Galilean inertial charts and the center-of-mass rest frame

For this nonrelativistic framework, a Newtonian event-coordinate domain is the Cartesian product $I\times {\mathbb{R}}^3$ (The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$), where $I\subseteq \mathbb{R}$ is an open time interval as in Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, measured in seconds, and ${\mathbb{R}}^3$ has the Euclidean inner product and norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ (the norm properties follow from Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation); lengths are measured in metres as in SI quantities and unit conventions. A Galilean inertial chart labels events in such a domain by coordinates $(t,\mathbf{x})$. If a second chart uses the translated time domain $I^{\mathrm{\prime }}=I+\tau$, the allowed coordinate changes are $t^{\mathrm{\prime }}=t+\tau ,{\mathbf{x}}^{\mathrm{\prime }}=R\mathbf{x}+\mathbf{a}-\mathbf{b}t,$ where $\tau$ is a constant time shift measured in seconds, $\mathbf{a}\in {\mathbb{R}}^3$ is a constant spatial translation measured in metres, $\mathbf{b}\in {\mathbb{R}}^3$ is a constant boost velocity measured in metres per second, and $R\in M_{3\times 3}(\mathbb{R})$ is a dimensionless constant matrix (The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations) satisfying $R^{\mathsf{T}}R=I_3$ using matrix transpose and product (The transpose $A^{\mathsf{T}}$ of a matrix, Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes), with $I_3$ the identity matrix. In the coordinate formula, $(R\mathbf{x}{)}_i:={\sum }_{k<3}{R}_{ik}{x}_k$ for $i<3$. Thus time is absolute in this model and no Lorentzian structure is used. Here “frame” means this coordinate chart, not an observer congruence or a basis/tetrad field.

For a finite point-mass representation with $N\in \mathbb{N}$ (The natural numbers $\mathbb{N}$ (von Neumann)), $N\ge 1$, masses $m_i>0$ in kg at positions ${\mathbf{x}}_i\in {\mathbb{R}}^3$ in metres for $i<N$, assume $M:={\sum }_{i<N}{m}_i>0$ (Finite sums and finite products, by recursion) and define the center-of-mass coordinates by $X_{\mathrm{c}\mathrm{m},j}:=M^{-1}{\sum }_{i<N}{m}_i{x}_{i,j}$ for $j<3$; these three real coordinates form ${\mathbf{X}}_{\mathrm{c}\mathrm{m}}$. The masses are invariant under the allowed chart changes. Other mechanical models must supply their total mass and center-of-mass map; this thermodynamic item does not construct those quantities for a continuum or field model. Every admitted mechanical model must also supply center-of-mass maps and velocities that transform under the displayed Galilean chart changes by ${\mathbf{X}}_{\mathrm{c}\mathrm{m}}^{\mathrm{\prime }}(t^{\mathrm{\prime }})=R{\mathbf{X}}_{\mathrm{c}\mathrm{m}}(t)+\mathbf{a}-\mathbf{b}t$ and ${\mathbf{v}}_{\mathrm{c}\mathrm{m}}^{\mathrm{\prime }}(t^{\mathrm{\prime }})=R{\mathbf{v}}_{\mathrm{c}\mathrm{m}}(t)-\mathbf{b}$, where $t^{\mathrm{\prime }}=t+\tau$; these are kinematic compatibility conditions, not dynamical theorems about arbitrary mechanical models. When a model supplies three coordinate functions $X_{\mathrm{c}\mathrm{m},j}:I\to \mathbb{R}$ for $j<3$ as in The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$, assemble its center-of-mass position as ${\mathbf{X}}_{\mathrm{c}\mathrm{m}}(t):=(X_{\mathrm{c}\mathrm{m},0}(t),X_{\mathrm{c}\mathrm{m},1}(t),X_{\mathrm{c}\mathrm{m},2}(t))\in {\mathbb{R}}^3$. At a time $t\in I$ where each component limit exists, define the velocity by $v_{\mathrm{c}\mathrm{m},j}(t)={\lim }_{h\to 0}(X_{\mathrm{c}\mathrm{m},j}(t+h)-X_{\mathrm{c}\mathrm{m},j}(t))\mathrm{/}h$ through $H_t:=\{h\in \mathbb{R}:t+h\in I,h\mathrm{\ne }0\}$, for each $j<3$, using the scalar limit of The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$; these three components form ${\mathbf{v}}_{\mathrm{c}\mathrm{m}}(t)$. A statewise instantaneous center-of-mass rest chart at a specified time $t$ is the Galilean chart obtained by the pure boost $\mathbf{b}={\mathbf{v}}_{\mathrm{c}\mathrm{m}}(t)$ with $R=I_3$, $\mathbf{a}=0$, and $\tau =0$. Its center-of-mass velocity vanishes at that event, but need not vanish at other times because the boost is constant. For an interval $J\subseteq I$ on which ${\mathbf{v}}_{\mathrm{c}\mathrm{m}}$ exists at every time, a single Galilean chart has zero center-of-mass velocity throughout $J$ if and only if ${\mathbf{v}}_{\mathrm{c}\mathrm{m}}$ is constant on $J$: in such a chart $R{\mathbf{v}}_{\mathrm{c}\mathrm{m}}(t)=\mathbf{b}$ for every $t\in J$, so ${\mathbf{v}}_{\mathrm{c}\mathrm{m}}(t)=R^{\mathsf{T}}\mathbf{b}$ is constant, and conversely a constant velocity ${\mathbf{v}}_0$ is made zero by the pure boost $\mathbf{b}={\mathbf{v}}_0$. For a fixed closed body with total mass $M>0$ in kg, its separately tracked translational kinetic energy in a selected inertial chart is $K_{\mathrm{c}\mathrm{m}}=\frac{1}{2}M\mathrm{\parallel }{\mathbf{v}}_{\mathrm{c}\mathrm{m}}{\mathrm{\parallel }}_2^2$ in joules; under a pure boost it is $K_{\mathrm{c}\mathrm{m}}^{\mathrm{\prime }}=\frac{1}{2}M\mathrm{\parallel }{\mathbf{v}}_{\mathrm{c}\mathrm{m}}-\mathbf{b}{\mathrm{\parallel }}_2^2$.

In this framework $U$ denotes the scalar internal-energy coordinate evaluated at a state in its statewise instantaneous center-of-mass rest chart. Thus an equilibrium preparation's internal energy is evaluated using the boost selected by its center-of-mass velocity at that preparation; a process using one chart with the center of mass at rest throughout its interval must separately satisfy the constant-velocity condition above. Bulk translational kinetic energy and any separately tracked external-field potential energy are excluded from $U$. For a given state, the model treats $U$ as invariant between its center-of-mass rest charts that differ only by spatial translation or orthogonal orientation, and fixes its numerical zero, including any retained or subtracted constituent rest-energy offsets, consistently across the composition changes it admits. This is a nonrelativistic modeling convention, not a derived energy-transformation law.