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A rigid-material model of two point masses on a line: the inertia matrix can be singular, and rotation about the line is invisible to the rotational term

Physical scope

Under the inherited Axiom of Countable Choice ${A C}_{ω}$ of the inertia and kinetic-splitting suppliers, a counterexample inside the rigid-material model branch of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope: two equal point masses at opposite points of a line through the reference origin give a centred mass distribution whose inertia matrix has a zero principal moment and is not invertible, and a rotation of the model about that line has zero rotational kinetic energy and zero angular momentum about the centre of mass. The item refutes only the claim that every positive-mass rigid-material model has positive definite inertia; it asserts no measured value and no failure of the model's other statements, which carry the degeneracy caveat.

Statement refuted

In the rigid-material model branch of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope, every model of positive total mass has a positive definite inertia matrix: equivalently, every principal moment of the body inertia matrix $I$ is strictly positive and $I$ is invertible.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ as specified in [A1]. The model branch and its inertia data (Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope, Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model), with the notation of Mass distributions on $R^{3}$ : total mass, first moment and inertia matrix. The model parameters are $m_{0} > 0$ in kilograms ( $[m_{0}] = μ$ ), $a > 0$ in metres ( $[a] = ℓ$ ), and a constant rate $ϖ \in R$ in radians per second ( $[ϖ] = τ^{- 1}$ ), with dimensionless radians and time $t$ in seconds (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Inertia entries and moments have dimension $μ ℓ^{2}$ , read in kilogram metres squared; rotational energy and centre-of-mass angular momentum have dimensions $μ ℓ^{2} τ^{- 2}$ and $μ ℓ^{2} τ^{- 1}$ , read in joules and kilogram metres squared per second. All values are model quantities, not measurements.

[A1]

The Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) is adopted: for every family $(X_{n})_{n \in N}$ of nonempty sets there is $f : N \to ⋃_{n} X_{n}$ with $f (n) \in X_{n}$ for all $n$ . This is the inherited hypothesis for Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model, Finiteness of the moments of a bounded finite Borel measure on $R^{3}$ , Symmetry, the quadratic identity $ω^{T} I ω = \int ∥ ω \times ρ ∥_{2}^{2} d ν$ , and the positive-definiteness criterion for the inertia matrix, Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values and Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion; the explicit finite-atomic calculation does not establish this axiom.

[L1]

Dirac measures and their weighted sums (The Dirac set function at a point, A Dirac set function is a probability measure, Nonnegative scalar multiples and countable weighted sums of measures are measures, Nonnegative scalar multiples and countable weighted sums of measures): a finite weighted sum of Dirac measures at distinct points with positive weights is a finite bounded measure on the Borel space of $R^{3}$ , and its integral on nonnegative measurable functions is computed by the finite-atomic formula of step 1.1.

[L2]

Integral rules (The nonnegative Lebesgue integral, The integral of a nonnegative simple function, Integral over a measurable subset, Integrable real and complex functions, and their integrals, The Lebesgue integral is linear on $L^{1} (μ)$ ): linearity of the integral on $L^{1}$ and the componentwise reading of vector integrals.

[L3]

Inertia data and principal axes (Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model, Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values, Symmetry, the quadratic identity $ω^{T} I ω = \int ∥ ω \times ρ ∥_{2}^{2} d ν$ , and the positive-definiteness criterion for the inertia matrix clauses 1-2): $I = \int (∥ ρ ∥_{2}^{2} I_{3} - ρ ρ^{T}) d ν$ , $I$ is symmetric, $ω^{T} I ω = \int ∥ ω \times ρ ∥_{2}^{2} d ν \geq 0$ , and the principal basis and the vanishing criterion of clause (d) of Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values are available.

[L4]

Kinetic splitting and angular velocity (Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion clauses 3-4, Body and spatial angular velocity of a rigid-material motion, and the material velocity field): on a motion with body angular velocity $ω$ , the rotational term is $\frac{1}{2} ω^{T} I ω$ and the angular momentum about the centre of mass is $L_{c} = R I ω$ .

[L5]

Cross-product and matrix algebra (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The vector space $M_{m \times n} (F) : = F^{m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, Laws of finite sums and finite products): $e_{3} \times t e_{3} = 0$ for every real $t$ and $∥ t e_{3} ∥_{2} = ∣ t ∣$ ; matrix-vector multiplication is componentwise in the standard basis.

Counterexample

technique · direct; define the model, compute its inertia matrix from the finite-atomic formula, and exhibit the zero moment and the unobservable rotation

1.1givenL1L2algebra

Finite-atomic integration formula: fix $m_{0} > 0$ , $a > 0$ and $ν = m_{0} δ_{a e_{3}} + m_{0} δ_{- a e_{3}}$ . For a nonnegative simple function $s$ , the defining simple integral gives $\int s d ν = m_{0} s (a e_{3}) + m_{0} s (- a e_{3})$ by [L1] and [L2]. If $f \geq 0$ is measurable and finite at the two atoms, $s_{0} = f (a e_{3}) χ_{{a e_{3}}} + f (- a e_{3}) χ_{{- a e_{3}}} \leq f$ is a nonnegative simple minorant. Its integral gives the lower bound $m_{0} f (a e_{3}) + m_{0} f (- a e_{3})$ , while every simple $s \leq f$ gives the corresponding upper bound. Taking the supremum proves $\int f d ν = m_{0} f (a e_{3}) + m_{0} f (- a e_{3})$ . For a nonnegative function infinite at an atom, simple minorants with arbitrary finite value there have unbounded integrals, giving the same identity with value $+ \infty$ . For a real measurable function finite at the atoms, applying the finite formula to its modulus establishes integrability, and positive/negative parts and linearity give the signed formula; the complex case follows by real/imaginary parts.

2.1step 1.1givenL1

The model. Take $m_{0} > 0$ , $a > 0$ and $ν : = m_{0} δ_{a e_{3}} + m_{0} δ_{- a e_{3}},$ a finite weighted sum of Dirac measures at the distinct points $a e_{3}$ and $- a e_{3}$ with positive weights. By [L1] $ν$ is a finite bounded measure; its total mass is $m = 2 m_{0} \in (0, \infty)$ and its first moment is $\int ρ d ν = m_{0} a e_{3} + m_{0} (- a e_{3}) = 0$ by the finite-atomic formula of step 1.1, so the reference coordinates are centred and $ν$ is a mass distribution of the model branch.

3.1A1step 1.1step 2.1L3L5algebra

Inertia matrix: by step 1.1 and [L3], $I = m_{0} (∥ a e_{3} ∥_{2}^{2} I_{3} - (a e_{3}) (a e_{3})^{T}) + m_{0} (∥ - a e_{3} ∥_{2}^{2} I_{3} - (- a e_{3}) (- a e_{3})^{T}) = 2 m_{0} a^{2} (I_{3} - e_{3} e_{3}^{T}),$ which is $d i a g (2 m_{0} a^{2}, 2 m_{0} a^{2}, 0)$ in the standard basis by [L5]. In particular $I e_{3} = 0$ with $e_{3} \neq 0$ , so $I$ is not invertible.

4.1A1step 3.1L3L5algebra

Principal data and the vanishing criterion: the displayed matrix is diagonal, so the principal moments are $2 m_{0} a^{2}$ , $2 m_{0} a^{2}$ and $0$ , and $e_{3}$ is a principal axis with vanishing moment. This is consistent with clause (d) of [L3]: $ν$ is supported on the line $R e_{3}$ of the principal axis $e_{3}$ , so the principal moment about $e_{3}$ vanishes; the model is degenerate, and the nondegeneracy hypothesis of the positivity statement fails.

4.2A1step 3.1L3L4L5algebra

A rotation about the line is invisible to the rotational term. Let the model move with constant-rate orientation path $R (t) = R_{z} (ϖ t)$ about the symmetry axis, so that the configuration path is $(c, R)$ with $c$ arbitrary of class $C^{2}$ ; by Body and spatial angular velocity of a rigid-material motion, and the material velocity field its body angular velocity is $ω = ϖ e_{3}$ (the axis $e_{3}$ is fixed by $R_{z}$ , and $R^{T} R^{'} = [ϖ e_{3}]_{\times}$ by the derivative clause of the coordinate-rotation matrices). Every material point lies on the rotation axis: $ω \times (\pm a e_{3}) = \pm ϖ (e_{3} \times a e_{3}) = 0$ by [L5], so the material velocity equals $c^{'}$ on both mass-carrying labels and the rotational term is $\frac{1}{2} ω^{T} I ω = \frac{1}{2} ϖ^{2} e_{3}^{T} I e_{3} = 0$ by [L4] and step 3.1. The angular momentum about the centre of mass is $L_{c} = R I ω = 0$ by [L4], so the motion's spin about the line produces neither rotational kinetic energy nor angular momentum.

5.1A1step 3.1step 4.1step 4.2L3L4∎

Conclusion. The model is a rigid-material model of positive total mass, and its inertia matrix $d i a g (2 m_{0} a^{2}, 2 m_{0} a^{2}, 0)$ has a zero principal moment, is not invertible, and fails to detect rotation about the line carrying the masses. The refuted statement — that every positive-mass model of the branch has positive definite inertia — is therefore false. The correct criterion is clause (d) of Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values: the moment about a principal axis vanishes exactly when the mass is supported on that axis; the model branch, the kinetic splitting and the principal-axis theorem all remain valid with their stated degeneracy caveats.

Remarks

What is refuted, and what is not. Only the universal positivity claim is refuted. The positivity criterion of Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values clause (d) is consistent with the example: the vanishing moment occurs exactly because the mass is supported on the principal axis. No statement of the model branch that carries the nondegeneracy hypothesis is affected.
The physical reading of the zero moment. The two point masses lie on the instantaneous rotation axis, so a rotation of the model about that line moves no material point; the model attributes no kinetic energy to a motion that does not move its mass. The angular momentum $L_{c} = R I ω$ loses the component of $ω$ along the line: $I ω = d i a g (2 m_{0} a^{2}, 2 m_{0} a^{2}, 0) ω$ , so the relation between $ω$ and $L_{c}$ is not injective. A consumer requiring $ω$ from $L_{c}$ , or requiring an invertible inertia matrix, must exclude or regularize this case explicitly.
No measured value. The parameters $m_{0}$ and $a$ are model data; the example asserts no measured inertia and no claim that any concrete body has collinear material points.

A rigid-material model of two point masses on a line: the inertia matrix can be singular, and rotation about the line is invisible to the rotational term

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

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