Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix

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.