Framework, quantities and geometry

Adopt as physical assumptions of the non-relativistic classical mechanics framework the following statements about spacetime.

(i) Spacetime data. Spacetime is the mathematical structure $(V,\tau ,g,A,\text{fixed orientation of }K)$ of Affine time space with Euclidean simultaneous slices and adapted charts: a four-dimensional real vector space $V$; a surjective linear map $\tau :V\to \mathbb{R}$, the absolute-time covector; the kernel $K=\ker \tau$, a three-dimensional real vector space, the spatial displacement space; a spatial metric $g$, a symmetric positive-definite bilinear form $g:K\times K\to \mathbb{R}$; a fixed orientation of $K$, that is, a chosen one of the two classes of ordered bases of $K$ under the equivalence generated by positive-determinant changes of basis; and an affine space $A$ modelled on $V$ (a nonempty set with a free and transitive $V$-action, with displacements as in Affine time space with Euclidean simultaneous slices and adapted charts). The elements of $A$ are the events of the model. The structure is the arena of the model; no equation of motion, force law or matter content is asserted by this clause.

(ii) Interpretation of the data. For a displacement $w\in V$, the number $\tau (w)$ is the elapsed time carried by $w$, read in seconds; for a displacement $k\in K$ inside a simultaneous slice, the number $g(k,k)$ is the squared spatial length of $k$, read in square metres; the square root $\sqrt{g(k,k)}$ is the length of $k$. The dimensions and units are those fixed by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, so that $[\tau (w)]=\tau$ and $[g(k,k)]={\mathrm{\ell }}^2$. The orientation of $K$ is the convention relative to which spatial bases are called positively oriented and against which orientation-preserving rotations are defined.

(iii) Absolute time, simultaneity and distance. Two events $p,q\in A$ are simultaneous when their displacement lies in $K$, that is, when $\tau (q-p)=0$. Simultaneity is an equivalence relation on $A$: it is reflexive because $\tau (0)=0$, symmetric because $\tau (p-q)=-\tau (q-p)$ and transitive because $\tau$ is additive. Its classes are the simultaneous slices $A_t=\{p\in A:\tau (p-o)=t\}$ of any adapted chart; the class does not depend on the choice of adapted chart, and the elapsed time between two events is the chart-independent number $\tau (q-p)$. Two events of a common slice carry the slice distance $d_g(p,q):=\sqrt{g(p-q,p-q)}$, which is defined because $p-q\in K$; distances are not defined across distinct slices by this structure.

(iv) Smooth structure. $A$ is read with the affine smooth structure determined by its affine coordinates: an adapted chart of Affine time space with Euclidean simultaneous slices and adapted charts is a bijection $A\to \mathbb{R}\times {\mathbb{R}}^3$ whose inverse is the map $(t,x)\mapsto o+tu+{\sum }_i{x}_i{e}_i$, and curves are declared $C^k$ when their coordinate functions in one (equivalently, every) adapted chart are $C^k$. The equivalence of the two readings is supplied by the affine transition law between adapted charts proved in Transformation between adapted charts and its kinematic consequences; this postulate adopts the resulting smooth structure as the one in which regularity of worldlines and of force maps is stated.

(v) What the structure excludes. The data contain no inner product on $V$, no Lorentzian metric, no light cone, no universal speed bound, no null cones and no causal classification of displacements other than the split into $\tau$-values; they contain no distinguished origin, no distinguished vector $u$ with $\tau (u)=1$ and no distinguished basis of $K$. Consequently the structure by itself distinguishes no inertial frame: the class of frames and the transformations between them are fixed separately, by the frame postulate Inertial frames, stationary observers and the adopted Galilean transformations between frames. No statement about the speed of any signal, about electromagnetic propagation or about gravitation is made here.

(vi) Status. These clauses are adopted assumptions of the model, not theorems and not empirical findings. They are not derived from the uniqueness theory of differential equations, from any symmetry principle or from experiment; the Newtonian tradition and Galilean relativity motivate them. The model makes no claim that the physical world is exactly a four-dimensional affine space with these data, and no numerical measurement is recorded.

• Adopted, not deduced. Clauses (i)-(iv) fix the arena and the reading of its data; clause (v) records what is deliberately absent. Nothing here selects a force law, an equation of motion, a matter model or a solution of any equation; those enter through the postulates and definitions of the pages that follow.

• Domain of intended use. The model is intended for the non-relativistic regime, in which speeds are small compared with the speed of light and a classical particle or continuum description, as appropriate to the application, and a classical description of the apparatus are adequate. The framework sets no numerical cutoff for that regime; an application must state the regime in which the model, its force law and its apparatus description are adequate. Relativistic, quantum and quantum-gravitational regimes are outside the scope of this model.

• Empirical content is downstream. Empirical significance lies in the comparison of predictions computed from a fully specified model with reported observations; those comparisons carry apparatus, calibration, sampling and uncertainty qualifications and are recorded where the framework treats them. This postulate itself makes no quantitative empirical claim, and agreement of a model with data would be evidence about the model, not a proof of this postulate.

• Absolute time, not absolute space. Absolute simultaneity and Euclidean slice geometry are part of the adopted model. The model does not contain a distinguished rest frame: two inertial frames are related by the transformations of Inertial frames, stationary observers and the adopted Galilean transformations between frames, and the structure alone cannot select one of them. This is a property of the adopted data, not a conclusion of relativistic kinematics, and no Lorentzian or light-cone structure is imported.

Adopt as physical assumptions of the framework the following statements about frames, in the spacetime structure of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure.

(i) Rotations. A rotation is a matrix $R\in SO(3)$, where

$SO(3)=\{R\in {\mathbb{R}}^{3\times 3}:R^{\mathsf{T}}R=I_3,\det R=1\}$

in the sense of The orthogonal group $O(3)$ and the rotation group $SO(3)$. A rotation acts on spatial column vectors by matrix multiplication; its entries are dimensionless under the radian convention of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework. Rotations are part of the formulation of spatial changes of basis; the definition of $SO(3)$ itself asserts nothing about physical space.

(ii) Inertial frames. An inertial frame is the global adapted affine chart of the spacetime data determined by

$o\in A,u\in V\text{ with }\tau (u)=1,(e_1,e_2,e_3)\text{ an ordered, }g\text{-orthonormal, positively oriented basis of }K,$

with adapted coordinates $(t,x)$ as in Affine time space with Euclidean simultaneous slices and adapted charts: the time coordinate is $t(p)=\tau (p-o)$ and the spatial coordinates are the coefficients of $p-o-tu\in K$ in the basis $(e_1,e_2,e_3)$. Existence and uniqueness of the coordinates, and the affine structure of the slices, are the mathematical content of Existence and uniqueness of adapted coordinates, and the affine structure of simultaneous slices and Affine time space with Euclidean simultaneous slices and adapted charts; no physical claim beyond the reading of clause (iii) is attached to them here.

An inertial frame is a coordinate chart: the triple $(o,u,(e_i))$. It is definitively not an observer congruence and not a time-varying basis field. In the model the basis $(e_i)$ is fixed once and for all; a family of bases depending on the time coordinate is a different structure, whose kinematics is defined separately where it is used, and is not an inertial frame. The physical reading of the frame is that $t$ is the model's time coordinate in seconds and $x$ the spatial position in metres, by the conventions of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.

(iii) Stationary observers. For $x_0\in {\mathbb{R}}^3$ the curve ${\alpha }_{x_0}:\mathbb{R}\to A$, ${\alpha }_{x_0}(t):=o+tu+{\sum }_{j=1}^3(x_0{)}_j{e}_j$, is the observer at rest in the frame at position $x_0$; the family $\{{\alpha }_{x_0}{\}}_{x_0\in {\mathbb{R}}^3}$ is the stationary observer congruence of the frame. Each curve has $t({\alpha }_{x_0}(s))=s$, is $C^{\mathrm{\infty }}$ in adapted coordinates, and in the frame's own spatial coordinates has constant coordinate triple $x_0$. Here an observer worldline means precisely a $C^2$ curve $\alpha :I\to A$ on a nondegenerate open real interval with $t(\alpha (s))=s$, used to describe an observer; the stationary curves just defined have $I=\mathbb{R}$. This kinematic definition assigns neither a particle label nor a mass or a dynamical law to an observer. A particle worldline will separately be assigned to a particle label in the particle postulate. The congruence is a derived object, determined by the frame and the spacetime data, not an additional primitive. Distinct frames can have the same family of parametrized curves: if $o^{\mathrm{\prime }}=o+{\sum }_j{c}_j{e}_j$ while $u^{\mathrm{\prime }}=u$ and $e_j^{\mathrm{\prime }}=e_j$, then ${\alpha }_{x_0}^{\mathrm{\prime }}(t)={\alpha }_{x_0+c}(t)$ for all $t$. Thus shifting only the spatial origin relabels this family. Changing $u$ changes the curve family, since every curve in it has the affine derivative $u$.

(iv) Allowed changes between frames. The model adopts the class of transformations between inertial frames

$t^{\mathrm{\prime }}=t+b,x^{\mathrm{\prime }}=Rx+vt+a,$

where $b\in \mathbb{R}$ is a time shift ($[b]=\tau$), $a\in {\mathbb{R}}^3$ a spatial translation ($[a_i]=\mathrm{\ell }$), $v\in {\mathbb{R}}^3$ a boost velocity ($[v_i]=\mathrm{\ell }{\tau }^{-1}$) and $R\in SO(3)$ a rotation ($[R_{ij}]=1$). Every pair of inertial frames is related by exactly one such transformation: that two adapted charts of the same data differ by a transformation of this form, with $R\in SO(3)$ because both bases are positively oriented, is the mathematical transformation law proved in Transformation between adapted charts and its kinematic consequences. Uniqueness follows by comparing the coordinate maps on all $(t,x)\in \mathbb{R}\times {\mathbb{R}}^3$: evaluating the time map fixes $b$; evaluating the spatial map at $(0,0)$ fixes $a$, at $(1,0)$ fixes $v$, and at $(0,e_j)$ for the three standard coordinate vectors fixes each column of $R$. The kinematic consequences of the law — the transformation of coordinate velocity and acceleration, and the frame-independence of zero coordinate acceleration — are proved as a statement about the model in Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes.

(v) Free motion is not asserted here. The framework's dynamics enters through the force and equation-of-motion postulates of the pages that follow; those pages adopt, as the content of Newton's first law, that there exist inertial frames in the sense of clause (ii) in which a particle with zero applied interaction has zero coordinate acceleration. This postulate introduces the frames and their changes but asserts no dynamics: it does not state that any particular motion has zero acceleration, and it does not privilege any one inertial frame — each of them is a legitimate arena for the formulation of the model's equations.

(vi) Kinematic symmetry is not conservation. The class (iv) is a class of coordinate changes. A quantity is conserved along a motion when its value as a function of the frame time is constant on the interval of that motion. No such constancy is asserted here, nor is it asserted that every mechanical system's description is invariant in form; a force law must also transform appropriately under (iv) to be a law of the model in every frame, and an external apparatus or background field can single out frames for the description of a concrete experiment. Conservation statements, when they hold, are theorems of later pages with their own hypotheses, not consequences of this postulate.

• Adopted, not deduced. Clauses (i)-(iv) adopt the frame notion and the transformation class; clauses (v)-(vi) delimit what is not asserted. The mathematical transformation law between adapted charts is cited, not reproved, and is used only through its exact statement and hypotheses.

• Domain of intended use. Inertial frames are the frames of the non-relativistic model of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure: spatial Euclidean geometry and absolute time. The transformations of clause (iv) are Galilean; no Lorentz transformation, no boost composition law of special relativity and no light-speed limit appears in this model.

• Empirical content. The classification of an apparatus frame as inertial, and the adequacy of the Galilean transformation class for a concrete experiment, are application-level judgements with stated approximations; this postulate asserts no measurement and no numerical value. Agreement of model predictions with observations is evidence about the model, not a proof of these assumptions.

Adopt as physical assumptions of the framework the following statements about particles and their motion, in the spacetime of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure and the frames of Inertial frames, stationary observers and the adopted Galilean transformations between frames.

(i) Particle labels and worldlines. A model of the framework has particle labels $i=1,\dots ,N$ for some $N\ge 1$. To each label and each inertial frame of the model — an adapted chart $(o,u,(e_1,e_2,e_3))$ with coordinates $(t,x)$ in the sense of Inertial frames, stationary observers and the adopted Galilean transformations between frames — there is assigned a worldline ${\gamma }_i:I\to A$ described in that frame, where $I\subseteq \mathbb{R}$ is a nondegenerate open interval — that is, an interval of the form $(a,b)$ with $-\mathrm{\infty }\le a<b\le +\mathrm{\infty }$ and at least two points — the curve ${\gamma }_i$ is of class $C^2$ in the adapted coordinates of the frame, and the parameter condition

$\tau ({\gamma }_i(t)-o)=t(t\in I)$

holds: the curve parameter is the elapsed time from the origin of the frame, so in the frame's coordinates ${\gamma }_i(t)=(t,x_i(t))$ with $x_i:I\to {\mathbb{R}}^3$ of class $C^2$. The parameter condition replaces the causal conditions of relativistic kinematics: no timelike condition on the velocity and no upper bound on its Euclidean norm is imposed, and the parameter is not free up to reparametrization.

(ii) Description in another frame. The same motion described in a second inertial frame is the curve obtained by the time reparametrization of the two frames: if $(t^{\mathrm{\prime }},x^{\mathrm{\prime }})$ are the coordinates of the second frame and $t^{\mathrm{\prime }}=t+b$ is the time component of the transformation between the two frames (Inertial frames, stationary observers and the adopted Galilean transformations between frames, Transformation between adapted charts and its kinematic consequences), then the description of the motion in the second frame is the curve ${\gamma }_{i,b}(t^{\mathrm{\prime }}):={\gamma }_i(t^{\mathrm{\prime }}-b)$, defined on $I+b$, and it again satisfies the parameter condition $\tau ({\gamma }_{i,b}(t^{\mathrm{\prime }})-o^{\mathrm{\prime }})=t^{\mathrm{\prime }}$ with respect to the second frame's origin $o^{\mathrm{\prime }}$. This is a consistency requirement of the model's frame language; the accompanying spatial transformation of the coordinate curve is the affine law of Transformation between adapted charts and its kinematic consequences, and its kinematic consequences are proved in Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes.

(iii) Regularity is frame-independent. The class of the worldline is independent of the frame used to describe it: by the transformation law of Transformation between adapted charts and its kinematic consequences, the coordinates of the same event in two inertial frames differ by $t^{\mathrm{\prime }}=t+b$, $x^{\mathrm{\prime }}=Rx+vt+a$ with constant $b,a,v$ and $R\in SO(3)$, so a coordinate curve is twice continuously differentiable in one inertial frame exactly when its reparametrized description is twice continuously differentiable in the other. The assumptions of this postulate are therefore read in any one inertial frame without loss.

(iv) Velocity and acceleration. In a fixed inertial frame the quantities $w_i:=d{x}_i\mathrm{/}dt$ and $a_i:=d^2{x}_i\mathrm{/}d{t}^2$ are the coordinate velocity and coordinate acceleration of the label, of dimensions $\mathrm{\ell }{\tau }^{-1}$ and $\mathrm{\ell }{\tau }^{-2}$ by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework; they are defined precisely, and their transformation under a change of inertial frame is recorded, in Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors.

(v) Fixed inertial mass. Each particle label $i$ carries a mass $m_i>0$, a primitive parameter of the model with dimension $\mu$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The mass is fixed: it is the same positive number at all times and in all inertial frames of this model; the index $i$ identifies a label within that model and asserts no cross-model identification or equality of mass parameters. It is not a function of the time coordinate, of the position or of the velocity, and no variable-mass model is a model of this clause. The mass enters the dynamics only through its later appearance in the equation of motion; this postulate attaches no dynamics to it. Distinct labels may carry distinct masses, and the masses are model data, not measured values.

(vi) Point-particle idealization. The model retains of each label only its position and its fixed mass, together with the label index: size, shape, spin, deformation, temperature and internal degrees of freedom are omitted, and the quantity $x_i(t)$ is the single point representing the label at time $t$. An application of the model must state when this idealization is adequate for the system at hand; for a body whose rotation, deformation or internal structure matters for the question asked, the point-particle model is not adequate and a richer model must be specified. The framework makes no claim that every physical body is a point particle.

(vii) Calibration caveat. An apparatus calibration may estimate a mass parameter of a concrete body under a stated measurement model, with its assumptions, sampling and uncertainty; such an estimate neither defines the mass parameter of a label nor verifies the equation of motion in which it occurs, and it is not deduced from this postulate. No measured value of any mass is asserted by this item.

• Adopted, not deduced. Clauses (i)-(vi) fix the model's particle ontology; the parameter condition and the fixed-mass clause are part of that ontology, not theorems. Clause (vii) delimits the interface with measurement. Nothing here asserts that a concrete object satisfies the idealization, and no empirical comparison is made.

• No dynamics. Neither a force law nor an equation of motion is asserted. In particular the model does not yet say what $a_i$ is; the dynamics is adopted on the later pages that introduce force, initial data and the equations of motion. Consequently no conservation statement and no prediction follows from this postulate alone.

• Domain of intended use. The non-relativistic regime of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure; particles moving at speeds small compared with the speed of light, in situations where the point-particle idealization and the classical description of the apparatus are adequate. The framework sets no numerical cutoff and records no measurement.

(a) Dimensions. Write $\mathrm{\ell }$, $\tau$, $\mu$ for the three independent dimension symbols of length, duration and mass. A dimension is an element of the free abelian group generated by $\mathrm{\ell },\tau ,\mu$, that is, a triple $(a,b,c)\in {\mathbb{Z}}^3$ written ${\mathrm{\ell }}^a{\tau }^b{\mu }^c$; products and inverses of dimensions are componentwise addition and negation, so that ${\mathrm{\ell }}^a{\tau }^b{\mu }^c\cdot {\mathrm{\ell }}^{a^{\mathrm{\prime }}}{\tau }^{b^{\mathrm{\prime }}}{\mu }^{c^{\mathrm{\prime }}}={\mathrm{\ell }}^{a+a^{\mathrm{\prime }}}{\tau }^{b+b^{\mathrm{\prime }}}{\mu }^{c+c^{\mathrm{\prime }}}$ and $({\mathrm{\ell }}^a{\tau }^b{\mu }^c{)}^{-1}={\mathrm{\ell }}^{-a}{\tau }^{-b}{\mu }^{-c}$. The dimensionless dimension is ${\mathrm{\ell }}^0{\tau }^0{\mu }^0$, abbreviated $1$.

(b) The dimension map. The conventions assign to the quantities of the framework a dimension $[X]$, subject to the rules

$[XY]=[X][Y],[X\mathrm{/}Y]=[X][Y{]}^{-1},\left[\frac{dX}{du}\right]=[X][u{]}^{-1},$

and to the requirement that a sum $X+Y$ is meaningful only when $[X]=[Y]$, in which case $[X+Y]=[X]$. The third rule applies to a differentiable dependence of a quantity $X$ on a quantity $u$ and covers iterated differentiation, so that $[d^{2}X\mathrm{/}d{u}^2]=[X][u{]}^{-2}$. The assignment is part of the conventions: the rules above are consistency requirements on it, not physical laws and not theorems.

(c) Reference units. The metre is the reference unit of length, the second the reference unit of duration and the kilogram the reference unit of mass. A quantity of dimension ${\mathrm{\ell }}^a{\tau }^b{\mu }^c$ is expressed in the unit ${\mathrm{m}}^a{\mathrm{s}}^b{\mathrm{k}\mathrm{g}}^c$: its value is a real multiple of that unit, and the conventions fix how the numerical value of a quantity is read (a coordinate value alone is a real number, and the unit declaration attaches its dimension). The coordinates of an adapted chart of Affine time space with Euclidean simultaneous slices and adapted charts are read with the time coordinate $t$ in seconds, $[t]=\tau$, and the spatial coordinate components $x_1,x_2,x_3$ in metres, $[x_i]=\mathrm{\ell }$; the slice distance $d_g$ of Affine time space with Euclidean simultaneous slices and adapted charts between simultaneous events is read in metres.

(d) Typed quantity conventions. The following constructions specify the quantities whose dimensions are recorded here, without adopting their dynamics. Fix a positive integer $N$, a nondegenerate open interval $I\subseteq \mathbb{R}$ with time parameter $t$ in seconds, position curves $x_i:I\to {\mathbb{R}}^3$ of class $C^2$ with components in metres, and constant parameters $m_i>0$ in kilograms, for $1\le i\le N$. Derivatives and integrals have the meanings of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral. These are conditional numerical constructions in a fixed spatial coordinate system; whether they represent a particular system is a separate model assumption.

1. Velocity and acceleration are the maps $w_i=x_i^{\mathrm{\prime }}:I\to {\mathbb{R}}^3$ and $a_i=x_i^{\mathrm{\prime }\mathrm{\prime }}:I\to {\mathbb{R}}^3$, of dimensions $\mathrm{\ell }{\tau }^{-1}$ and $\mathrm{\ell }{\tau }^{-2}$ componentwise.
2. The mass operator is the linear map $\mathsf{M}:({\mathbb{R}}^3{)}^N\to ({\mathbb{R}}^3{)}^N$, $\mathsf{M}(z_1,\dots ,z_N)=(m_1{z}_1,\dots ,m_N{z}_N)$, with dimension $\mu$; the momentum of label $i$ is $p_i=m_i{w}_i:I\to {\mathbb{R}}^3$, with component dimension $\mu \mathrm{\ell }{\tau }^{-1}$.
3. A force input is a specified map $F:I\times O\times {\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N}$, where $O\subseteq {\mathbb{R}}^{3N}$ is open, its arguments are time, positions and velocities, and each output component has dimension $\mu \mathrm{\ell }{\tau }^{-2}$. No regularity or equation equating $F$ with $\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}$ is adopted here. A later model must state both if needed.
4. For a specified scalar potential energy map $U:I\times O\to \mathbb{R}$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ and a tuple $x(t)\in O$, the kinetic energy is $T(t)=\frac{1}{2}{\sum }_{i=1}^N{m}_i{\sum }_{j=1}^3(w_i(t){)}_j^2$, and the mechanical energy is $E(t)=T(t)+U(t,x(t))$. Both are real-valued and have dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$; no conservation or applicability to every interaction is asserted.
5. Given a scalar map $L:I\times O\times {\mathbb{R}}^{3N}\to \mathbb{R}$ of the energy dimension and $t_0<t_1$ in $I$, assume $t\mapsto L(t,x(t),w(t))$ is integrable on $[t_0,t_1]$. Its action on this curve segment is the real number $S[x;t_0,t_1]={\int }_{t_0}^{t_1}L(t,x(t),w(t))dt$, with dimension $\mu {\mathrm{\ell }}^2{\tau }^{-1}$ (the integration convention multiplies the integrand's dimension by duration). No variational principle is imposed.
6. Relative to the chosen spatial origin, define the angular momentum $J_i:I\to {\mathbb{R}}^3$ by $J_i(t)=x_i(t)\times p_i(t)$, where for $r,s\in {\mathbb{R}}^3$ the operation is explicitly $r\times s=(r_2{s}_3-r_3{s}_2,r_3{s}_1-r_1{s}_3,r_1{s}_2-r_2{s}_1)$. Its components have dimension $\mu {\mathrm{\ell }}^2{\tau }^{-1}$. These quantities depend on the chosen coordinates; dimensions alone assert no transformation or conservation law.
7. A rotation matrix here means a real $3\times 3$ matrix $R$ satisfying $R^{\mathsf{T}}R=I_3$ and $\det R=1$; its entries are dimensionless. A radian angle is an arc length divided by its positive radius, hence dimensionless. A generalized coordinate means a component of a specified local coordinate chart on a configuration space; each such component has its own declared dimension, and only quantities of equal dimension are added.

(e) Conventions only. No numerical value of any constant or parameter of a model is fixed by this item: in particular no mass, length or duration of any concrete system is asserted, and no measured value is recorded. Two numerical values that are equal but carry different dimensions are not thereby the same quantity. Where a treatment or a later item suppresses a dimensionful constant by a natural-unit convention, that convention is read as shorthand for the corresponding SI-dimensioned statement with the constant restored, and it changes no dimension fixed here.

• The dimension symbols and the rules of (b) are bookkeeping of the conventions. They constrain how the framework writes its equations; they neither predict a value nor restrict which models are admissible.

• The dimension map is a homomorphism-type convention on products, quotients and derivatives only. It is not extended here to sums of unequal dimension (which are excluded), to transcendental functions (whose arguments must be dimensionless where they occur) or to arbitrary nonlinear operations; each later item that introduces such an operation declares the dimensions of its outputs.

• This item belongs to the physical framework because it fixes how physical quantities are read, but it adopts no physical law. The physical content of the framework begins with the spacetime postulate Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure and the frame and worldline assumptions that follow it.

Let $N\ge 1$ particle labels with fixed masses $m_1,\dots ,m_N>0$ be given as in Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses, and fix one inertial frame of the model with spatial coordinates $x_i\in {\mathbb{R}}^3$ for each label $i$. Identify the space of all position tuples with ${\mathbb{R}}^{3N}=({\mathbb{R}}^3{)}^N$ and write its points as $x=(x_1,\dots ,x_N)$; each component $x_i$ has dimension $\mathrm{\ell }$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

(a) Unconstrained branch. The unconstrained configuration domain is an open set $Q\subseteq {\mathbb{R}}^{3N}$, chosen as part of the model. A tuple $x\in Q$ is an admissible configuration: an assignment of one position to each label at one time. A singular-force exclusion is the removal of the collision set

$\mathrm{\Delta }:=\{x\in {\mathbb{R}}^{3N}:\text{ there are }i\mathrm{\ne }j\text{ with }{x}_i=x_j\}$

from the domain, which is done when the model's force law is not defined at coincident positions: $\mathrm{\Delta }$ is closed (it is the union of the finitely many closed sets $\{x_i=x_j\}$, each the zero set of the continuous map $x\mapsto x_i-x_j$), so $Q:={\mathbb{R}}^{3N}\setminus \mathrm{\Delta }$ is open and is a legitimate configuration domain. Whether the exclusion is part of the model is a modelling decision recorded with the force law; the framework itself only requires $Q$ to be open and stated.

(b) Holonomic constraint branch. Let $O\subseteq {\mathbb{R}}^{3N}$ be an open ambient set, let $k\in \mathbb{N}$ with $1\le k\le 3N-1$, and let

$G=(G_1,\dots ,G_k):O⟶{\mathbb{R}}^k$

be a smooth map, each component carrying a declared dimension and unit fixed by the model (a component can always be rescaled by a nonzero constant without changing its zero set, and the rescaling is part of the declaration). Assume:

1. Nonempty level set: $G^{-1}(0)\mathrm{\ne }\mathrm{\varnothing }$;
2. Rank condition: the differential $DG(x):{\mathbb{R}}^{3N}\to {\mathbb{R}}^k$ is surjective, equivalently has rank $k$, at every $x\in G^{-1}(0)$.

Then $0$ is a regular value of $G$ in the sense of Regular and critical points and values, and the constrained configuration space is $Q:=G^{-1}(0)$. By A regular level set is an embedded submanifold — applied to the smooth map $G$ from the open subset $O$ of ${\mathbb{R}}^{3N}$ to ${\mathbb{R}}^k$, whose regular value $0$ has nonempty fibre — $Q$ is an embedded submanifold of ${\mathbb{R}}^{3N}$ of dimension $3N-k$ (Embedded submanifolds and slice charts). The degrees of freedom of this branch of the model is the number $\dim Q=3N-k\mathrm{.}$ In the sense of Constrained configurations, the mass matrix and the mass metric, $Q$ together with the inclusion $X:Q\hookrightarrow {\mathbb{R}}^{3N}$ is then a particle configuration, and the mass metric of that definition is available with the masses $m_i$; the configuration is constrained to $Q$ for all times in this branch.

(c) Time-dependent holonomic constraints. If the constraint map depends on time,

$G:\mathbb{R}\times O⟶{\mathbb{R}}^k,(t,x)⟼G(t,x),$

with $G$ smooth and $D_{x}G(t_0,x)$ of rank $k$ at every $x$ of the nonempty set $G(t_0,\cdot {)}^{-1}(0)$, then for each fixed $t_0\in \mathbb{R}$ the instantaneous configuration space $Q_{t_0}:=G(t_0,\cdot {)}^{-1}(0)$ is an embedded submanifold of dimension $3N-k$ by the same theorem applied to $G(t_0,\cdot )$. The family $\{Q_{t_0}\}$ varies with $t_0$; a curve $t\mapsto q(t)$ with $G(t,q(t))=0$ is not thereby a curve in one fixed manifold, and the kinematic treatment of such constraints (velocities, the reaction forces that maintain them and the distinction between rheonomic and scleronomic constraints) belongs to the framework's constraint development and is not part of this definition.

(d) Mass operator. In the ambient coordinates define the mass operator $\mathsf{M}:{\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N}$ by block multiplication,

$\mathsf{M}(x_1,\dots ,x_N):=(m_1{x}_1,\dots ,m_N{x}_N);$

its matrix in the standard basis is block diagonal with blocks $m_i{I}_3$, the mass matrix of Constrained configurations, the mass matrix and the mass metric. Its dimension is $\mu$: it multiplies positions of dimension $\mathrm{\ell }$ by masses of dimension $\mu$. On the constrained branch, for the inclusion $X:Q\hookrightarrow {\mathbb{R}}^{3N}$ and tangent vectors $h,z\in T_{q}Q$, the mass metric of that item is ${\mathcal{G}}_q(h,z):=⟨d{X}_q(h),\mathsf{M}d{X}_q(z){⟩}_{3N}={\sum }_{i=1}^N{m}_i{h}_i\cdot z_i,$ where $h_i,z_i\in {\mathbb{R}}^3$ are the blocks of $d{X}_q(h),d{X}_q(z)$ under the canonical Euclidean tangent identification specified in that supplier. It restricts the mass-weighted ambient bilinear form to $T_{q}Q$, rather than the unweighted Euclidean inner product.

(e) Velocity constraints are a different object. A velocity-map constraint on the open ambient set $O$ is a pair of smooth maps $A:\mathbb{R}\times O\to {\mathbb{R}}^{k^{\mathrm{\prime }}\times 3N},b:\mathbb{R}\times O\to {\mathbb{R}}^{k^{\mathrm{\prime }}},1\le k^{\mathrm{\prime }}\le 3N,$ with $A(t,x)$ of constant rank $k^{\mathrm{\prime }}$ everywhere in the declared domain, imposing $A(t,x(t))\dot{x}(t)+b(t,x(t))=0$ on a $C^1$ curve $x:J\to O$, where $J\subseteq \mathbb{R}$ is a nondegenerate open interval, the parameter $t$ is frame time, and $\dot{x}(t):=x^{\mathrm{\prime }}(t)$ is the componentwise derivative in the convention of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework. Equivalently $A(t,x)$ is a linear map ${\mathbb{R}}^{3N}\to {\mathbb{R}}^{k^{\mathrm{\prime }}}$. Each row's output unit is declared, and its corresponding component of $b$ has that same unit. The allowed velocities at $(t,x)$ form the affine set $\{v:A(t,x)v+b(t,x)=0\}$; its differences form $\ker A(t,x)$.

The law is locally holonomic-equivalent at $(t_0,x_0)$ if on some open neighbourhood $V\subseteq \mathbb{R}\times O$ of that point there are a $C^2$ map $g:V\to {\mathbb{R}}^{k^{\mathrm{\prime }}}$ with $D_{x}g$ of rank $k^{\mathrm{\prime }}$ and a $C^2$ matrix map $C:V\to {\mathbb{R}}^{k^{\mathrm{\prime }}\times k^{\mathrm{\prime }}}$ with $\det C\mathrm{\ne }0$ such that $A=C{D}_{x}g,b=C{\mathrm{\partial }}_{t}g\text{on }V\mathrm{.}$ Thus the velocity law is locally the derivative of the regular positional constraints $g(t,x)=\text{constant}$. It is locally nonholonomic at the specified point when no such neighbourhood and pair $(g,C)$ exists; a nonholonomic model in this local sense declares a point (or a region of points) with this failure. For a homogeneous time-independent law this rules out such a regular-level-set representation throughout a neighbourhood. It need not rule out an individual embedded submanifold $S\subseteq O$ with $T_{x}S=\ker A(x)$ for every $x\in S$, using the ambient tangent identification of Constrained configurations, the mass matrix and the mass metric; for an affine time-dependent law the criterion also includes the drift $b$. Absence of a single global pair alone does not establish local nonintegrability. The rank of $A$ counts instantaneous velocity restrictions, not the dimension of a positional configuration manifold; accessibility requires its own definition. Rank changes require a separate model.

• The definition is a stage, not a law. Nothing here says how $x$ varies with time; the equation of motion, the force law and the reaction forces of the constrained branch are adopted later in the framework, each with its own hypotheses. The dimension count of clause (b) is a statement about $G^{-1}(0)$ as a set, proved from the established regular-level theorem; it presupposes exactly the nonemptiness and rank hypotheses displayed there.

• Choice of coordinates. The definition uses the redundant particle positions $x\in {\mathbb{R}}^{3N}$ as coordinates, not a set of generalized coordinates of $Q$; describing a constrained configuration by $3N-k$ coordinates requires charts of the embedded submanifold, and the transformation between such charts is part of the differential geometry of $Q$. The distinction between the redundant description and a chart is not erased by the count $\dim Q$.

• Dimensions of constraint components. The dimension of a constraint component is part of the model's declaration, exactly as the unit of a physical quantity is: a constraint expressed in metres and the same constraint multiplied by a second quantity have different components and the same zero set. The rescaling freedom is recorded so that a model can choose convenient units, and it changes no physical content of the constraint.

Let an inertial frame $(o,u,(e_1,e_2,e_3))$ of the model with adapted coordinates $(t,x)$ be fixed, and let $\gamma :I\to A$ be the worldline of a particle label described in this frame as in Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses; thus $\gamma$ is of class $C^2$, parametrized so that $t(\gamma (s))=s$ for all $s\in I$, and

$\gamma (s)=o+su+{\sum }_{j=1}^3{\xi }_j(s)e_j$

for the unique coordinate functions ${\xi }_j:I\to \mathbb{R}$ of the point $\gamma (s)$ in the adapted chart (Affine time space with Euclidean simultaneous slices and adapted charts, Existence and uniqueness of adapted coordinates, and the affine structure of simultaneous slices).

(a) The coordinate curve. The coordinate curve of $\gamma$ in the frame is

$x_{\gamma }:=({\xi }_1,{\xi }_2,{\xi }_3):I⟶{\mathbb{R}}^3\mathrm{.}$

It is of class $C^2$ by the regularity clause of Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses, and it determines $\gamma$ by the display above, since the parameter condition fixes the time coordinate. It has dimension $[{\xi }_j]=\mathrm{\ell }$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

(b) Coordinate velocity and acceleration. The derivative of a vector-valued map is taken componentwise: for the map $x_{\gamma }$ above, its derivative at $s$ is

$x_{\gamma }^{\mathrm{\prime }}(s):=({\xi }_1^{\mathrm{\prime }}(s),{\xi }_2^{\mathrm{\prime }}(s),{\xi }_3^{\mathrm{\prime }}(s)),$

where each ${\xi }_j^{\mathrm{\prime }}(s)={\lim }_{h\to 0}({\xi }_j(s+h)-{\xi }_j(s))\mathrm{/}h$ is the ordinary derivative of a real function. The coordinate velocity of the label in the frame is

$w(s):=x_{\gamma }^{\mathrm{\prime }}(s)\in {\mathbb{R}}^3,$

with components $w_j={\xi }_j^{\mathrm{\prime }}$, and the coordinate acceleration is the coordinate velocity of the velocity,

$a(s):=w^{\mathrm{\prime }}(s)=x_{\gamma }^{\mathrm{\prime }\mathrm{\prime }}(s)\in {\mathbb{R}}^3\mathrm{.}$

Their dimensions are $[w_j]=\mathrm{\ell }{\tau }^{-1}$ and $[a_j]=\mathrm{\ell }{\tau }^{-2}$, read in metres per second and metres per second squared. The triple $w(s)$ is the triple of numbers read in the frame's axes: it is a coordinate description, not an element of spacetime.

(c) Spatial velocity and acceleration vectors. For the affine curve $\gamma :I\to A$, subtraction of events uses the affine displacement in $V$. Define its derivative by the limit in the finite-dimensional vector space $V$, ${\gamma }^{\mathrm{\prime }}(s):={\lim }_{h\to 0}\frac{\gamma (s+h)-\gamma (s)}{h}=u+{\sum }_j{\xi }_j^{\mathrm{\prime }}(s)e_j,$ and define ${\gamma }^{\mathrm{\prime }\mathrm{\prime }}(s)$ as the derivative of the $V$-valued map ${\gamma }^{\mathrm{\prime }}$. This is the canonical identification of each affine tangent space with $V$; the $C^2$ coordinate regularity guarantees both limits. The frame's time direction $u$ and spatial basis identify two elements of the spatial displacement space $K$ attached to the motion:

$\underset{‾}{v}(s):={\gamma }^{\mathrm{\prime }}(s)-u={\sum }_{j=1}^3{w}_j(s)e_j\in K,\underset{‾}{a}(s):={\gamma }^{\mathrm{\prime }\mathrm{\prime }}(s)={\sum }_{j=1}^3{a}_j(s)e_j\in K\mathrm{.}$

Here ${\gamma }^{\mathrm{\prime }}(s)-u$ indeed lies in $K=\ker \tau$, because $\tau ({\gamma }^{\mathrm{\prime }}(s))=\frac{d}{ds}\tau (\gamma (s)-o)=1$ by linearity of $\tau$ and the parameter condition. The vectors $\underset{‾}{v}(s)$ and $\underset{‾}{a}(s)$ are the spatial velocity vector and spatial acceleration vector of the label in the frame; the velocity depends on the frame time direction $u$, whereas the acceleration is an intrinsic element of $K$ unchanged by an inertial frame change (time shifts have derivative $1$). Their coordinates in the frame's basis $(e_1,e_2,e_3)$ are the triples $w(s)$ and $a(s)$.

(d) Speed. The speed of the label in the frame at time $s$ is the Euclidean norm $\mathrm{\parallel }w(s){\mathrm{\parallel }}_2=\sqrt{{\sum }_j{w}_j(s{)}^2}$, of dimension $\mathrm{\ell }{\tau }^{-1}$; equivalently $\mathrm{\parallel }\underset{‾}{v}(s){\mathrm{\parallel }}_2=\sqrt{g(\underset{‾}{v}(s),\underset{‾}{v}(s))}$, the norm induced by the spatial metric, since the basis is $g$-orthonormal.

(e) Transformation under a change of inertial frame (recorded). Let a second inertial frame be given, with coordinates $(t^{\mathrm{\prime }},x^{\mathrm{\prime }})$ and transformation parameters $(b,a,v,R)$ as in Inertial frames, stationary observers and the adopted Galilean transformations between frames, and let $\eta (t^{\mathrm{\prime }}):=x^{\mathrm{\prime }}(\gamma (t^{\mathrm{\prime }}-b))\in {\mathbb{R}}^3$ be the description of the motion in the second frame. Then

${\eta }^{\mathrm{\prime }}(t^{\mathrm{\prime }})=Rw(t^{\mathrm{\prime }}-b)+v,{\eta }^{\mathrm{\prime }\mathrm{\prime }}(t^{\mathrm{\prime }})=Ra(t^{\mathrm{\prime }}-b),$

that is, in the differentiated variables and abbreviated notation $w^{\mathrm{\prime }}=Rw+v$ and $a^{\mathrm{\prime }}=Ra$. This law is clause 2 of the mathematical transformation law of Transformation between adapted charts and its kinematic consequences, recorded here in the model's notation; the definition itself asserts neither law. The model-level statement of the law, including the frame-independence of the property $a\equiv 0$, is proved as a theorem later on this page.

• Velocity is frame-dependent; acceleration rotates. By clause (e) the coordinate velocity changes under a boost, $w^{\mathrm{\prime }}=Rw+v$, while the coordinate acceleration transforms by the rotation alone, $a^{\mathrm{\prime }}=Ra$; the speed is invariant under the rotation part of a frame change and changes under boosts. These are kinematic properties of the model's frames and hold before any dynamics is introduced.

• Spatial vectors and their coordinates. $\underset{‾}{v}={\gamma }^{\mathrm{\prime }}-u$ changes when the frame time direction changes. By contrast $\underset{‾}{a}={\gamma }^{\mathrm{\prime }\mathrm{\prime }}$ has the same value in $K$ in every inertial frame at the corresponding event; its coordinate triple rotates by clause (e). A change of basis changes coordinates without changing this affine vector.

• No dynamical content. Nothing in this item says how $a(s)$ is determined: the force law and the equation of motion are adopted on the later pages of the framework. In particular the identification of $ma$ with a force is not made here.

Adopt the spacetime assumptions of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure, the frame assumptions of Inertial frames, stationary observers and the adopted Galilean transformations between frames and the worldline assumptions of Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses. Let

$F=(o,u,(e_1,e_2,e_3)),F^{\mathrm{\prime }}=(o^{\mathrm{\prime }},u^{\mathrm{\prime }},(e_1^{\mathrm{\prime }},e_2^{\mathrm{\prime }},e_3^{\mathrm{\prime }}))$

be two inertial frames of the model with adapted coordinates $(t,x)$ and $(t^{\mathrm{\prime }},x^{\mathrm{\prime }})$, and let $\gamma :I\to A$ be a worldline of a particle label described in $F$, with coordinate curve $x:I\to {\mathbb{R}}^3$ and coordinate velocity and acceleration $w=x^{\mathrm{\prime }}$ and $a=x^{\mathrm{\prime }\mathrm{\prime }}$ in the sense of Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors. Then:

1. There are $b\in \mathbb{R}$, $a_0,v\in {\mathbb{R}}^3$ and $R\in SO(3)$ (The orthogonal group $O(3)$ and the rotation group $SO(3)$) such that for every event $p\in A$

$t^{\mathrm{\prime }}(p)=t(p)+b,x^{\mathrm{\prime }}(p)=Rx(p)+vt(p)+a_0\mathrm{.}$

The parameters have dimensions $[b]=\tau$, $[a_{0,j}]=\mathrm{\ell }$ and $[v_j]=\mathrm{\ell }{\tau }^{-1}$, and $R$ is dimensionless (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework); they are the ones that occur in the adopted class of frame changes of Inertial frames, stationary observers and the adopted Galilean transformations between frames.

2. Let $\eta :I+b\to {\mathbb{R}}^3$ be the coordinate curve of the same label in $F^{\mathrm{\prime }}$, defined by $\eta (t^{\mathrm{\prime }}):=x^{\mathrm{\prime }}(\gamma (t^{\mathrm{\prime }}-b))$ for $t^{\mathrm{\prime }}\in I+b$. Then $\eta$ is of class $C^2$ and

${\eta }^{\mathrm{\prime }}(t^{\mathrm{\prime }})=Rw(t^{\mathrm{\prime }}-b)+v,{\eta }^{\mathrm{\prime }\mathrm{\prime }}(t^{\mathrm{\prime }})=Ra(t^{\mathrm{\prime }}-b)(t^{\mathrm{\prime }}\in I+b)\mathrm{.}$

3. The coordinate acceleration vanishes identically in one frame exactly when it does in the other:

$a\equiv 0\text{ on }I⟺{\eta }^{\mathrm{\prime }\mathrm{\prime }}\equiv 0\text{ on }I+b\mathrm{.}$

Equivalently, the property "the coordinate curve is an affine function of the frame time" is an inertial-frame-independent property of the motion.

4. Write $G(b,a_0,v,R)$ for the transformation $(t,x)\mapsto (t+b,Rx+vt+a_0)$ of the coordinate space. Then

$G(b_2,a_2,v_2,R_2)\circ G(b_1,a_1,v_1,R_1)=G(b_1+b_2,R_2{a}_1+v_2{b}_1+a_2,R_2{v}_1+v_2,R_2{R}_1)$

and

$G(b,a_0,v,R{)}^{-1}=G(-b,-R^{\mathsf{T}}(a_0-vb),-R^{\mathsf{T}}v,R^{\mathsf{T}}),$

with all parameters again in $\mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3\times SO(3)$. Consequently the Galilean transformations form a group of transformations of the coordinate space, and the relation on inertial frames "the two charts differ by a Galilean transformation" is an equivalence relation.