Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame

Fix a model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: $N\ge 1$ particle labels with fixed masses $m_1,\dots ,m_N>0$ (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)), a time window $I_0\subseteq \mathbb{R}$, an open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ with its collision set removed when the model's force law is singular there (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), and a force map $F:I_0\times Q\times ({\mathbb{R}}^3{)}^N⟶({\mathbb{R}}^3{)}^N$ with the standing continuity and state-local-Lipschitz hypotheses of clause (iii) of that postulate. Fix one inertial frame of the model with adapted coordinates $(t,x)$ (Inertial frames, stationary observers and the adopted Galilean transformations between frames), and let $J\subseteq I_0$ be an interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). All sums below range over $i=1,\dots ,N$ (Laws of finite sums and finite products), and $u\times v$, $⟨u,v⟩$ and $\parallel u{\parallel }_2$ are the cross product, the Euclidean inner product and the Euclidean norm on ${\mathbb{R}}^3$ (The cross product in ${\mathbb{R}}^3$, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

(a) Curve families and motions. A curve family of the model is a map $x:J\to Q$, $x=(x_1,\dots ,x_N)$, of class $C^2$ on $J$ (Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors); its coordinate velocities are $w_i:=x_i^{\mathrm{\prime }}$ on $J$, written $w=(w_1,\dots ,w_N):J\to ({\mathbb{R}}^3{)}^N$. A curve family is a motion of the model when $m_i{\ddot{x}}_i(t)=F_i(t,x(t),w(t))(i=1,\dots ,N,t\in J),$ the notion of clause (c) of The associated first-order system, motions, maximal motions and the Newtonian flow of a force model. Every quantity of this definition is defined for an arbitrary curve family; the balance theorems of this page state which of their properties use the equations of motion. Along a curve family the force map is evaluated at $(t,x(t),w(t))\in I_0\times Q\times ({\mathbb{R}}^3{)}^N$; when the model invokes the external-plus-pair presentation of clause (i) of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch, the maps $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}$ and $F_{ij}$ are evaluated at their declared arguments on the motion's state graph, where the presentation is an identity for $F_i$.

(b) Total mass, total momentum, centre of mass, relative positions. Define $M:={\sum }_i{m}_i>0,p_i:=m_i{w}_i,P(t):={\sum }_i{p}_i(t),X(t):=M^{-1}{\sum }_i{m}_i{x}_i(t),{\rho }_i:=x_i-X\mathrm{.}$ Here $p_i$ is the momentum of label $i$, $P$ the total momentum, $X$ the centre of mass and ${\rho }_i$ the relative position of label $i$. The dimensions are $[M]=\mu$, $[p_i]=[P]=\mu \mathrm{\ell }{\tau }^{-1}$ (read in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$) and $[X]=[{\rho }_i]=\mathrm{\ell }$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Since $x_i$ is $C^2$, each $p_i$ is $C^1$ and each of $P$, $X$, ${\rho }_i$ is differentiable, as a finite sum of differentiable curves (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral); write $V:=X^{\mathrm{\prime }}$ for the centre-of-mass velocity. The algebraic identities ${\sum }_i{m}_i{\rho }_i=0$, $P=MV$ and the splitting of the kinetic sum ${\sum }_i{m}_i\parallel w_i{\parallel }_2^2$ are proved in the kinetic-splitting theorem of this page; they are not part of this definition.

(c) Angular momentum, torque, internal torque sum. Let a reference curve $o:J\to {\mathbb{R}}^3$ of class $C^1$ be given: a point about which moments are taken, allowed to move (a constant map $o\equiv o_0$ is the fixed-origin case). Define the angular momentum about $o$ and the total torque about $o$ by $J_o(t):={\sum }_i(x_i(t)-o(t))\times p_i(t),T_o(t):={\sum }_i(x_i(t)-o(t))\times F_i(t,x(t),w(t)),$ with dimensions $[J_o]=\mu {\mathrm{\ell }}^2{\tau }^{-1}$ (read in $\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}$) and $[T_o]=\mu {\mathrm{\ell }}^2{\tau }^{-2}$ (read in $\mathrm{N}\mathrm{m}$). When the model invokes the external-plus-pair presentation of clause (i) of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch, define in addition the external torque and the internal torque sum $T_o^{\mathrm{e}\mathrm{x}\mathrm{t}}(t):={\sum }_i(x_i(t)-o(t))\times F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}(t,x_i(t),w_i(t)),I(t):={\sum }_{i<j}(x_i(t)-x_j(t))\times F_{ij}(t,x_i(t),x_j(t),w_i(t),w_j(t))\mathrm{.}$ The sum $I$ does not involve $o$; it is defined whenever the pair presentation is declared. The identity $T_o=T_o^{\mathrm{e}\mathrm{x}\mathrm{t}}+I$ uses the antisymmetry $F_{ji}=-F_{ij}$ of clause (ii)1 of that postulate to reindex the ordered pairs and is proved as step 1.1 of the angular-momentum balance of this page.

(d) Kinetic energy, power, work. Define the kinetic energy and the power of the total force along the curve family by $K(t):=\frac{1}{2}{\sum }_i{m}_i\parallel w_i(t){\parallel }_2^2,\mathcal{P}(t):={\sum }_i⟨F_i(t,x(t),w(t)),w_i(t)⟩,$ of dimensions $[K]=\mu {\mathrm{\ell }}^2{\tau }^{-2}$ (read in joules) and $[\mathcal{P}]=\mu {\mathrm{\ell }}^2{\tau }^{-3}$ (read in watts), and for $t_0<t_1$ in $J$ the work of the total force over $[t_0,t_1]$ by $W([t_0,t_1]):={\int }_{t_0}^{t_1}\mathcal{P}(t)dt,$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ (read in joules); $\mathcal{P}$ is continuous on $J$ because $F$, $x$ and $w$ are continuous, so the Riemann integral exists (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, see the page the-riemann-integral). The potential energy is not defined by this item: it belongs to the potential branch of the model and is defined in the potential-and-mechanical-energy item of this page.

(e) Frame and reference-point dependence. The quantities $p_i$, $P$, $X$, ${\rho }_i$, $V$, $K$, $\mathcal{P}$ and $W$ are read in the one inertial frame fixed above, and $J_o$, $T_o$, $T_o^{\mathrm{e}\mathrm{x}\mathrm{t}}$, $I$ are read about the reference curve $o$ in that frame. They are not invariant under a change of frame or reference point: their transformation laws under a change of inertial frame are stated and proved in the frame-transformation item of this page, and every conservation statement involving them names the frame and the reference point in which it is made.

(f) Status. This item is a definition: it introduces symbols, their domains, regularity, dimensions and frame dependence from the model's data. It adopts no force law beyond the force branch already fixed, asserts no balance of any of these quantities and records no measured value.

• Curve families versus motions. The quantities are defined on any $C^2$ curve family in $Q$, so the algebraic identities among them (the splitting of Total momentum equals total mass times centre-of-mass velocity, and the kinetic energy splits into centre-of-mass and relative parts (two-body reduced mass)) hold without the equations of motion; the balance theorems Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces, Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case, Power of the total force and the work-energy theorem: $K^{\mathrm{\prime }}=\mathcal{P}$ and $K(t_1)-K(t_0)=W([t_0,t_1])$ along a motion and Mechanical energy balance for a potential presentation: $E^{\mathrm{\prime }}={\mathrm{\partial }}_{t}U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form use the equations of motion explicitly, and each states its hypotheses on the model's interaction structure.

• The reference curve and the fixed origin. For a constant reference curve $o\equiv o_0$ the quantity $J_o$ is the angular momentum about the fixed point $o_0$; for the zero reference curve it is ${\sum }_i{x}_i\times p_i$, the expression of the mathematical supplier Total momentum and total angular momentum balances for a finite system with antisymmetric pair forces; pairwise central forces make the internal torque vanish. The point $o(t)$ is a coordinate triple in the fixed frame, not a worldline of the model; a moving reference curve is used only to state the moving-origin correction.

• The internal torque sum is not a smallness statement. $I$ collects one term per unordered pair of labels; its vanishing is a hypothesis on the evaluated pair forces (for instance the centrality clause (ii)2 of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch implies it), not a consequence of the equations of motion.