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

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.