Cm Math Rigid

Let $I\subseteq \mathbb{R}$ be an interval with at least two points, let $t_0\in I$ and let $A,B:I\to M_3(\mathbb{R})$ be differentiable at $t_0$, meaning (The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations) that every entry function $t\mapsto A(t{)}_{ij}$ and $t\mapsto B(t{)}_{ij}$ is differentiable at $t_0$ in the sense of The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set; write $A^{\mathrm{\prime }}(t_0)$ for the matrix with entries $A^{\mathrm{\prime }}(t_0{)}_{ij}:=(t\mapsto A(t{)}_{ij}{)}^{\mathrm{\prime }}(t_0)$, and similarly for $B$.

1. $A+B$ and $A^{\mathsf{T}}$ are differentiable at $t_0$, with $(A+B{)}^{\mathrm{\prime }}(t_0)=A^{\mathrm{\prime }}(t_0)+B^{\mathrm{\prime }}(t_0)$ and $(A^{\mathsf{T}}{)}^{\mathrm{\prime }}(t_0)=(A^{\mathrm{\prime }}(t_0){)}^{\mathsf{T}}$.
2. $AB$ is differentiable at $t_0$, with $(AB{)}^{\mathrm{\prime }}(t_0)=A^{\mathrm{\prime }}(t_0)B(t_0)+A(t_0)B^{\mathrm{\prime }}(t_0)$.
3. If $A(t)\in SO(3)$ for every $t\in I$ (The orthogonal group $O(3)$ and the rotation group $SO(3)$), then the matrix $A(t_0{)}^{\mathsf{T}}{A}^{\mathrm{\prime }}(t_0)$ is skew-symmetric, i.e. $(A^{\mathsf{T}}{A}^{\mathrm{\prime }}{)}^{\mathsf{T}}=-A^{\mathsf{T}}{A}^{\mathrm{\prime }}$ at $t_0$, and likewise $A^{\mathrm{\prime }}(t_0)A(t_0{)}^{\mathsf{T}}$ is skew-symmetric. In particular the trace of each of these two matrices vanishes.

Use the standard basis $(e_1,e_2,e_3)$ of ${\mathbb{R}}^3$ and coordinates $\omega ={\omega }_1{e}_1+{\omega }_2{e}_2+{\omega }_3{e}_3$ (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$, A finite list $v:n\to V$ is an ordered basis if and only if every $x\in V$ equals ${\sum }_{i<n}{\lambda }_i{v}_i$ for exactly one $\lambda :n\to F$; those scalars are the coordinates of $x$ in that ordered basis). Define the map $[\cdot {]}_{\times }:{\mathbb{R}}^3\to M_3(\mathbb{R})$ by $[\omega {]}_{\times }:=\left(\begin{array}{ccc}0& -{\omega }_3& {\omega }_2\\ {\omega }_3& 0& -{\omega }_1\\ -{\omega }_2& {\omega }_1& 0\end{array}\right)\mathrm{.}$ Then:

1. $[\omega {]}_{\times }z=\omega \times z$ for all $\omega ,z\in {\mathbb{R}}^3$, the cross product being that of The cross product in ${\mathbb{R}}^3$.
2. $[\omega {]}_{\times }^{\mathsf{T}}=-[\omega {]}_{\times }$ for every $\omega$; the map $\omega \mapsto [\omega {]}_{\times }$ is linear and injective, and its image is exactly the set of skew-symmetric matrices of $M_3(\mathbb{R})$. Consequently every skew-symmetric $\mathrm{\Omega }\in M_3(\mathbb{R})$ equals $[\omega {]}_{\times }$ for exactly one $\omega$, namely $\omega =({\mathrm{\Omega }}_{32},{\mathrm{\Omega }}_{13},{\mathrm{\Omega }}_{21})$.
3. For every $R\in SO(3)$ (The orthogonal group $O(3)$ and the rotation group $SO(3)$) and every $\omega \in {\mathbb{R}}^3$, $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }\mathrm{.}$

Let $\mathcal{B}({\mathbb{R}}^3)$ be the Borel sigma-algebra of ${\mathbb{R}}^3$ (The Borel sigma-algebra of a topological space) and let $\nu$ be a measure on the measurable space $({\mathbb{R}}^3,\mathcal{B}({\mathbb{R}}^3))$ (Measures on sigma-algebras, Measure spaces) that is finite, $\nu ({\mathbb{R}}^3)<+\mathrm{\infty }$, and bounded, meaning that there is a real $R>0$ with $\nu ({\mathbb{R}}^3\setminus {\bar{B}}_R(0))=0,$ the closed ball being that of Open ball, closed ball and sphere in a metric space. Throughout, integrals are Lebesgue integrals against $\nu$ in the sense of Integral over a measurable subset and Integrable real and complex functions, and their integrals, and for a measurable vector- or matrix-valued function the integral is defined componentwise: $\int Fd\nu :=(\int F_{1}d\nu ,\int F_{2}d\nu ,\int F_{3}d\nu )$ and $(\int Md\nu {)}_{jk}:=\int M_{jk}d\nu$, whenever every displayed scalar integral is finite.

A mass distribution on ${\mathbb{R}}^3$ is a bounded finite Borel measure $\nu$ with $m:=\nu ({\mathbb{R}}^3)\in (0,+\mathrm{\infty })$; the number $m$ is its total mass. Writing points of ${\mathbb{R}}^3$ as $\rho$ and using the Euclidean inner product $⟨\cdot ,\cdot ⟩$ and norm $\parallel \rho {\parallel }_2=\sqrt{⟨\rho ,\rho ⟩}$ of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, define:

1. the first moment $\mu (\nu ):=\int \rho d\nu \in {\mathbb{R}}^3$, read componentwise as $\mu (\nu {)}_j=\int {\rho }_{j}d\nu$;
2. the inertia matrix $I(\nu )\in M_3(\mathbb{R})$ (The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations) by its entries $I(\nu {)}_{jk}:=\int (\parallel \rho {\parallel }_2^2{\delta }_{jk}-{\rho }_j{\rho }_k)d\nu$, where ${\delta }_{jk}=1$ if $j=k$ and ${\delta }_{jk}=0$ otherwise; equivalently $I(\nu )=\int (\parallel \rho {\parallel }_2^2{I}_3-\rho {\rho }^{\mathsf{T}})d\nu$ with $I_3$ the identity matrix and $\rho {\rho }^{\mathsf{T}}$ the matrix product of Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes.

The mass distribution is centred when $\mu (\nu )=0$. Under the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), each of the integrals above is a finite real number: on ${\bar{B}}_R(0)$ one has $\mathrm{\mid }{\rho }_j\mathrm{\mid }\le R$ and $\mathrm{\mid }{\rho }_j{\rho }_k\mathrm{\mid }\le R^2$, and $\nu ({\mathbb{R}}^3)<+\mathrm{\infty }$; the precise statement and proof are Finiteness of the moments of a bounded finite Borel measure on ${\mathbb{R}}^3$ ↗, which depends on this definition only for its notation. The assumption is used there exactly once, to obtain the Borel measurability of the polynomial integrands from the published continuity criterion; every later item that invokes this finiteness carries the assumption explicitly.

• Nothing physical is asserted. "Mass distribution", "total mass" and "inertia matrix" are names for the mathematical objects just defined. No postulate that a physical body is described by such a measure, no identification of $\nu$ with a measured mass, and no mechanical law is part of this definition; the physical reading is supplied by the classical-mechanics items that consume it, with their own hypotheses.

• Generality. The measure need not be absolutely continuous with respect to Lebesgue measure and need not be atomless: finitely many point masses, a surface density, and a volume density are all special cases. The bounded-support hypothesis makes every polynomial moment of degree at most two finite; without it the second moment could diverge.

• Why the first moment matters. Being centred is a condition on the coordinates in which the measure is expressed: it says that the origin of the body coordinate system has been placed at the centre of mass of the distribution. A physical consumer that uses this definition must state that its body coordinates are so placed; the general translation behaviour of $\mu$ and $I$ is not asserted here.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $\nu$ be a measure on $({\mathbb{R}}^3,\mathcal{B}({\mathbb{R}}^3))$ (Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix) that is finite and bounded: $\nu ({\mathbb{R}}^3)<+\mathrm{\infty }$ and $\nu ({\mathbb{R}}^3\setminus {\bar{B}}_R(0))=0$ for some $R>0$ (Open ball, closed ball and sphere in a metric space). Then:

1. for all $j,k\in \{1,2,3\}$ the functions $\rho \mapsto {\rho }_j$ and $\rho \mapsto {\rho }_j{\rho }_k$ are Borel measurable and $\nu$-integrable, with $\int \mid {\rho }_j\mid d\nu \le R\nu ({\mathbb{R}}^3)$ and $\int \mid {\rho }_j{\rho }_k\mid d\nu \le R^2\nu ({\mathbb{R}}^3)$;
2. consequently $m:=\nu ({\mathbb{R}}^3)$ and the scalar integrals $\int {\rho }_{j}d\nu$ and $\int (\parallel \rho {\parallel }_2^2{\delta }_{jk}-{\rho }_j{\rho }_k)d\nu$ are finite, with $\mid \int {\rho }_{j}d\nu \mid \le Rm$ and $\mid \int (\parallel \rho {\parallel }_2^2{\delta }_{jk}-{\rho }_j{\rho }_k)d\nu \mid \le 2R^{2}m$. When $m>0$, these integrals are precisely the components of the first moment $\mu (\nu )$ and inertia matrix $I(\nu )$ of Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix. When $m=0$, all these integrals vanish; the zero measure is not called a mass distribution.

Let $I\subseteq \mathbb{R}$ be an interval with at least two points (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), let $\nu$ be a mass distribution on ${\mathbb{R}}^3$ with total mass $m$ (Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix), let $c:I\to {\mathbb{R}}^3$ be differentiable and let $R:I\to SO(3)$ be differentiable (The orthogonal group $O(3)$ and the rotation group $SO(3)$), differentiability meaning entrywise differentiability of $R$ in the sense of The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set with derivative $R^{\mathrm{\prime }}$; write $c^{\mathrm{\prime }}$ for the derivative of $c$.