Dimensions and SI unit conventions of the non-relativistic classical mechanics framework

(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.