Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope

Adopt as physical assumptions of the framework the following rigid-material model branch, in the spacetime and frames of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure and Inertial frames, stationary observers and the adopted Galilean transformations between frames.

(i) Reference mass distribution. A rigid-material model carries, besides the framework's time and frame structure, a reference mass distribution: a measure $\nu$ on the Borel space $({\mathbb{R}}^3,\mathcal{B}({\mathbb{R}}^3))$ that is a mass distribution in the sense of Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix — finite and bounded, with $\nu ({\mathbb{R}}^3\setminus {\bar{B}}_R(0))=0$ for some real $R>0$, and of total mass $m:=\nu ({\mathbb{R}}^3)\in (0,+\mathrm{\infty })$ — whose reference coordinates are centred, $\int \rho d\nu (\rho )=0\in {\mathbb{R}}^3\mathrm{.}$ The label variable $\rho$ is a coordinate of the model's reference placement and has dimension $\mathrm{\ell }$, each component read in metres; the total mass $m$ has dimension $\mu$, read in kilograms, and $R$ has dimension $\mathrm{\ell }$ and is the model's reference size (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Under the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, used only through Finiteness of the moments of a bounded finite Borel measure on ${\mathbb{R}}^3$ to make the polynomial moments of $\nu$ of degree at most two finite, the first and second moments of $\nu$ are finite real numbers. The measure $\nu$ and the number $m$ are model data: no distribution of mass of any concrete body and no measured mass is asserted.

(ii) Time window and motions. A model carries a time window $I_0\subseteq \mathbb{R}$, a nondegenerate open interval $(a,b)$ with $-\mathrm{\infty }\le a<b\le +\mathrm{\infty }$ and at least two points, read in seconds. Fix an inertial frame of the model, that is the adapted chart $(o,u,(e_1,e_2,e_3))$ with coordinates $(t,x)$ of Inertial frames, stationary observers and the adopted Galilean transformations between frames. A motion of the model in this frame is a map $x:I_0\times {\mathbb{R}}^3\to {\mathbb{R}}^3$ for which there exist a map $c:I_0\to {\mathbb{R}}^3$ and a map $R:I_0\to SO(3)$, both of class $C^2$ (componentwise derivatives, Affine time space with Euclidean simultaneous slices and adapted charts for the coordinates and The orthogonal group $O(3)$ and the rotation group $SO(3)$ for the target), such that $x(t,\rho )=c(t)+R(t)\rho (t\in I_0,\rho \in {\mathbb{R}}^3)\mathrm{.}$ Adopt: the motions of the model are the maps of this form, and the pair $(c,R)$ is the configuration path of the motion. The map $x(t,\cdot ):{\mathbb{R}}^3\to {\mathbb{R}}^3$ is the placement at time $t$ and is the rigid placement of Rigid placements, body and spatial angular velocity, and material velocities with translation $c$ and rotation $R$; its material velocity is ${\mathrm{\partial }}_{t}x(t,\rho )=c^{\mathrm{\prime }}(t)+R^{\mathrm{\prime }}(t)\rho$, defined by differentiability of $c$ and of the entries of $R$. The time window $I_0$ is the interval on which the model's configuration path is considered; the model's time coordinate is the frame time $t$.

(iii) Rigidity idealization. The idealization adopted is that the model describes a body that does not deform, and its content is the representation (ii): a placement of the form $x=c+R\rho$ with $R\in SO(3)$ preserves every mutual distance, since $R(t)$ preserves the Euclidean norm (The orthogonal group $O(3)$ and the rotation group $SO(3)$), so for all $\rho ,{\rho }^{\mathrm{\prime }}\in {\mathbb{R}}^3$ the number $\parallel x(t,\rho )-x(t,{\rho }^{\mathrm{\prime }}){\parallel }_2=\parallel \rho -{\rho }^{\mathrm{\prime }}{\parallel }_2$ is independent of $t$. The model class (ii) is adopted as the definition of a rigid-material placement; a motion of material points that preserves all mutual distances but whose label set fails to determine a rotation is not claimed to be of this form, and the general characterization of distance-preserving placements is neither asserted nor used on this page. For a nondegenerate labeling — a label set containing the origin and two linearly independent material vectors — distinct configurations $(c,R)$ and $(\bar{c},\bar{R})$ give distinct placements, by A labeling containing two independent vectors has trivial $SO(3)$-stabilizer; the configuration data of (ii) therefore do not overcount for such a body. Degenerate labelings and degenerate mass distributions are not excluded by this postulate; the later items of the pair state where invertibility of the inertial data must be assumed.

(iv) Centre of mass and orientation. Because the reference coordinates are centred, the identity $\int x(t,\rho )d\nu (\rho )=mc(t)$ holds for every motion, by linearity of the integral and $\int \rho d\nu =0$; the physical reading is that $c(t)$ is the position of the centre of mass of the model in the placement at time $t$. The rotation $R(t)$ is the model's orientation: it carries reference coordinates into the frame's spatial coordinates, and the body-fixed frame at time $t$ is the ordered orthonormal basis $(R(t)e_1,R(t)e_2,R(t)e_3)$ of the spatial space. The reading of $R(t)$ as the orientation of the placement uses the nondegeneracy of (iii): for a nondegenerate labeling the material points determine $R(t)$ uniquely, so the orientation is an attribute of the body's placement and not of the choice of label coordinates.

(v) Frame changes. The description of a motion in a second inertial frame is obtained by the change of coordinates $t^{\mathrm{\prime }}=t+b$, $x^{\mathrm{\prime }}=Qx+ut+a$ of Inertial frames, stationary observers and the adopted Galilean transformations between frames clause (iv), applied to every event of the motion; here $Q\in SO(3)$ and $u\in {\mathbb{R}}^3$ are the rotation and the boost velocity of that clause, written $R$ and $v$ there and renamed here to avoid a clash with the orientation path $R(t)$ of clause (ii) and with the material velocity field of the pair's definition. With the same $\rho$ labelling a material point, the description in the second frame is $\tilde{x}(t^{\mathrm{\prime }},\rho ):=Qx(t^{\mathrm{\prime }}-b,\rho )+u(t^{\mathrm{\prime }}-b)+a\mathrm{.}$ This description is again a motion of the model, with configuration path $\tilde{c}(t^{\mathrm{\prime }})=Qc(t^{\mathrm{\prime }}-b)+u(t^{\mathrm{\prime }}-b)+a,\tilde{R}(t^{\mathrm{\prime }})=QR(t^{\mathrm{\prime }}-b),$ because $SO(3)$ is closed under multiplication (The orthogonal group $O(3)$ and the rotation group $SO(3)$); this is a consistency requirement of the model's frame language, exactly as for the particle worldlines of Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (ii). The derived transformation laws of the angular velocity, of the inertia data and of the kinetic splitting are theorems of this pair's page, not part of this postulate.

(vi) No dynamics. This postulate fixes the kinematic class of the model and its mass data only. It asserts no force law, no torque law, no equation of motion for the pair $(c,R)$, and no balance or conservation law; in particular the model does not by itself constrain the configuration path, and the effect of forces and torques on a body is a separate model assumption. It also asserts no measured value of any mass, length, duration or orientation.

(vii) Scope and exclusions. The branch excludes deformation, internal degrees of freedom such as temperature, chemical state or an independent spin variable, variable mass, ablation and impact events, contact, friction and dissipative constitutive laws, and any relativistic or quantum description. It excludes bodies whose mass distribution is not a bounded finite Borel measure of the stated kind. A rigid-material model is not a system of finitely many particle labels: the continuum measure $\nu$ carries the mass data of the model, and the situations of point masses are the special cases of measures supported on finitely many points. All physical statements about the model weight material points by $\nu$; properties of $\nu$-null sets of labels are not physical properties of the body.

• Rigidity is an adopted assumption, not a derived law. Clause (iii) declares the model class; no theorem of this pair proves that any physical body is rigid, and no measured shape or mass is recorded. A statement of the branch is a conditional consequence of this postulate, the cited mathematics and the item's own hypotheses.

• The mass measure is fixed in time. The measure $\nu$ is attached to the model once and does not depend on the time coordinate or on the frame; the motor of the motion is the pair $(c,R)$ alone. This is what excludes variable-mass and deformable descriptions from the branch.

• What the pair proves. The definitions of this pair fix the configuration space ${\mathbb{R}}^3\times SO(3)$, the angular velocity and the inertia data of the model; its theorems prove the inertial identities of the motion (total momentum, kinetic splitting, angular momentum about the centre of mass), the principal-axis statements and the frame-covariance laws, each under its stated hypotheses. None of these completes the model: forces, torques and the corresponding equations of motion are supplied elsewhere.

• Degenerate shapes are carried, not hidden. The postulate admits mass distributions supported on a line through the origin. For such a shape the inertia tensor is not invertible and the relation between angular velocity and angular momentum loses injectivity; the pair's theorems state the criterion for invertibility, and the companion page exhibits the failure.

• Frames are inertial and fixed. The branch uses the inertial frames of Inertial frames, stationary observers and the adopted Galilean transformations between frames and their allowed changes; time-dependent bases, rotating frames and non-inertial observers are different structures and belong to other pages of the track.