Rigid-body dynamics and stability

Adopt as physical assumptions of the framework the following external-torque branch of the rigid-material model, in the setting of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope.

(i) Setting. A model of the branch carries the data of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope: a reference mass distribution $\nu$ with total mass $m>0$ and centred reference coordinates, a time window $I_0$, an inertial frame with adapted coordinates $(t,x)$ (Inertial frames, stationary observers and the adopted Galilean transformations between frames), 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). Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), used only through Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion to make the defining integrals of the model's inertial quantities finite. Along the motion the body angular velocity $\omega$ and the body inertia matrix $I$ are those of Body and spatial angular velocity of a rigid-material motion, and the material velocity field and Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model, and the angular momentum about the centre of mass is $L_c(t):=\int (x(t,\rho )-c(t))\times ({\mathrm{\partial }}_{t}x(t,\rho )-c^{\mathrm{\prime }}(t))d\nu (\rho ),$ the finite vector of clause 1 of Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion, which by clause 4 of that item satisfies the identity $L_c(t)=R(t)I\omega (t)(t\in I_0)\mathrm{.}$

(ii) External torque datum. An external torque datum of the branch on the time window $I_0$ is a continuous map $\tau :I_0⟶{\mathbb{R}}^3,[\tau ]=\mu {\mathrm{\ell }}^2{\tau }^{-2},$ the dimension being read in newton metres (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Its intended physical reading is: $\tau (t)$ is the total external torque about the centre of mass at time $t$, the resultant moment about $c(t)$ of the forces external to the model that act on the body at time $t$: weight in a declared gravitational field, forces exerted by supports, pivots, axles or contacts, thrusts and drives, applied couples, and frictional torques. Which external agencies occur and which datum they jointly produce is model data fixed by the model's further assumptions; this postulate does not derive $\tau$ from any force system, and the statements of the pair use $\tau$ only through the balance below.

(iii) The adopted balance. Along every motion of a model of the branch with external torque datum $\tau$, adopt the external-torque balance $L_c^{\mathrm{\prime }}(t)=\tau (t)(t\in I_0)\mathrm{.}$ This is an adopted dynamical law of the model. Through the identity $L_c=RI\omega$ of clause (i) it is one vector equation relating the configuration path, its first two time derivatives and the datum; it does not by itself determine $(c,R)$, and it is not a consequence of the kinematic definitions of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope.

(iv) Torque-free motions. A model of the branch whose datum vanishes, $\tau \equiv 0$ on $I_0$, is called torque-free; for such a model the balance reads $L_c^{\mathrm{\prime }}\equiv 0$ on $I_0$. Torque-free is a property of the declared datum, not of the mass distribution: every model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope may be used with the zero datum, and the statements of this page about torque-free motions are conditional on that use.

(v) Relation to the particle branch. The framework's angular-momentum balance for finitely many particle labels is Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case: for a model that declares the external-plus-pair presentation with antisymmetric pair forces, the angular momentum about a fixed reference point has derivative equal to the total torque about that point, the internal torque sum cancelling when the evaluated pair forces are central. The rigid-material model has no particle labels; its mass data are the measure $\nu$, and all its physical statements weight material points by $\nu$. That balance is therefore not available verbatim for the model of clause (i), and clause (iii) is adopted here as the rigid-material analogue. No claim that clause (iii) follows from Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case, or from any other item of the framework, is made by this postulate or proved anywhere on this page.

(vi) No other balances. This postulate asserts no balance for the total momentum $m{c}^{\mathrm{\prime }}$ and no equation of motion for the centre of mass: an external force datum is not part of the branch, the curve $c$ remains free except as restricted by further model assumptions (for instance a pivot constraint adopted on this page), and the theorems of the pair use only the balance (iii). In particular the branch neither asserts nor denies that the centre of masses moves uniformly, and it asserts no conservation of the rotational kinetic energy; the power identity for the rotational term is a theorem of this page, not an assumption.

(vii) Frame consistency. Let a second inertial frame be related to the chosen one 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 $Q\in SO(3)$, and let $\tilde{x}(t^{\mathrm{\prime }},\rho ):=Qx(t^{\mathrm{\prime }}-b,\rho )+u(t^{\mathrm{\prime }}-b)+a$ be the description of the same motion in the second frame (Behaviour of the rigid-material configuration, angular velocity, inertia data and inertial quantities under a change of inertial frame clause 1). Adopt: this description is again a motion of the same model of the branch, and an external torque datum of the model transforms between the two descriptions by $\tilde{\tau }(t^{\mathrm{\prime }}):=Q\tau (t^{\mathrm{\prime }}-b),$ the transformation of a spatial vector under the rotation $Q$ alone, with no contribution from the boost $u$, the translation $a$ or the time shift $b$. The consistency of the balance (iii) with this transformation law — namely that the description in the second frame satisfies the balance with datum $\tilde{\tau }$, because ${\tilde{L}}_c(t^{\mathrm{\prime }})=Q{L}_c(t^{\mathrm{\prime }}-b)$ by clause 5 of Behaviour of the rigid-material configuration, angular velocity, inertia data and inertial quantities under a change of inertial frame and differentiation commutes with the constant rotation $Q$ and the time shift — is recorded and proved as a clause of the frame-covariance theorem of this page; it is not an assumption of the present postulate.

(viii) Scope and exclusions. The branch excludes deformable bodies, internal degrees of freedom, variable mass, ablation, impact and contact events, and every constitutive law for friction, rolling or slipping; any such effect that is present must be declared as part of the datum $\tau$ of clause (ii) with its own model, and a datum is a model object, not a measurement. The branch also excludes relativistic and quantum descriptions. Wherever a statement of the pair needs the inverse $I^{-1}$ it assumes the model is nondegenerate in the sense of Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model clause (a), and that hypothesis is stated explicitly in the theorem that uses it. The postulate asserts no measured torque, no numerical value, and no claim that any concrete body or apparatus is described by the branch; whether a given datum is adequate for a given apparatus is decided by comparing the model's conditional predictions with reported observations under the apparatus's stated regime.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), used only through the integrals of Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion; the theorems below neither use nor imply any other choice principle. Adopt a model of the rigid-material branch Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope with the external-torque branch Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass: a reference mass distribution $\nu$ with total mass $m$ and centred reference coordinates, a time window $I_0$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), an inertial frame, 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), 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), and a continuous external torque datum $\tau :I_0\to {\mathbb{R}}^3$ with respect to which the adopted balance of Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass clause (iii) holds along the motion. Then:

1. Euler's equations with torque. Writing $l:=I\omega$, one has $l^{\mathrm{\prime }}=R^{\mathsf{T}}\tau -\omega \times I\omega$ and $I{\omega }^{\mathrm{\prime }}(t)+\omega (t)\times I\omega (t)=R(t{)}^{\mathsf{T}}\tau (t)(t\in I_0)\mathrm{.}$
2. Principal-axis components. If $(e_1,e_2,e_3)$ is a positively oriented orthonormal eigenbasis of $I$ with $I{e}_j=I_j{e}_j$ (Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values clause (a)) and $\omega ={\sum }_j{\omega }_j{e}_j$, $R^{\mathsf{T}}\tau ={\sum }_j{\tau }_j{e}_j$, then the equations are componentwise $I_1{\omega }_1^{\mathrm{\prime }}=(I_2-I_3){\omega }_2{\omega }_3+{\tau }_1,I_2{\omega }_2^{\mathrm{\prime }}=(I_3-I_1){\omega }_3{\omega }_1+{\tau }_2,I_3{\omega }_3^{\mathrm{\prime }}=(I_1-I_2){\omega }_1{\omega }_2+{\tau }_3\mathrm{.}$
3. Rotational power identity. The rotational kinetic energy $K_{\mathrm{r}\mathrm{o}\mathrm{t}}(t):=\frac{1}{2}\omega (t{)}^{\mathsf{T}}I\omega (t)$ is differentiable on $I_0$ with $\frac{d}{dt}{K}_{\mathrm{r}\mathrm{o}\mathrm{t}}(t)=\omega (t)\cdot R(t{)}^{\mathsf{T}}\tau (t)(t\in I_0);$ in particular, if $\tau \equiv 0$, then $K_{\mathrm{r}\mathrm{o}\mathrm{t}}$ is constant on $I_0$, and $K_{\mathrm{r}\mathrm{o}\mathrm{t}}$ coincides with the rotational term $\frac{1}{2}{\omega }^{\mathsf{T}}I\omega$ of the model's kinetic energy in clause 3 of Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion.
4. Torque-free reduction and nondegeneracy. Suppose $\tau \equiv 0$ and the model is nondegenerate in the sense of Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model clause (a), so that $I$ is invertible with $I^{-1}l=\omega$. Then, writing $\omega$ as a function of $l$, $l^{\mathrm{\prime }}=l\times (I^{-1}l),{\omega }^{\mathrm{\prime }}=F(\omega ):=-I^{-1}(\omega \times I\omega );$ the second form exhibits the torque-free body angular velocity as a solution of the autonomous field $F$ on ${\mathbb{R}}^3$.
5. Frame covariance. Let a second inertial frame be related to the first by $t^{\mathrm{\prime }}=t+b$, $x^{\mathrm{\prime }}=Qx+ut+a$ with $b\in \mathbb{R}$, $a,u\in {\mathbb{R}}^3$, $Q\in SO(3)$ (Inertial frames, stationary observers and the adopted Galilean transformations between frames clause (iv)), and let $\tilde{x}(t^{\mathrm{\prime }},\rho ):=Qx(t^{\mathrm{\prime }}-b,\rho )+u(t^{\mathrm{\prime }}-b)+a$, $\tilde{\tau }(t^{\mathrm{\prime }}):=Q\tau (t^{\mathrm{\prime }}-b)$ be the transformed motion and datum of Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass clause (vii). Then the balance holds in the second frame, and $I{\tilde{\omega }}^{\mathrm{\prime }}(t^{\mathrm{\prime }})+\tilde{\omega }(t^{\mathrm{\prime }})\times I\tilde{\omega }(t^{\mathrm{\prime }})=\tilde{R}(t^{\mathrm{\prime }}{)}^{\mathsf{T}}\tilde{\tau }(t^{\mathrm{\prime }})(t^{\mathrm{\prime }}\in I_0+b),$ where $\tilde{\omega }(t^{\mathrm{\prime }})=\omega (t^{\mathrm{\prime }}-b)$ and $\tilde{R}(t^{\mathrm{\prime }})=QR(t^{\mathrm{\prime }}-b)$; the body-frame torque is unchanged: $\tilde{R}(t^{\mathrm{\prime }}{)}^{\mathsf{T}}\tilde{\tau }(t^{\mathrm{\prime }})=R(t^{\mathrm{\prime }}-b{)}^{\mathsf{T}}\tau (t^{\mathrm{\prime }}-b)$.

The statement is conditional on the adopted model and torque balance; it asserts no force law for $\tau$, no conservation of $L_c$ and no measured value.

• What is a theorem and what is an assumption. The balance $L_c^{\mathrm{\prime }}=\tau$ is adopted in Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass; clauses 1-4 here are its consequences in the body frame and in the torque-free case, and clause 5 records that the formulation is consistent with the framework's frame changes. No clause produces $\tau$ from a force system.

• Where the nondegeneracy is used. Clauses 1-3 and 5 hold for every model of the branch, including degenerate ones; only clause 4 uses the invertibility of $I$, and the items that invert $I$ on this page state that hypothesis explicitly.

• The rotational power identity. Clause 3 says that the rate of change of the rotational kinetic energy equals the power $\omega \cdot R^{\mathsf{T}}\tau$ developed by the external torque; in the torque-free branch the rotational kinetic energy is a constant of the motion. This is a consequence of the adopted balance, not a separate assumption, and it does not by itself determine the motion.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), used only through the integrals of Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion. Adopt a model of the rigid-material branch Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope that is nondegenerate in the sense of Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model clause (a) (no line through the origin carries the whole mass), and use it in the torque-free external-torque branch of Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass clause (iv): the datum vanishes, so along every motion of the resulting model the adopted balance reads $L_c^{\mathrm{\prime }}=0$ on the time window $I_0$. Let $\omega$ be the body angular velocity, $I$ the body inertia matrix and $l:=I\omega$ (Body and spatial angular velocity of a rigid-material motion, and the material velocity field, Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model). Then:

1. Conservation of angular momentum. The angular momentum about the centre of mass $L_c(t)=R(t)l(t)$ of Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion clause 4 is constant on $I_0$; in particular $\parallel L_c{\parallel }_2$ is constant.
2. Energy-momentum invariants and principal components. The scalars $C:=⟨l,l⟩=\parallel L_c{\parallel }_2^2,E:=\frac{1}{2}{\omega }^{\mathsf{T}}I\omega =\frac{1}{2}⟨l,I^{-1}l⟩$ are constant on $I_0$, and $l$ satisfies the torque-free equation $l^{\mathrm{\prime }}=l\times (I^{-1}l)\mathrm{.}$ If $(e_1,e_2,e_3)$ is a positively oriented orthonormal eigenbasis of $I$ with $I{e}_j=I_j{e}_j$ and $l={\sum }_j{l}_j{e}_j$ (Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values clauses (a)-(b)), then $C={\sum }_j{l}_j^2,E={\sum }_j\frac{l_j^2}{2I_j},$ and the component equations are $l_1^{\mathrm{\prime }}=(\frac{1}{I_3}-\frac{1}{I_2})l_2{l}_3,l_2^{\mathrm{\prime }}=(\frac{1}{I_1}-\frac{1}{I_3})l_1{l}_3,l_3^{\mathrm{\prime }}=(\frac{1}{I_2}-\frac{1}{I_1})l_1{l}_2\mathrm{.}$
3. The energy invariant is the rotational kinetic energy. $E$ equals the rotational term $\frac{1}{2}{\omega }^{\mathsf{T}}I\omega$ of the model's kinetic energy in clause 3 of Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion; and with $I_{\min }:={\min }_j{I}_j>0$ one has $\parallel \omega (t){\parallel }_2^2\le \frac{2E}{I_{\min }}(t\in I_0)\mathrm{.}$ In particular both $l$ and $\omega$ are bounded on $I_0$.
4. The spin field is globally solvable. The autonomous field $F(l)=l\times (I^{-1}l)$ on ${\mathbb{R}}^3$ is locally Lipschitz in the state variable (Local Lipschitz continuity in the state variable, locally uniform in time and parameters), and every solution of $l^{\mathrm{\prime }}=F(l)$ has a unique maximal extension defined on all of $\mathbb{R}$, and that extension satisfies $\parallel l(t){\parallel }_2=\parallel l(0){\parallel }_2$ for every $t$. Consequently the $l$-path of a torque-free motion is the restriction to $I_0$ of a solution defined on all of $\mathbb{R}$, and its path is contained in the compact level set $\mathrm{\Lambda }(C,E):=\{z\in {\mathbb{R}}^3:⟨z,z⟩=C,\frac{1}{2}⟨z,I^{-1}z⟩=E\},$ which is nonempty because it contains the datum at any chosen time of the motion.

The statement is conditional on the adopted model, its nondegeneracy and the zero-torque balance; it asserts no orientation of the body in space, no measured value and no empirical law.

• The scalar invariants. The two invariant functions displayed are $C=\parallel l{\parallel }_2^2$ and $E=\frac{1}{2}⟨l,I^{-1}l⟩$; they need not be functionally independent in every model. For spherical inertia $I=\alpha I_3$ with $\alpha >0$, one has $E=C\mathrm{/}(2\alpha )$. The components $l_j=I_j{\omega }_j$ are coordinates, not additional conserved quantities. The statement proves the displayed invariants and boundedness; it does not parameterize their level sets, compute elliptic functions or claim periodicity.

• What is not claimed. No orientation statement is made: the conservation of $L_c$ bounds the body-frame data $l$ and $\omega$, but the orientation $R(t)$ itself is not confined by these invariants, and the pages that follow record this distinction explicitly.

• The countable-choice hypothesis. ${\mathrm{A}\mathrm{C}}_{\omega }$ enters only through [L1] and [L2], which make the defining integrals of the model's inertial quantities finite; the algebraic identities themselves are choice-free once those integrals exist.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), carried through Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope clause (i) and Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model for finite mass moments, through Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion and Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass clause (i) for the finite inertial integrals, and through the double-cover supplier of The quaternion rotation map $\rho$: conjugation on the imaginary quaternions, the unit sphere $S^3$, and quaternion lifts (its second remark) for the surjectivity used to choose an initial unit quaternion. Adopt a nondegenerate model of the rigid-material branch Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope: a reference mass distribution $\nu$ with total mass $m$, centred reference coordinates and no line through the origin carrying the whole mass (Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model clause (a)). Fix an orthonormal principal basis $(e_1,e_2,e_3)$ with $I{e}_j=I_j{e}_j$ and $I_j>0$ (Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values clauses (a) and (d)), use the model in the torque-free branch of Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass clause (iv), and fix an index $j\in \{1,2,3\}$ and a rate $\mathrm{\Omega }\in \mathbb{R}$. Use the model's nondegenerate open time window $I_0$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length and Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope clause (ii)), and fix a time $t_0\in I_0$ and an orientation $R_0\in SO(3)$. Then:

1. Existence of the uniform rotation. There is a path $R:I_0\to SO(3)$ of class $C^1$ with $R(t_0)=R_0$ and $R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)=[\mathrm{\Omega }{e}_j{]}_{\times }(t\in I_0),$ that is, with constant body angular velocity $\omega \equiv \mathrm{\Omega }{e}_j$ (Body and spatial angular velocity of a rigid-material motion, and the material velocity field clause (a)).
2. It is a torque-free motion. For every such path $R$ and every curve $c:I_0\to {\mathbb{R}}^3$ of class $C^2$, the map $x(t,\rho ):=c(t)+R(t)\rho$ is a motion of the torque-free model of the branch: its angular momentum about the centre of mass is $L_c(t)=\mathrm{\Omega }{I}_{j}R(t)e_j,$ it is constant on $I_0$, and the adopted balance $L_c^{\mathrm{\prime }}=0$ holds. (Here $R$ is automatically of class $C^2$, because $R^{\mathrm{\prime }}=R[\mathrm{\Omega }{e}_j{]}_{\times }$ is the product of the $C^1$ curve $R$ with a constant matrix, so the configuration path $(c,R)$ has the regularity required of a motion.)
3. Geometry and invariants. In the motions of clause 2 one has $l=I\omega =\mathrm{\Omega }{I}_j{e}_j$ constant, $C=\parallel L_c{\parallel }_2^2={\mathrm{\Omega }}^2{I}_j^2,E=\frac{1}{2}{\omega }^{\mathsf{T}}I\omega =\frac{1}{2}{\mathrm{\Omega }}^2{I}_j;$ if $\mathrm{\Omega }\mathrm{\ne }0$, define the spatial line of the body-fixed axis by ${\mathrm{\ell }}_j(t):=\{c(t)+sR(t)e_j:s\in \mathbb{R}\}$ and its unit direction by $n_j(t):=R(t)e_j$. Then $n_j(t)=L_c\mathrm{/}(\mathrm{\Omega }{I}_j)$ for every $t$, so this direction is fixed in the inertial frame and the spatial angular velocity ${\omega }_{\mathrm{s}}(t):=R(t)\omega (t)=\mathrm{\Omega }{n}_j$ of Body and spatial angular velocity of a rigid-material motion, and the material velocity field clause (b) is constant; the line ${\mathrm{\ell }}_j(t)$ need not be fixed because $c$ is arbitrary. If $\mathrm{\Omega }=0$, then $R$ is constant on $I_0$.
4. Converse classification. Conversely, let a motion of a nondegenerate torque-free model have constant body angular velocity $\omega \equiv {\omega }_0$ on its time window. Then $I{\omega }_0=\lambda {\omega }_0$ for some $\lambda \in \mathbb{R}$: either ${\omega }_0=0$ and the orientation path is constant, or ${\omega }_0\mathrm{\ne }0$ and ${\omega }_0$ lies along a principal axis; in the latter case an orthonormal principal basis can be chosen with one vector $e_j={\omega }_0\mathrm{/}\mathrm{\parallel }{\omega }_0{\mathrm{\parallel }}_2$ and $\mathrm{\Omega }=\mathrm{\parallel }{\omega }_0{\mathrm{\parallel }}_2\mathrm{\ne }0$, and the conclusions of clauses 2 and 3 hold in that adapted basis. It can be made positively oriented by negating a different basis vector. No alignment with the initially fixed basis is asserted when eigenvalues repeat.

The statement is conditional on the adopted model and zero-torque balance; it asserts no force, no measurement and no empirical law, and the existence clause is a statement about the model's kinematic class, not about nature.