Rigid-body configuration and inertia

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.

Adopt the model data, the time window $I_0$, the inertial frame and the notation of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope: the reference mass distribution $\nu$ with total mass $m$ and centred reference coordinates, and a motion $x(t,\rho )=c(t)+R(t)\rho$ with configuration path $(c,R)$ of class $C^2$.

(a) Configuration and configuration space. The configuration of the model at time $t$ in the frame is the pair $(c(t),R(t))$; the pair $(c,R)$ is a configuration path. The configuration space of the model is the set $\mathcal{C}:={\mathbb{R}}^3\times SO(3),$ read through the identification of $M_3(\mathbb{R})$ with ${\mathbb{R}}^9$ (The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations) so that $\mathcal{C}\subseteq {\mathbb{R}}^{12}$. The configuration determines the placement $x=c+R\cdot$ at that time, and the reference placement is $(c,R)=(0,I_3)$. Under the identification of $M_3(\mathbb{R})$ with ${\mathbb{R}}^9$, the group $SO(3)$ is a compact three-dimensional embedded submanifold of $M_3(\mathbb{R})$ and a subgroup of the general linear group ($SO(3)$ is a compact three-dimensional embedded submanifold of $M_3(\mathbb{R})$, with tangent space the matrices $R\mathrm{\Omega }$ for skew $\mathrm{\Omega }$ clauses 1 and 3, $SO(3)$ is a subgroup of the general linear group, with $R^{-1}=R^{\mathsf{T}}$); consequently $\mathcal{C}={\mathbb{R}}^3\times SO(3)$ is a six-dimensional smooth manifold with the product structure. Its dimension $3+3=6$ is read as three translation coordinates of $c$ and three orientation degrees of freedom of $R$; the model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope has no other configuration degrees of freedom, since deformation and internal variables are excluded by that postulate.

(b) Centre of mass, orientation, body frame. Because $\int \rho d\nu =0$, the configuration's translation is the position of the centre of mass of the model, $\int x(t,\rho )d\nu (\rho )=mc(t)$. The rotation $R(t)$ is the orientation of the model at time $t$: the linear map that carries reference coordinates into the frame's spatial coordinates, so that a material point whose reference coordinate is $\rho$ is found at the spatial position $x(t,\rho )$. 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, obtained by applying the orientation to the standard basis; the basis is $g$-orthonormal because $R(t)\in O(3)$ preserves the Euclidean inner product (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, The orthogonal group $O(3)$ and the rotation group $SO(3)$). The material point of reference coordinate $\rho$ has spatial position $x(t,\rho )$ and reference coordinates $\rho$.

(c) Physical labels and faithfulness. The physical label set is $S:=\mathrm{supp}\nu$, defined here as the complement of the union of all open $\nu$-null subsets of ${\mathbb{R}}^3$. Thus $\rho \in S$ exactly when every open neighbourhood of $\rho$ has positive mass. This is a closed set carrying all mass: the rational open boxes form a countable basis (${\mathbb{Q}}^n$ is a countable dense subset of ${\mathbb{R}}^n$, and rational open boxes form a countable basis, $n=3$), so its complement is the union of the countably many null basis boxes. Disjointifying this union and using countable additivity (Measures on sigma-algebras) gives $\nu ({\mathbb{R}}^3\setminus S)=0$. In particular $S\mathrm{\ne }\mathrm{\varnothing }$ because $m>0$.

If two configurations satisfy $c+R\rho =\bar{c}+\bar{R}\rho$ for every $\rho \in S$, integration against the centred measure gives $mc=m\bar{c}$, hence $c=\bar{c}$. Their exact orientation ambiguity is therefore ${\bar{R}}^{-1}R\in {\mathrm{Stab}}_{SO(3)}(S):=\{H\in SO(3):H\rho =\rho \text{ for every }\rho \in S\},$ the pointwise stabilizer of A labeling containing two independent vectors has trivial $SO(3)$-stabilizer. Conversely, $c=\bar{c}$ and this stabilizer condition give equality of the placements on $S$ by multiplication. A nondegenerate physical labeling is one whose $S$ contains two linearly independent vectors; its stabilizer is $\{I_3\}$ by that lemma, so the physical placement determines $(c,R)$ uniquely. The same injectivity conclusion holds for any auxiliary label set containing the origin and two independent vectors, by evaluation at the origin followed by the lemma, as in Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope clause (iii).

A degenerate physical labeling has no two independent vectors. If $S$ contains a nonzero vector $u$, then $S\subseteq \mathbb{R}u$, and its stabilizer is exactly $\{H\in SO(3):Hu=u\}$, the rotations fixing the line $\mathbb{R}u$ pointwise; the orientation is determined only up to right composition with this subgroup. If $S$ contains no nonzero vector, then $S=\{0\}$ and its stabilizer is all of $SO(3)$. More generally a singleton support $\{q\}$ must have $q=0$, since centring gives $mq=0$. Thus a centred point mass has no distinguished orientation axis. These are all the degenerate cases: after choosing any nonzero $u\in S$, failure of independence forces every vector in $S$ to be a multiple of $u$. Configuration pairs remain model data in every case, including orientation data invisible in the placed physical labels.

(d) Description in another inertial frame. Let a second inertial frame be related to the first by the transformation $t^{\mathrm{\prime }}=t+b$, $x^{\mathrm{\prime }}=Qx+ut+a$ of Inertial frames, stationary observers and the adopted Galilean transformations between frames clause (iv), with $b\in \mathbb{R}$, $a,u\in {\mathbb{R}}^3$ and $Q\in SO(3)$ (Transformation between adapted charts and its kinematic consequences clause 1); the letters $Q$ and $u$ are the rotation and the boost velocity written $R$ and $v$ in Inertial frames, stationary observers and the adopted Galilean transformations between frames, renamed to avoid clashes with the orientation path $R(t)$ and the material velocity field of clause (e). The description of the same motion in the second frame assigns to the same material label $\rho$ the events of the motion in the new coordinates, $\tilde{x}(t^{\mathrm{\prime }},\rho ):=Qx(t^{\mathrm{\prime }}-b,\rho )+u(t^{\mathrm{\prime }}-b)+a\mathrm{.}$ Substituting $x(t,\rho )=c(t)+R(t)\rho$ and using distributivity of the matrix product over sums expresses the description as a motion of the same model in the second frame, with configuration path $\tilde{c}(t^{\mathrm{\prime }})=Qc(t^{\mathrm{\prime }}-b)+u(t^{\mathrm{\prime }}-b)+a,\tilde{R}(t^{\mathrm{\prime }})=QR(t^{\mathrm{\prime }}-b),$ where $\tilde{R}(t^{\mathrm{\prime }})\in SO(3)$ for every $t^{\mathrm{\prime }}$ because $SO(3)$ is closed under matrix multiplication ($SO(3)$ is a subgroup of the general linear group, with $R^{-1}=R^{\mathsf{T}}$); the pair $(\tilde{c},\tilde{R})$ is again of class $C^2$. This description is a convention forced by the frame change, not a dynamical statement; the transformation laws of the velocity field, the angular velocity and the inertia data are proved in the frame-covariance theorem of this page.

(e) Material velocities. For a motion with configuration path $(c,R)$, the material velocity of the label $\rho$ at time $t$ is the derivative ${\mathrm{\partial }}_{t}x(t,\rho ):=c^{\mathrm{\prime }}(t)+R^{\mathrm{\prime }}(t)\rho ,$ which exists by differentiability of $c$ and of the entries of $R$ (Rigid placements, body and spatial angular velocity, and material velocities). The velocity field of the motion at time $t$ is the map $p\mapsto v(t,p)$ assigning to a spatial point the velocity of the material point currently placed there, defined on the image of the placement. Neither the configuration nor the placement by itself asserts a dynamics: the pair $(c,R)$ is a description of the position and orientation of the body, and the material velocity is its kinematic derivative.

• Configuration pairs are not coordinates. The description in (a) names the global objects of the configuration space. Local coordinates on $SO(3)$, their singularities and the global character of the orientation are treated in The global orientation and its charts: Euler angles, the quaternion double cover and singularities of parametrizations.

• Six degrees of freedom, by parametrization and by faithfulness. The dimension count in (a) uses the manifold structure of $SO(3)$; the nondegeneracy clause (c) is what makes the count a statement about the placements of a labelled body rather than a count of parameters, in the sense of A labeling containing two independent vectors has trivial $SO(3)$-stabilizer.

• The convention on $\nu$ is part of the interface. A consumer that uses a different origin of reference coordinates, or a non-centred mass distribution, must state its own translation bookkeeping; the model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope fixes the origin at the centre of mass, and Parallel-axis theorem: the inertia tensor about a displaced point of a centred mass distribution is the mathematical supplier for inertia tensors about displaced points.

• No measured or empirical content. The configuration data are model quantities; the item asserts no measured orientation, position, size or mass, and no claim that any body is rigid.

Adopt the model, inertial frame and notation of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope and Configuration, configuration space and orientation of a rigid-material model in an inertial frame, and let a motion $x(t,\rho )=c(t)+R(t)\rho$ have configuration path $(c,R)$ of class $C^2$ on the time window $I_0$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

(a) Body angular velocity. By Rigid placements, body and spatial angular velocity, and material velocities, the matrix $R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)$ is skew-symmetric at every $t\in I_0$ (Derivative rules for differentiable matrix-valued curves, and skewness of $R^{\mathsf{T}}{R}^{\mathrm{\prime }}$ for a curve in $SO(3)$ clause 3), and by The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$ clause 2 there is exactly one vector $\omega (t)\in {\mathbb{R}}^3$ with $R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)=[\omega (t){]}_{\times }\mathrm{.}$ The map $\omega :I_0\to {\mathbb{R}}^3$ is the body angular velocity of the motion. Since $R$ is of class $C^2$, the entries of $R^{\mathsf{T}}{R}^{\mathrm{\prime }}$ are differentiable and hence $\omega$ is of class $C^1$; its derivative is the body angular acceleration. The dimension is $[\omega ]={\tau }^{-1}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework), read in radians per second under the radian convention, in which angles are dimensionless.

(b) Spatial angular velocity. The spatial angular velocity of the motion is ${\omega }_{\mathrm{s}}(t):=R(t)\omega (t)\in {\mathbb{R}}^3,$ of the same dimension ${\tau }^{-1}$. By the conjugation clause of The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$, $R^{\mathrm{\prime }}(t)R(t{)}^{\mathsf{T}}=[{\omega }_{\mathrm{s}}(t){]}_{\times }$; this is the space-frame rate matrix of the motion.

(c) Material velocity field. For every material label $\rho$, ${\mathrm{\partial }}_{t}x(t,\rho )=c^{\mathrm{\prime }}(t)+R(t)(\omega (t)\times \rho ),$ because $R^{\mathrm{\prime }}(t)=R(t)[\omega (t){]}_{\times }$ and $[\omega {]}_{\times }\rho =\omega \times \rho$ (Rigid placements, body and spatial angular velocity, and material velocities, The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$ clause 1). Writing $p=x(t,\rho )$ for the spatial position of the label and using $R(\omega \times \rho )=(R\omega )\times (R\rho )$, the same identity reads ${\mathrm{\partial }}_{t}x(t,\rho )=c^{\mathrm{\prime }}(t)+{\omega }_{\mathrm{s}}(t)\times (p-c(t)):$ the material velocity field is that of a translation $c^{\mathrm{\prime }}(t)$ composed with a rotation of angular velocity ${\omega }_{\mathrm{s}}(t)$ about the centre of mass $c(t)$. The motion is a pure translation at time $t$ exactly when $\omega (t)=0$, equivalently $R^{\mathrm{\prime }}(t)=0$; then every material velocity equals $c^{\mathrm{\prime }}(t)$.

(d) Uniqueness of the velocity data. The velocity field determines $c^{\mathrm{\prime }}(t)$ and ${\omega }_{\mathrm{s}}(t)$ uniquely: if a motion has material velocity field $p\mapsto v(t,p)=c^{\mathrm{\prime }}(t)+{\omega }_{\mathrm{s}}(t)\times (p-c(t))$, then evaluating at $p=c(t)$ gives $c^{\mathrm{\prime }}(t)=v(t,c(t))$, and for $u\in {\mathbb{R}}^3$ the difference $v(t,c(t)+u)-v(t,c(t))={\omega }_{\mathrm{s}}(t)\times u$ is a linear map of $u$; testing it on two linearly independent vectors determines ${\omega }_{\mathrm{s}}(t)$ by the injectivity clause of The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$, and then $\omega (t)=R(t{)}^{\mathsf{T}}{\omega }_{\mathrm{s}}(t)$. The pair $(c^{\mathrm{\prime }}(t),{\omega }_{\mathrm{s}}(t))$ is therefore an intrinsic description of the velocity field; the body angular velocity is its description in the body-fixed frame, and the choice of reference coordinates is part of the model data fixed in Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope.

(e) Instantaneous rotation axis. If ${\omega }_{\mathrm{s}}(t)\mathrm{\ne }0$, the line $\{c(t)+s{\omega }_{\mathrm{s}}(t):s\in \mathbb{R}\}$ is the instantaneous rotation axis of the motion at time $t$: it is the set of spatial points whose material velocity equals $c^{\mathrm{\prime }}(t)$, as follows from ${\omega }_{\mathrm{s}}\times ({\omega }_{\mathrm{s}}s)=0$. Conversely, write $w={\omega }_{\mathrm{s}}(t)$ and $z=p-c(t)$. If $w\times z=0$, choose a coordinate $i$ with $w_i\mathrm{\ne }0$; the cross-product coordinate equations (The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$ clause 1) give $w_i{z}_j=w_j{z}_i$ for every $j$, hence $z=(z_i\mathrm{/}{w}_i)w$. This proves that the zero relative-velocity set is exactly the displayed line. The velocity field is constant on lines parallel to ${\omega }_{\mathrm{s}}(t)$ and rotates about this axis; if also $c^{\mathrm{\prime }}(t)=0$ at that instant, every point of the axis is instantaneously at rest and the motion is an instantaneous rotation about it. No claim is made that the axis is fixed in time or in space.

(f) Conventions and scope. The convention fixed here is the body-fixed one of the cited mathematical definition: $\mathrm{\Omega }=R^{\mathsf{T}}{R}^{\mathrm{\prime }}=[\omega {]}_{\times }$ and $R^{\mathrm{\prime }}=R\mathrm{\Omega }$. A consumer using the space-fixed convention must state the change, which is the rotation by $R$ recorded in (b); the two vectors agree only when $R(t)$ fixes $\omega (t)$. Both vectors are kinematic quantities of the motion: no torque, no angular momentum and no equation of motion for $\omega$ is asserted, and no measured angular velocity is recorded. The behaviour of $\omega$ under a change of inertial frame is proved in the frame-covariance theorem of this page.

• The angular velocity is a vector of the spatial space in (b) and of the body in (a). The body vector is the coordinate description of the spatial vector in the body-fixed frame; the identity ${\omega }_{\mathrm{s}}=R\omega$ is the translation between the two descriptions. The distinction matters when a consumer compares the model with data attached to the body, such as principal axes.

• What is used and what is not. Only differentiability of $R$ is needed for existence and uniqueness of $\omega$; the twice differentiability in clause (a) is used for the angular acceleration and, on the dynamics pages of the track, for rate balances. No smoothness beyond $C^2$ across the time window is asserted.

• Degenerate shapes. Clauses (c)-(e) concern the motion of the body as a placement and do not use the mass distribution. The mass distribution enters through the inertia data of Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model and through the angular-momentum identity of Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion, where a mass distribution supported on a line produces a non-invertible inertia matrix.

Adopt the model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope: the reference mass distribution $\nu$, its total mass $m=\nu ({\mathbb{R}}^3)$ and its centred reference coordinates $\int \rho d\nu (\rho )=0$. Assume 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 real numbers. Fix an inertial frame and a motion $x(t,\rho )=c(t)+R(t)\rho$ with configuration path $(c,R)$ of class $C^2$ (Configuration, configuration space and orientation of a rigid-material model in an inertial frame).

(a) Body inertia matrix. The body inertia matrix of the model is the inertia matrix of the reference mass distribution, $I:=I(\nu )=\int (\parallel \rho {\parallel }_2^2{I}_3-\rho {\rho }^{\mathsf{T}})d\nu (\rho )\in M_3(\mathbb{R}),$ with entries $I_{jk}=\int (\parallel \rho {\parallel }_2^2{\delta }_{jk}-{\rho }_j{\rho }_k)d\nu$ as in Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix; the matrix is expressed in the standard basis of the reference coordinates. Its entries are finite real numbers, it is symmetric, and for every $\omega \in {\mathbb{R}}^3$, ${\omega }^{\mathsf{T}}I\omega =\int \parallel \omega \times \rho {\parallel }_2^{2}d\nu (\rho )\ge 0$ by Symmetry, the quadratic identity ${\omega }^{\mathsf{T}}I\omega =\int \parallel \omega \times \rho {\parallel }_2^{2}d\nu$, and the positive-definiteness criterion for the inertia matrix clauses 1-2; hence $I$ is symmetric and its quadratic form is positive semidefinite. The dimension is $[I]=\mu {\mathrm{\ell }}^2$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework), read in kilogram metres squared. The model is nondegenerate when no line through the origin carries the whole mass, that is $\nu ({\mathbb{R}}^3\setminus \mathbb{R}\omega )>0$ for every $\omega \mathrm{\ne }0$, and degenerate otherwise; the connection between this criterion and the strict positivity and invertibility of $I$ is proved in the principal-axes theorem of this page.

(b) Spatial inertia tensor at the centre of mass. The spatial inertia tensor of the model at time $t$ is $I(t):=R(t)IR(t{)}^{\mathsf{T}}\in M_3(\mathbb{R}),$ the matrix of the same quadratic form in the frame's spatial coordinates: for every $n\in {\mathbb{R}}^3$, $n^{\mathsf{T}}I(t)n=(R(t{)}^{\mathsf{T}}n{)}^{\mathsf{T}}I(R(t{)}^{\mathsf{T}}n)$. Its entries are $C^2$ functions of $t$, since $R$ is of class $C^2$ and $I$ is constant; the matrix rotates with the body and returns to $I$ in the body-fixed frame, where $R=I_3$.

(c) Inertia tensor about a spatial point. For a spatial point $p\in {\mathbb{R}}^3$ let $a(t,p):=R(t{)}^{\mathsf{T}}(p-c(t))\in {\mathbb{R}}^3$ be the reference coordinate of $p$, that is the label of the material point of the reference placement that is placed at $p$ at time $t$. The inertia tensor of the model about $p$ at time $t$ is $I(t,p):=R(t)I_{\nu }(a(t,p))R(t{)}^{\mathsf{T}}\in M_3(\mathbb{R}),$ where $I_{\nu }(a)=\int (\parallel \rho -a{\parallel }_2^2{I}_3-(\rho -a)(\rho -a{)}^{\mathsf{T}})d\nu (\rho )$ is the inertia tensor of the mass distribution about the point $a$ of the reference coordinates (The inertia tensor of a finite bounded Borel measure about any point of ${\mathbb{R}}^3$). Since the label of a spatial point is uniquely determined by the placement, this is well defined; for the centre of mass, $a(t,c(t))=0$ and $I(t,c(t))=I(t)$. The dimension is again $\mu {\mathrm{\ell }}^2$, read in kilogram metres squared.

(d) Moment of inertia about an axis. Let $\mathrm{\ell }$ be a spatial line at time $t$, given by a point $p\in {\mathbb{R}}^3$ and a unit direction $n\in {\mathbb{R}}^3$ ($\parallel n{\parallel }_2=1$). The moment of inertia of the model about $\mathrm{\ell }$ is ${\mathrm{\Theta }}_{\mathrm{\ell }}(t):=n^{\mathsf{T}}I(t,p)n\mathrm{.}$ Its invariance under the choices entering the description of the line and its identification with the integral defining the physical moment of inertia about the line are statements, not conventions; they are proved in clause (c) of the principal-axes theorem of this page. For a body-fixed axis, that is a line through the centre of mass whose direction in reference coordinates is a constant unit vector $n_0$, the moment is $n_0^{\mathsf{T}}I{n}_0=\int \parallel n_0\times \rho {\parallel }_2^{2}d\nu (\rho ),$ by clause (a), and is independent of time, because $I$ and $n_0$ are model data.

(e) What is constant and what moves. The body inertia matrix $I$ is determined by the model's reference mass distribution alone and does not depend on the time, the frame or the motion; the spatial inertia tensor $I(t)$ depends on the motion through the orientation $R(t)$; the inertia tensor $I(t,p)$ about a spatial point depends on the motion through both the orientation and the position of $p$. Principal axes and principal moments of $I$ are the subject of the principal-axes theorem of this page.

• The physical reading of ${\mathrm{\Theta }}_{\mathrm{\ell }}$. The number $n^{\mathsf{T}}I(t,p)n$ is what a consumer reads as the rotational inertia of the body about the line $\mathrm{\ell }$; the identity with the integral of the squared distance from the line is the content of the principal-axes theorem of this page, and is not assumed here.

• Degenerate shapes. If $\nu$ is supported on a line through the origin, then $I$ has a zero eigenvalue, is not invertible, and the relation between angular velocity and angular momentum is not injective; the criterion is proved in Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values clause (d) below, and the companion page exhibits a two-point-mass body of this kind. The model branch admits such shapes; a consumer that inverts $I$ must state the nondegeneracy hypothesis.

• No measured inertia. The matrix $I$ is model data computed from $\nu$; the item asserts no measured moment of inertia and no claim that any body has a given inertia tensor.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), used only to make the integrals below finite. Adopt the model and notation of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope: a reference mass distribution $\nu$ with total mass $m$ and centred reference coordinates, a fixed inertial frame, and a motion $x(t,\rho )=c(t)+R(t)\rho$ with configuration path $(c,R)$ of class $C^2$ on the time window $I_0$, body angular velocity $\omega$ (Body and spatial angular velocity of a rigid-material motion, and the material velocity field) and body inertia matrix $I$ (Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model). Define the total momentum, kinetic energy and angular momentum about the centre of mass of the model along the motion by the componentwise integrals $P(t):=\int {\mathrm{\partial }}_{t}x(t,\rho )d\nu (\rho ),K(t):=\frac{1}{2}\int \parallel {\mathrm{\partial }}_{t}x(t,\rho ){\parallel }_2^{2}d\nu (\rho ),$ $L_c(t):=\int (x(t,\rho )-c(t))\times ({\mathrm{\partial }}_{t}x(t,\rho )-c^{\mathrm{\prime }}(t))d\nu (\rho ),$ with dimensions $\mu \mathrm{\ell }{\tau }^{-1}$, $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ and $\mu {\mathrm{\ell }}^2{\tau }^{-1}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Then for every $t\in I_0$:

1. the three integrands are $\nu$-integrable, so $P(t)$, $K(t)$ and $L_c(t)$ are a well-defined finite vector and finite real numbers;
2. $P(t)=m{c}^{\mathrm{\prime }}(t)$;
3. $K(t)={\displaystyle \frac{1}{2}}m\parallel c^{\mathrm{\prime }}(t){\parallel }_2^2+{\displaystyle \frac{1}{2}}\omega (t{)}^{\mathsf{T}}I\omega (t)$;
4. $L_c(t)=R(t)I\omega (t)$.

The identities hold for every $C^2$ configuration path; no equation of motion, force law or conservation statement is used or asserted.

• Instantaneous identities. The three displayed identities are identities at each time for the stated motion; they do not say that either term of the kinetic splitting is conserved, and no equation of motion for $(c,R)$ enters. The terms are the model's translational kinetic energy $\frac{1}{2}m\parallel c^{\mathrm{\prime }}{\parallel }_2^2$ and rotational kinetic energy $\frac{1}{2}{\omega }^{\mathsf{T}}I\omega$.

• Frame dependence of the two terms. The centre-of-mass velocity $c^{\mathrm{\prime }}$ and hence the translational term depend on the inertial frame; the rotational term $\frac{1}{2}{\omega }^{\mathsf{T}}I\omega$ is built from the body angular velocity and the body inertia matrix and is therefore unchanged by a change of inertial frame. The precise transformation laws are proved in Behaviour of the rigid-material configuration, angular velocity, inertia data and inertial quantities under a change of inertial frame on this page.

• The countable-choice hypothesis. ${\mathrm{A}\mathrm{C}}_{\omega }$ is used only to make the integrals finite through [L1] and [L3]; the algebraic identities themselves are choice-free once the integrals exist. A consumer that needs the identities under weak hypotheses must carry this assumption.

• Degenerate shapes remain allowed. The identities hold for every mass distribution of the branch, including those supported on a line; in that case $I\omega$ is not injective in $\omega$, so the angular momentum does not determine the body angular velocity, as the companion page's counterexample exhibits.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), used only to make the integrals finite. Adopt the model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope and its inertia data (Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model): the reference mass distribution $\nu$ with total mass $m$ and centred reference coordinates, the body inertia matrix $I$, the spatial inertia tensor $I(t)=R(t)IR(t{)}^{\mathsf{T}}$ of a motion $x(t,\rho )=c(t)+R(t)\rho$, and the inertia tensor $I(t,p)$ about a spatial point $p$ (Configuration, configuration space and orientation of a rigid-material model in an inertial frame). Then:

(a) Principal basis. There is an orthonormal basis $(e_1,e_2,e_3)$ of ${\mathbb{R}}^3$ and real numbers $I_1,I_2,I_3$, the principal axes and principal moments of the model, with $I{e}_j=I_j{e}_j(j=1,2,3)\mathrm{.}$

(b) Quadratic form and rotational term. For every $\omega ={\sum }_j{\omega }_j{e}_j\in {\mathbb{R}}^3$ one has $I\omega ={\sum }_j{I}_j{\omega }_j{e}_j$ and ${\omega }^{\mathsf{T}}I\omega ={\sum }_j{I}_j{\omega }_j^2=\int \parallel \omega \times \rho {\parallel }_2^{2}d\nu (\rho )\ge 0,$ and each principal moment is $I_j=e_j^{\mathsf{T}}I{e}_j=\int \parallel e_j\times \rho {\parallel }_2^{2}d\nu (\rho )$.

(c) Axis moments. For every unit vector $n\in {\mathbb{R}}^3$, $n^{\mathsf{T}}In=\int \parallel n\times \rho {\parallel }_2^{2}d\nu (\rho );$ for every spatial point $p$ and unit vector $n$, with $x=x(t,\cdot )$ the placement at time $t$, $n^{\mathsf{T}}I(t,p)n=\int \parallel n\times (x(t,\rho )-p){\parallel }_2^{2}d\nu (\rho )\mathrm{.}$ Consequently the number ${\mathrm{\Theta }}_{\mathrm{\ell }}(t)=n^{\mathsf{T}}I(t,p)n$ of Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model depends only on the spatial line $\mathrm{\ell }$ $\{p+sn:s\in \mathbb{R}\}$ and not on the choice of the point $p$ on it nor on the sign of the unit direction $n$.

(d) Vanishing and positivity. Each principal moment satisfies $I_j\ge 0$, and $I_j=0$ exactly when $\nu$ is supported on the line $\mathbb{R}{e}_j$ of the principal axis, that is $\nu ({\mathbb{R}}^3\setminus \mathbb{R}{e}_j)=0$. If the model is nondegenerate, that is if $\nu ({\mathbb{R}}^3\setminus \mathbb{R}\omega )>0$ for every $\omega \mathrm{\ne }0$, then every $I_j>0$ and $I$ is invertible. If $\nu$ is supported on some line through the origin (such a line exists whenever the support of a centred $\nu$ is contained in a line, because the mean of $\nu$ is the origin), then $I$ is not invertible.

(e) Extremal moments. For every unit vector $n={\sum }_j{c}_j{e}_j$ one has $n^{\mathsf{T}}In={\sum }_j{I}_j{c}_j^2$ and therefore ${\min }_j{I}_j\le n^{\mathsf{T}}In\le {\max }_j{I}_j,$ with the value ${\max }_j{I}_j$ attained at a unit eigenvector for a maximal principal moment and the value ${\min }_j{I}_j$ attained at a unit eigenvector for a minimal principal moment. Hence the largest and smallest principal moments are the maximum and minimum of the rotational-inertia quadratic form over the unit sphere.

The identities are conditional on the model data and the stated choice assumption; no equation of motion and no conservation statement is used or asserted.