Configuration, configuration space and orientation of a rigid-material model in an inertial frame

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.