y)\mathrm{/}\mid x-y{\mid }^3$; a concrete instance of that differentiation is verified on the companion page. The adopted model satisfies the antisymmetry and centrality options of (ii) where they are invoked.

(vi) The mass identification is an empirical assumption. The identification

$m_i^g=m_i(i=1,\dots ,N)$

of gravitational couplings with inertial masses is not part of clauses (i)–(v) and is not a consequence of the equation of motion. It is an additional empirical assumption, adopted only when a model states it, and it is the subject of the weak equivalence principle; the sources record its empirical support and its historical status, and this page records no measurement. Likewise, the inverse-square form in (v) is an adopted law, not a consequence of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance.

(vii) Status. Clauses (i)–(v) are adopted modelling assumptions that restrict the force maps of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance; they add no dynamics beyond that postulate, and every theorem of this page remains conditional on these adopted hypotheses. The balances of momentum, angular momentum and energy, when asserted later, are theorems with their own hypotheses on this structure and are not made here.

• Why the restrictions are separated. Antisymmetry, centrality and the existence of a potential are stated as separate restrictions. A model's declared assumptions must be jointly consistent with its force maps and decomposition; no logical independence or realizability of every combination is asserted. Keeping the hypotheses explicit prevents the later balance theorems from silently assuming more than their proofs use.

• Open subsystems. The pair form is declared for the labels of the model. If a physical system is described by a proper subset of its bodies, interactions with the remaining bodies are external forces in the sense of clause (i), and the pair properties of the full system say nothing about them.

• No inverse-square from the second law. The gravitational potential and the inverse-square force are additional adopted laws; the second law alone determines no force law, and this page proves no property of nature from the form of the equation of motion.

• Dimensions of the gravitational couplings. Assign each independent gravitational coupling the mass dimension $[m_i^g]=\mu$, read in kilograms. Then the potential in clause (v) has dimension

  $[G_{\mathrm{N}}{m}_i^g{m}_j^g\mathrm{/}\mathrm{\mid }{x}_i-x_j\mathrm{\mid }]=({\mathrm{\ell }}^3{\mu }^{-1}{\tau }^{-2}){\mu }^2{\mathrm{\ell }}^{-1}=\mu {\mathrm{\ell }}^2{\tau }^{-2}\mathrm{.}$

  as required for energy. If a model additionally adopts $m_i^g=m_i$, this assignment is dimensionally consistent with the inertial mass dimension $[m_i]=\mu$; dimensional agreement does not establish that empirical identification.

Let a force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance be given: time window $I_0\subseteq \mathbb{R}$ nondegenerate and open, configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ open (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator clause (a)), fixed masses $m_1,\dots ,m_N>0$ (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)), and a force map $F$ with the standing continuity and state-local-Lipschitz hypotheses of clause (iii) of that postulate.

(a) State domain and inverse-mass operator. The state domain of the model is

$D:=I_0\times Q\times ({\mathbb{R}}^3{)}^N\subseteq \mathbb{R}\times {\mathbb{R}}^{3N}\times {\mathbb{R}}^{3N},$

an open subset of $\mathbb{R}\times {\mathbb{R}}^{6N}$; a point $(t,x,v)\in D$ is a time together with the state $z=(x,v)$ of positions $x\in Q$ and coordinate velocities $v\in ({\mathbb{R}}^3{)}^N$ of the labels (Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The inverse-mass operator is the linear map

${\mathsf{M}}^{-1}:{\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N},{\mathsf{M}}^{-1}(u_1,\dots ,u_N):=(\frac{u_1}{m_1},\dots ,\frac{u_N}{m_N}),$

the inverse of the mass operator $\mathsf{M}$ of clause (d) of Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator.

(b) Associated first-order system. The associated first-order field of the model is

$\mathcal{F}:D⟶{\mathbb{R}}^{3N}\times {\mathbb{R}}^{3N},\mathcal{F}(t,x,v):=(v,{\mathsf{M}}^{-1}F(t,x,v)),$

and the associated first-order system is $z^{\mathrm{\prime }}=\mathcal{F}(t,z)$ on the open domain $D$, in the sense of First-order systems, initial value problems, and solutions on intervals; here $z=(x,v)\in {\mathbb{R}}^{6N}$. The map $\mathcal{F}$ satisfies the Picard–Lindelöf hypotheses of Picard-Lindelöf local existence and uniqueness for first-order systems: it is continuous on $D$, and it is locally Lipschitz in the state variable $z$ locally uniformly in time, because its first component $z\mapsto v$ is linear and its second is the composition of the force map $F$ — which is locally Lipschitz in $(x,v)$ locally uniformly in time by Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iii) — with the linear map ${\mathsf{M}}^{-1}$.

(c) Motions. Let $J\subseteq \mathbb{R}$ 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 $C^2$ such that $t\in I_0$ for every $t\in J$ and

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

Equivalently: the state curve $z=(x,\dot{x}):J\to Q\times {\mathbb{R}}^{3N}$ is a solution of the associated first-order system on $J$. Indeed, if $x$ is a motion then $z=(x,x^{\mathrm{\prime }})$ is $C^1$ with $z^{\mathrm{\prime }}=(x^{\mathrm{\prime }},x^{\mathrm{\prime }\mathrm{\prime }})=(x^{\mathrm{\prime }},{\mathsf{M}}^{-1}F(t,x,x^{\mathrm{\prime }}))=\mathcal{F}(t,z)$ and $(t,z(t))\in D$; conversely, if $z=(x,v)$ is a solution with $(t,z(t))\in D$ for all $t$, then $x^{\mathrm{\prime }}=v$ and $v^{\mathrm{\prime }}={\mathsf{M}}^{-1}F(t,x,v)$, so $v$ is $C^1$, $x$ is $C^2$, and multiplying the second equation by the masses returns the displayed second-order equation. Motions of the labels of the model are exactly these curves by Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iv).

(d) Initial data and maximal motion. An initial datum of the model is a triple $(t_0,x_0,v_0)$ with $t_0\in I_0$, $x_0\in Q$ and $v_0\in ({\mathbb{R}}^3{)}^N$; write $z_0:=(x_0,v_0)$. A maximal motion with this datum is a motion $x:J_{\max }\to Q$ with $x(t_0)=x_0$ and $\dot{x}(t_0)=v_0$ that has no extension to a motion of the model on a strictly larger interval. By the Picard–Lindelöf hypotheses verified in (b) and the maximal-solution theorem Every Picard–Lindelöf initial value problem has one maximal solution on an open interval (with Extensions, maximal solutions, maximal intervals, and global solutions): every initial datum has exactly one maximal motion, its maximal interval $J_{\max }$ is an open interval containing $t_0$ and contained in $I_0$, and every motion with the same datum is the restriction of the maximal one. The restriction statement transfers from the associated first-order system to motions through the equivalence in (c).

(e) Newtonian flow. The Newtonian flow of the model is the map

$\mathrm{\Phi }:\mathrm{\Omega }⟶Q\times {\mathbb{R}}^{3N},\mathrm{\Omega }:=\{(t,t_0,z_0):t_0\in I_0,z_0\in Q\times {\mathbb{R}}^{3N},t\in J_{\max }(t_0,z_0)\},$

defined by $\mathrm{\Phi }(t;t_0,z_0):=z(t)$, where $t\mapsto z(t)=(x(t),\dot{x}(t))$ is the state curve of the maximal motion with datum $(t_0,z_0)$. By (d) the assignment is well defined: the maximal motion is unique, so $\mathrm{\Phi }$ is a function on $\mathrm{\Omega }$ and not merely a relation. For fixed $t_0$ and $z_0$ the curve $t\mapsto \mathrm{\Phi }(t;t_0,z_0)$ is defined exactly on the maximal interval $J_{\max }(t_0,z_0)$.

• The flow is a solution family, not a completeness claim. Whether $J_{\max }(t_0,z_0)$ is all of $I_0$ is a property of the particular force map and datum, not part of this definition; the criterion for a finite endpoint is the subject of Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain.

• Regularity of the flow is a separate question. This definition asserts no continuity, differentiability or measurability of $\mathrm{\Phi }$ in any variable; such statements would be separate results with their own hypotheses on the differentiability of $F$.

• Frame covariance. A motion in the sense of (c) is defined with respect to one fixed inertial frame; the transformation of the description between inertial frames and of the force map is clause (vi) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, and under $t^{\mathrm{\prime }}=t+b$ the maximal interval becomes $J_{\max }+b$. Maximality is preserved on the transferred domain: any proper extension there would pull back by the inverse affine transformation to a proper extension of the original motion.

Adopt the force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: time window $I_0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, masses $m_1,\dots ,m_N>0$, and a force map $F$ continuous and locally Lipschitz in the state variable, locally uniformly in time, on $D=I_0\times Q\times ({\mathbb{R}}^3{)}^N$. Let $\mathcal{F}(t,x,v)=(v,{\mathsf{M}}^{-1}F(t,x,v))$ be the associated first-order field and let motions and maximal motions be as in The associated first-order system, motions, maximal motions and the Newtonian flow of a force model. Fix an initial datum $(t_0,x_0,v_0)\in I_0\times Q\times ({\mathbb{R}}^3{)}^N$ and write $z_0=(x_0,v_0)$.

1. Local existence. There is $h>0$ and there is a motion $x$ of the model on $[t_0-h,t_0+h]$ with $x(t_0)=x_0$ and $\dot{x}(t_0)=v_0$; moreover such a motion is exactly one of class $C^2$ satisfying $m_i{\ddot{x}}_i(t)=F_i(t,x(t),\dot{x}(t))$ for all $i$ and all $t\in [t_0-h,t_0+h]$.
2. Uniqueness. If $x:J\to Q$ and $\tilde{x}:\tilde{J}\to Q$ are motions of the model with the same initial datum, $x(t_0)=\tilde{x}(t_0)=x_0$ and $\dot{x}(t_0)=\dot{\tilde{x}}(t_0)=v_0$, then $x=\tilde{x}$ on $J\cap \tilde{J}$.
3. Quantitative cylinder. Let $h,r>0$ satisfy $C:=[t_0-h,t_0+h]\times \bar{B}(z_0,r)\subseteq D$, put $M:=\sup \{\parallel \mathcal{F}(t,z){\parallel }_2:(t,z)\in C\}$ and let $\mathrm{\Lambda }$ be a state-Lipschitz constant of $\mathcal{F}$ on $C$. If $hM\le r$ and $\mathrm{\Lambda }h<1$, then there is exactly one motion on $[t_0-h,t_0+h]$ through the datum whose state graph $(t,(x(t),\dot{x}(t)))$ lies in $C$, and every motion through the datum agrees with it on the intersection of its interval with $[t_0-h,t_0+h]$.
4. Maximal motion. There is exactly one maximal motion with the datum: its maximal interval $J_{\max }$ is an open interval containing $t_0$ and contained in $I_0$, and every motion with the datum is its restriction.

The statement is conditional on the adopted force model: clause 1 locates a solution only near $t_0$ and bounds no interval length in terms of the data, clause 4 asserts no completeness of $J_{\max }$ in $I_0$, and no empirical content is claimed.

• What the theorem does not give. No growth bound, no lower bound on $h$ uniform over data, and no global existence in time; the interval length in clause 1 depends on the local Lipschitz data of $F$ near the datum through the cylinder construction of [L3]. Escape from the domain at a finite time is the subject of Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain.

• The regularity hypotheses are exactly used. Clause (iii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance supplies continuity and the state-local Lipschitz property, which are the two hypotheses of [L3]; neither is strengthened nor weakened here, and the companion counterexample on the examples page shows that continuity alone does not give clause 2.

Adopt the force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with time window $I_0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, masses $m_1,\dots ,m_N>0$ and force map $F$ continuous and locally Lipschitz in the state variable, locally uniformly in time; let $D=I_0\times Q\times ({\mathbb{R}}^3{)}^N$, $\mathcal{F}(t,x,v)=(v,{\mathsf{M}}^{-1}F(t,x,v))$, $U:=Q\times {\mathbb{R}}^{3N}$, and let motions, maximal motions and the state curve $z=(x,\dot{x})$ be as in The associated first-order system, motions, maximal motions and the Newtonian flow of a force model. Fix an initial datum $(t_0,x_0,v_0)\in I_0\times Q\times ({\mathbb{R}}^3{)}^N$, write $z_0=(x_0,v_0)$, and let $x_{\max }:J_{\max }\to Q$ be the maximal motion with this datum, with state curve $z=(x_{\max },{\dot{x}}_{\max }):J_{\max }\to U$ and $J_{\max }=(a,b)$ its maximal interval, $-\mathrm{\infty }\le a<b\le +\mathrm{\infty }$.

1. Existence and uniqueness of the maximal motion. $J_{\max }$ is an open interval containing $t_0$ and contained in $I_0$, and every motion with the datum is a restriction of $x_{\max }$.
2. Breakdown criterion. If $b<+\mathrm{\infty }$, then for every compact $K\subseteq D$ and every $\eta >0$ with $(b-\eta ,b)\cap J_{\max }\mathrm{\ne }\mathrm{\varnothing }$ there is $t\in (b-\eta ,b)\cap J_{\max }$ with $(t,z(t))\mathrm{\notin }K$. Equivalently: if there are a compact $K\subseteq D$ and $\delta >0$ with $(t,z(t))\in K$ for all $t\in (b-\delta ,b)\cap J_{\max }$, then $x_{\max }$ extends to a motion of the model on a strictly larger interval than $J_{\max }$ — so $J_{\max }$ cannot have right endpoint $b$.
3. Autonomous case. Suppose $F$ does not depend on its time argument: $F(t,x,v)=F_0(x,v)$ for a map $F_0:U\to ({\mathbb{R}}^3{)}^N$ satisfying the standing regularity of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iii). Then the associated field is autonomous, $\mathcal{F}(t,x,v)=(v,{\mathsf{M}}^{-1}{F}_0(x,v))$, the equations do not use the time coordinate, and the model is read on the extended time window $\mathbb{R}$ (the standing hypotheses are unchanged by the extension). For this clause, take the same autonomous force law as a model on the extended time window $I_0=\mathbb{R}$, and let $J_{\max }=(a,b)$ be the maximal interval for that extended-time model. If $b<+\mathrm{\infty }$ and there are a compact $K_0\subseteq U$ and $\delta >0$ with $z(t)\in K_0$ for all $t\in (b-\delta ,b)\cap J_{\max }$, then the state curve extends past $b$ as a solution of the same model, contradicting maximality; thus no finite maximal endpoint has such a terminal compact trap. In particular, if one compact $K_0\subseteq U$ contains $z(t)$ for every $t\in J_{\max }\cap [t_0,+\mathrm{\infty })$, then $b=+\mathrm{\infty }$ and the motion is defined for all forward times $t\ge t_0$.
4. Reading. Suppose the time window is all of $\mathbb{R}$ (the time-dependent case included) and $b<+\mathrm{\infty }$. Then for every compact $K_0\subseteq U$ and every $\delta >0$ there is $t\in (b-\delta ,b)\cap J_{\max }$ with $z(t)\mathrm{\notin }{K}_0$: near a finite breakdown the state leaves every compact subset of the open state domain $U=Q\times {\mathbb{R}}^{3N}$. Consequently either the state is unbounded on every terminal interval, meaning that for every $M,\delta >0$ some $t\in (b-\delta ,b)\cap J_{\max }$ has $\parallel z(t){\parallel }_2>M$, or $Q\mathrm{\ne }{\mathbb{R}}^{3N}$ and the positions approach the excluded configurations in the following sense: for every $\epsilon ,\delta >0$ there are $t\in (b-\delta ,b)\cap J_{\max }$ and $y\in {\mathbb{R}}^{3N}\setminus Q$ with $\parallel x_{\max }(t)-y{\parallel }_2<\epsilon$. If $Q={\mathbb{R}}^{3N}$, the unbounded-state alternative necessarily holds.