How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Momentum, work and energy balances

1 · Prerequisites

2 · Summary

This page defines the mechanical quantities of the force model of cm02-forces-initial-data-and-newtonian-flow and proves their balance statements in one inertial frame. The definition Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame fixes, for any $C^{2}$ curve family of the model, the total mass $M$ , the total momentum $P$ , the centre of mass $X$ and the relative positions $ρ_{i}$ , the angular momentum $J_{o}$ about a fixed or moving reference curve together with the total, external and internal torques, the kinetic energy $K$ , the power $P$ of the total force and its work $W$ over a time interval, with their dimensions and their frame dependence. The algebraic content of these symbols is 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): $P = M V$ , the mass-weighted relative positions sum to zero, and the kinetic sum splits into centre-of-mass and relative parts, with the two-body reduced-mass form.

The balance theorems are conditional on the interaction structure the model declares. For a model with the external-plus-pair presentation and antisymmetric pair forces, Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces gives $\dot{P} = \sum_{i} F_{i}^{e x t}$ and the centre-of-mass equation $M X^{''} = \sum_{i} F_{i}^{e x t}$ , so that the total momentum is constant on an interval exactly when the total external force vanishes there. Under the same presentation, Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case proves the internal-torque identity $T_{o} = T_{o}^{e x t} + I$ , the moving-origin balance $J_{o}^{'} = T_{o}^{e x t} + I - o^{'} \times P$ , the vanishing of the internal torque sum $I$ when the evaluated pair forces are central, and the conservation of $J_{o}$ about a fixed point with vanishing total torque; centrality is sufficient but not necessary, as the companion page's counterexample exhibits. Independently of any pair structure, Power of the total force and the work-energy theorem: $K^{'} = P$ and $K (t_{1}) - K (t_{0}) = W ([t_{0}, t_{1}])$ along a motion proves $K^{'} = P$ and the work-energy theorem $K (t_{1}) - K (t_{0}) = W ([t_{0}, t_{1}])$ .

The potential branch is defined in Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets: a $C^{2}$ potential $U$ with $F_{i} = - \nabla_{x_{i}} U$ (optionally with declared residual forces), the potential and mechanical energy along a curve family, the non-uniqueness of the potential under addition of a function of time alone, the closed/exact distinction for time- and velocity-independent configuration-space force fields — with the Poincaré lemma on star-shaped subdomains, no global potential from closedness alone, and equipotential level sets — and local potentials. Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form then gives $E^{'} = \partial_{t} U (t, x (t)) + \sum_{i} ⟨ d_{i}, w_{i} ⟩$ , the conservation of $E$ when the representative is time-independent and the evaluated residual forces vanish along the motion (in particular, when the residual-force maps vanish throughout their state domains), and the integrated work form. The frame behaviour of all these quantities — affine centre-of-mass law, the boost law $\tilde{P} = R P + M v$ , the kinetic and mechanical boost terms, the angular-momentum boost, and the transformation of torque, power and work through the transferred force map — is Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy.

The closing audit Status audit and scope of the balance page: conditional conservation, open systems, frames, potential normalization and empirical boundaries records which hypotheses each conservation statement needs, the open-system boundary inherited from Status audit and model boundaries of the force page: open systems, frame selection, breakdown and empirical scope, the frame-relativity of the conserved values, the normalization of the potential, and the empirical status of model parameters. Worked checks of the laws on explicit data, the non-central cancellation counterexample, the closed-field-without-global-potential counterexample and a driven-energy computation are collected on cm03-momentum-work-and-energy-balances-examples.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

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

Physical scope

Vocabulary and dimension conventions for the mechanical quantities of a force model in one inertial frame: the total mass, total momentum, centre of mass, relative positions and centre-of-mass velocity; the angular momentum, total torque, external torque and internal torque sum about a reference curve; the kinetic energy, the power of the total force and the work over a time interval. The item defines these symbols from the model's particles and force map and fixes their frames; it asserts no balance, no conservation law and no measured value.

Definition

Fix a model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: $N \geq 1$ 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 $I_{0} \subseteq R$ , an open configuration domain $Q \subseteq R^{3 N}$ 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 $F : I_{0} \times Q \times (R^{3})^{N} ⟶ (R^{3})^{N}$ 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 $R$ : 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 $∥ u ∥_{2}$ are the cross product, the Euclidean inner product and the Euclidean norm on $R^{3}$ (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

(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 $w_{i} : = x_{i}^{'}$ on $J$ , written $w = (w_{1}, \dots, w_{N}) : J \to (R^{3})^{N}$ . 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 $(t, x (t), w (t)) \in I_{0} \times Q \times (R^{3})^{N}$ ; 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 $F_{i}^{e x t}$ and $F_{i j}$ 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 $M : = \sum_{i} m_{i} > 0, p_{i} : = m_{i} w_{i}, P (t) : = \sum_{i} p_{i} (t), X (t) : = M^{- 1} \sum_{i} m_{i} x_{i} (t), ρ_{i} : = x_{i} - X .$ Here $p_{i}$ is the momentum of label $i$ , $P$ the total momentum, $X$ the centre of mass and $ρ_{i}$ the relative position of label $i$ . The dimensions are $[M] = μ$ , $[p_{i}] = [P] = μ ℓ τ^{- 1}$ (read in $k g m s^{- 1}$ ) and $[X] = [ρ_{i}] = ℓ$ (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 $P$ , $X$ , $ρ_{i}$ 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 $V : = X^{'}$ for the centre-of-mass velocity. The algebraic identities $\sum_{i} m_{i} ρ_{i} = 0$ , $P = M V$ and the splitting of the kinetic sum $\sum_{i} m_{i} ∥ w_{i} ∥_{2}^{2}$ 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 $o : J \to R^{3}$ 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 $[J_{o}] = μ ℓ^{2} τ^{- 1}$ (read in $k g m^{2} s^{- 1}$ ) and $[T_{o}] = μ ℓ^{2} τ^{- 2}$ (read in $N m$ ). 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 $T_{o}^{e x t} (t) : = \sum_{i} (x_{i} (t) - o (t)) \times F_{i}^{e x t} (t, x_{i} (t), w_{i} (t)), I (t) : = \sum_{i < j} (x_{i} (t) - x_{j} (t)) \times F_{i j} (t, x_{i} (t), x_{j} (t), w_{i} (t), w_{j} (t)) .$ The sum $I$ does not involve $o$ ; it is defined whenever the pair presentation is declared. The identity $T_{o} = T_{o}^{e x t} + I$ uses the antisymmetry $F_{j i} = - F_{i j}$ 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 $K (t) : = \frac{1}{2} \sum_{i} m_{i} ∥ w_{i} (t) ∥_{2}^{2}, P (t) : = \sum_{i} ⟨ F_{i} (t, x (t), w (t)), w_{i} (t) ⟩,$ of dimensions $[K] = μ ℓ^{2} τ^{- 2}$ (read in joules) and $[P] = μ ℓ^{2} τ^{- 3}$ (read in watts), and for $t_{0} < t_{1}$ in $J$ the work of the total force over $[t_{0}, t_{1}]$ by $W ([t_{0}, t_{1}]) : = \int_{t_{0}}^{t_{1}} P (t) d t,$ of dimension $μ ℓ^{2} τ^{- 2}$ (read in joules); $P$ 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 $p_{i}$ , $P$ , $X$ , $ρ_{i}$ , $V$ , $K$ , $P$ and $W$ are read in the one inertial frame fixed above, and $J_{o}$ , $T_{o}$ , $T_{o}^{e x t}$ , $I$ 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.

Remarks

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: $K^{'} = P$ and $K (t_{1}) - K (t_{0}) = W ([t_{0}, t_{1}])$ along a motion and Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ 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.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

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)

Physical scope

The algebraic identities of a curve family of a force model in one inertial frame: the total momentum is the total mass times the centre-of-mass velocity, the mass-weighted relative positions sum to zero together with their derivatives, and the total kinetic energy equals the centre-of-mass kinetic energy plus the mass-weighted relative kinetic energy, with the two-body reduced-mass form. The result is a conditional identity for the model's masses and curves and asserts no conservation law, no force structure and no measured value.

Statement

Adopt the force model, the inertial frame and the notation of Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame: masses $m_{1}, \dots, m_{N} > 0$ , a curve family $x : J \to Q$ of class $C^{2}$ on an interval $J$ with nonempty interior, coordinate velocities $w_{i} = x_{i}^{'}$ , and the quantities $M = \sum_{i} m_{i}, P = \sum_{i} m_{i} w_{i}, X = M^{- 1} \sum_{i} m_{i} x_{i}, ρ_{i} = x_{i} - X, V = X^{'}, K = \frac{1}{2} \sum_{i} m_{i} ∥ w_{i} ∥_{2}^{2} .$ Then:

$P = M V$ on $J$ , that is $X^{'} = M^{- 1} P$ , and the identities $\sum_{i} m_{i} ρ_{i} = 0, \sum_{i} m_{i} ρ_{i}^{'} = 0$ hold on $J$ ;
for every $t \in J$ , $K (t) = \frac{1}{2} M ∥ V (t) ∥_{2}^{2} + \frac{1}{2} \sum_{i} m_{i} ∥ ρ_{i}^{'} (t) ∥_{2}^{2};$
if $N = 2$ , then with $r : = x_{1} - x_{2}$ and $μ : = m_{1} m_{2} / M$ one has, for every $t \in J$ , $K (t) = \frac{1}{2} M ∥ V (t) ∥_{2}^{2} + \frac{1}{2} μ ∥ r^{'} (t) ∥_{2}^{2} .$

The statement is an identity for the stated curves and masses in the fixed frame; it uses no equation of motion, no property of the force map and no conservation claim. The values are frame-dependent; the transformation laws are the frame-transformation item of this page.

Facts & Assumptions

Given: A force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, one inertial frame, a $C^{2}$ curve family $x : J \to Q$ , and the masses $m_{i} > 0$ .

[L1]

Model and notation (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame): $M$ , $P$ , $X$ , $ρ_{i}$ , $V$ and $K$ are defined by the displayed finite sums from $x$ and the masses; $x_{i}$ is $C^{2}$ ; $w_{i} = x_{i}^{'}$ ; dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.

[L2]

Mathematical supplier (Centre of mass, the identity $\sum_{i} m_{i} ρ_{i} = 0$ , and the splitting of the kinetic sum $\sum_{i} m_{i} ∥ x_{i}^{'} ∥_{2}^{2}$ into centre-of-mass and relative parts (with the two-body reduced-mass case)): for $N \geq 1$ , an interval $I$ with at least two points, masses $m_{i} > 0$ and curves $x_{i} : I \to R^{3}$ differentiable at every point, the centre $X = M^{- 1} \sum_{i} m_{i} x_{i}$ and the relative positions $ρ_{i} = x_{i} - X$ are differentiable with $X^{'} = M^{- 1} \sum_{i} m_{i} x_{i}^{'}$ and $ρ_{i}^{'} = x_{i}^{'} - X^{'}$ ; the identities $\sum_{i} m_{i} ρ_{i} = 0$ and $\sum_{i} m_{i} ρ_{i}^{'} = 0$ hold; and for every point $\sum_{i} m_{i} ∥ x_{i}^{'} ∥_{2}^{2} = M ∥ X^{'} ∥_{2}^{2} + \sum_{i} m_{i} ∥ ρ_{i}^{'} ∥_{2}^{2},$ with the two-body case $\sum_{i} m_{i} ∥ ρ_{i}^{'} ∥_{2}^{2} = μ ∥ r^{'} ∥_{2}^{2}$ for $r = x_{1} - x_{2}$ , $μ = m_{1} m_{2} / M$ .

[L3]

The Euclidean norm satisfies $∥ u ∥_{2}^{2} = ⟨ u, u ⟩$ and scalar factors pull out of finite sums (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Laws of finite sums and finite products).

Proof

technique · direct; apply the mathematical supplier to the model's curves and masses and translate its notation through the definition

1.1givenL2

The interval $J$ has nonempty interior, hence at least two points, and each $x_{i}$ is differentiable on $J$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral); so the hypotheses of [L2] are met with $I : = J$ . Its first clause gives differentiability of $X$ and each $ρ_{i}$ , the formula $X^{'} = M^{- 1} \sum_{i} m_{i} x_{i}^{'}$ , the formula $ρ_{i}^{'} = x_{i}^{'} - X^{'}$ , and the identities $\sum_{i} m_{i} ρ_{i} = 0$ and $\sum_{i} m_{i} ρ_{i}^{'} = 0$ on $J$ .

2.1step 1.1L1algebra

Writing $x_{i}^{'} = w_{i}$ by [L1], the formula of step 1.1 reads $X^{'} = M^{- 1} \sum_{i} m_{i} w_{i} = M^{- 1} P$ , so $P = M V$ and $X^{'} = M^{- 1} P$ ; together with step 1.1 this is clause 1.

3.1step 2.1L1L2L3algebra

Clause 2: multiply the splitting identity of [L2] by $\frac{1}{2}$ and substitute $x_{i}^{'} = w_{i}$ and $X^{'} = V$ ; the left side is $K$ by [L1], and the two right-hand terms are $\frac{1}{2} M ∥ V ∥_{2}^{2}$ and $\frac{1}{2} \sum_{i} m_{i} ∥ ρ_{i}^{'} ∥_{2}^{2}$ .

4.1step 3.1L2algebra∎

Clause 3: for $N = 2$ , the two-body clause of [L2] gives $\sum_{i} m_{i} ∥ ρ_{i}^{'} ∥_{2}^{2} = μ ∥ r^{'} ∥_{2}^{2}$ with $r = x_{1} - x_{2}$ and $μ = m_{1} m_{2} / M$ ; substituting into step 3.1 gives the displayed two-body splitting.

Remarks

The splitting is instantaneous. Clause 2 is an identity at each time between the kinetic sums, not a decomposition of the motion; the two terms are not separately conserved in general, and no force hypothesis enters.
Frame dependence. The identity holds in every inertial frame with that frame's $X$ and $ρ_{i}$ . Under a Galilean change, $V$ gains the boost velocity while each relative velocity $ρ_{i}^{'}$ is rotated without a boost; thus the centre-of-mass kinetic term generally changes and the relative kinetic term is unchanged. The total kinetic-energy law is Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy clause 2.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces

Physical scope

The momentum balance of a model that declares the external-plus-pair force presentation with antisymmetric pair forces: along every motion the total momentum changes at the rate of the total external force, it is constant on an interval exactly when the total external force vanishes there, and the centre of mass obeys the single-particle equation with the total mass and the total external force. The result is a conditional deduction from the adopted force and interaction posts; it asserts no conservation for models without the antisymmetry presentation and no measured value.

Statement

Adopt the force model and one inertial frame of Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, and suppose in addition that 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, $F_{i} (t, z) = F_{i}^{e x t} (t, x_{i}, v_{i}) + \sum_{j \neq i} F_{i j} (t, x_{i}, x_{j}, v_{i}, v_{j}),$ with the antisymmetry clause (ii)1 of that postulate, $F_{i j} (t, x_{i}, x_{j}, v_{i}, v_{j}) = - F_{j i} (t, x_{j}, x_{i}, v_{j}, v_{i})$ on the common domain. Let $x : J \to Q$ be a motion of the model on an interval $J$ with nonempty interior, with quantities $M$ , $P$ , $X$ , $V$ as in the definition, and write $σ (t) : = \sum_{i} F_{i}^{e x t} (t, x_{i} (t), w_{i} (t)) (t \in J)$ for the total external force along the motion. Then:

$P$ is of class $C^{1}$ on $J$ and $\dot{P} (t) = σ (t) (t \in J) :$ the total momentum changes at the rate of the total external force, the internal pair forces cancelling in the sum;
if $σ \equiv 0$ on $J$ , then $P$ is constant on $J$ and $V = M^{- 1} P$ is constant on $J$ ;
$X$ is of class $C^{2}$ on $J$ and $M X^{''} (t) = σ (t) (t \in J) :$ the centre of mass obeys the equation of a single particle of mass $M$ under the total external force.

The statement assumes the pair presentation and its antisymmetry; for a model whose force map is not presented through such pairs it asserts nothing, and the pair forces of the presentation need not be central (centrality is used only in the angular-momentum balance of this page). No empirical content is claimed.

Facts & Assumptions

Given: A force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance that invokes the external-plus-pair presentation of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch with the antisymmetry clause, one inertial frame, and a motion $x : J \to Q$ .

[L1]

Notions and notation (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model): $w_{i} = x_{i}^{'}$ , $p_{i} = m_{i} w_{i}$ , $P = \sum_{i} p_{i}$ , $M = \sum_{i} m_{i}$ , $X = M^{- 1} \sum_{i} m_{i} x_{i}$ , $V = X^{'}$ ; a motion satisfies $m_{i} {\ddot{x}}_{i} (t) = F_{i} (t, x (t), w (t))$ for all $i$ and $t \in J$ ; the external-plus-pair presentation is an identity for $F_{i}$ on its declared domain, and its pair maps satisfy the antisymmetry of clause (ii)1.

[L2]

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 clause 1): if $x_{i} : I \to R^{3}$ are twice differentiable, $p_{i} = m_{i} x_{i}^{'}$ and there are maps $f_{i}, f_{i j} : I \to R^{3}$ with $f_{i j} = - f_{j i}$ and $p_{i}^{'} = f_{i} + \sum_{j \neq i} f_{i j}$ , then $P = \sum_{i} p_{i}$ is differentiable on $I$ with $P^{'} = \sum_{i} f_{i}$ .

[L3]

A continuous real function on an interval whose derivative vanishes at every interior point is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant). Differentiability, sums and scalar multiples of curves behave as in The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral; finite sums are linear (Laws of finite sums and finite products).

Proof

technique · direct; evaluate the model's pair presentation along the motion and apply the mathematical balance identity

1.1givenL1

Define, for $t \in J$ , $f_{i} (t) : = F_{i}^{e x t} (t, x_{i} (t), w_{i} (t)), f_{i j} (t) : = F_{i j} (t, x_{i} (t), x_{j} (t), w_{i} (t), w_{j} (t)) (i \neq j) .$ These are well-defined functions $J \to R^{3}$ : the evaluated arguments lie in the stated domains of the pair presentation and of the model's state domain, and the presentation is an identity for $F_{i}$ there. By the antisymmetry clause, $f_{i j} (t) = - f_{j i} (t)$ for all $i \neq j$ and $t \in J$ .

2.1step 1.1L1L2algebra

Each $p_{i} = m_{i} w_{i}$ is differentiable on $J$ with $p_{i}^{'} = m_{i} {\ddot{x}}_{i}$ ; by the equation of motion and the presentation this equals $F_{i} (t, x (t), w (t)) = f_{i} (t) + \sum_{j \neq i} f_{i j} (t) (t \in J) .$ Thus the data of [L2] are met on the interval $J$ with the maps $f_{i}$ , $f_{i j}$ of step 1.1, and [L2] yields that $P$ is differentiable on $J$ with $P^{'} (t) = \sum_{i} f_{i} (t) = σ (t);$ $P^{'}$ is continuous because the $f_{i}$ are continuous, so $P$ is $C^{1}$ on $J$ , and clause 1 is proved.

3.1step 2.1L3algebra

Clause 2: if $σ \equiv 0$ on $J$ , then $P^{'} = 0$ at every interior point of $J$ by step 2.1, and $P$ is continuous on $J$ ; [L3] gives that $P$ is constant on $J$ ; then $V = M^{- 1} P$ is constant on $J$ as a constant multiple.

4.1step 2.1L1algebra∎

Clause 3: by clause 1 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) applied to the motion (which is a $C^{2}$ curve family), $X$ is differentiable on $J$ with $X^{'} = M^{- 1} P$ ; since $P$ is $C^{1}$ by step 2.1, $X^{'}$ is $C^{1}$ , so $X$ is $C^{2}$ with $X^{''} = M^{- 1} P^{'} = M^{- 1} σ,$ that is $M X^{''} = σ$ , which is clause 3.

Remarks

The hypothesis is the presentation, not the physics. The cancellation $\sum_{i} \sum_{j \neq i} f_{i j} = 0$ is a consequence of the declared antisymmetry of the model's pair maps; a force map for which no such presentation is declared has no momentum balance of this form. In particular, the framework does not claim that every interaction admits an antisymmetric pair presentation (Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clause (i)).
Open systems. For a proper subsystem, forces to labels outside the declared set appear among the $F_{i}^{e x t}$ and need satisfy no antisymmetry; the subsystem's momentum may be exchanged with the environment. Clause 1 is a balance, and clause 2 gives conservation when the evaluated total external force vanishes. This is the open-system boundary recorded in Status audit and model boundaries of the force page: open systems, frame selection, breakdown and empirical scope.
No centrality is used. The balance of this item holds for velocity- and position-dependent antisymmetric pair forces alike; centrality is required only for the internal-torque statement of Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case, and is sufficient but not necessary there.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case

Physical scope

The angular momentum balance of a model that declares the external-plus-pair presentation with antisymmetric pair forces: the internal torque sum is independent of the reference point and the balance about a possibly moving reference point carries the correction minus odot cross P; when the evaluated pair forces are central the internal torque sum vanishes; about a fixed point with central evaluated pair forces and vanishing external torque the angular momentum is constant. The result is conditional on the declared pair structure and centrality where invoked; it asserts no necessity of centrality and no measured value.

Statement

Adopt the force model and one inertial frame of Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, and suppose that 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 with the antisymmetry clause (ii)1 (as in the momentum balance of this page). Let $x : J \to Q$ be a motion on an interval $J$ with nonempty interior and let $o : J \to R^{3}$ be a $C^{1}$ reference curve, with $J_{o}$ , $T_{o}$ , $T_{o}^{e x t}$ , $I$ and $P$ as in the definition. Then:

Internal torque identity. For every $t \in J$ , $\sum_{i} (x_{i} (t) - o (t)) \times \sum_{j \neq i} F_{i j} (t, x_{i} (t), x_{j} (t), w_{i} (t), w_{j} (t)) = I (t), so T_{o} (t) = T_{o}^{e x t} (t) + I (t);$
Balance. $J_{o}$ is differentiable on $J$ and $J_{o}^{'} (t) = T_{o}^{e x t} (t) + I (t) - o^{'} (t) \times P (t) = T_{o} (t) - o^{'} (t) \times P (t) (t \in J);$
Centrality. Suppose that for every $i < j$ and every $t \in J$ the evaluated pair force is parallel to $x_{i} (t) - x_{j} (t)$ , that is, there is a real $λ_{i j} (t)$ with $F_{i j} (t, x_{i} (t), x_{j} (t), w_{i} (t), w_{j} (t)) = λ_{i j} (t) (x_{i} (t) - x_{j} (t)) .$ Then $I \equiv 0$ on $J$ , so $J_{o}^{'} (t) = T_{o}^{e x t} (t) - o^{'} (t) \times P (t) (t \in J);$ if moreover $o$ is constant, then $J_{o}^{'} = T_{o}^{e x t}$ ;
Conservation. If $o$ is constant and $T_{o} \equiv 0$ on $J$ — equivalently $T_{o}^{e x t} \equiv - I$ there, by clause 1 — then $J_{o}$ is constant on $J$ . In particular, if the evaluated pair forces are central (clause 3) and $T_{o}^{e x t} \equiv 0$ on $J$ — the case of a model with no external forces — then $T_{o} \equiv 0$ and $J_{o}$ is constant on $J$ .

Clause 3 allows cancellation: the vanishing of $I$ is a hypothesis on the evaluated pair forces and does not require centrality in general; the companion page exhibits antisymmetric non-central pair forces whose internal torque sum vanishes identically. No empirical content is claimed.

Facts & Assumptions

[L1]

Notation and notions (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model): $p_{i} = m_{i} w_{i}$ , $P = \sum_{i} p_{i}$ , $J_{o} = \sum_{i} (x_{i} - o) \times p_{i}$ , $T_{o}^{e x t} = \sum_{i} (x_{i} - o) \times F_{i}^{e x t}$ , $I = \sum_{i < j} (x_{i} - x_{j}) \times F_{i j}$ (all evaluated along the motion), $T_{o} = \sum_{i} (x_{i} - o) \times F_{i}$ ; the equation of motion is $m_{i} {\ddot{x}}_{i} = F_{i} (t, x, w)$ ; the presentation writes $F_{i} = F_{i}^{e x t} + \sum_{j \neq i} F_{i j}$ and its pair maps satisfy $F_{j i} = - F_{i j}$ .

[L2]

Product rule (Differentiation of sums, scalar multiples, inner products, cross products and squared norms of differentiable curves in $R^{3}$ clause 3): if $u, v : I \to R^{3}$ are differentiable at $t_{0}$ , then $u \times v$ is differentiable at $t_{0}$ with derivative $u^{'} (t_{0}) \times v (t_{0}) + u (t_{0}) \times v^{'} (t_{0})$ ; sums and scalar multiples of differentiable curves are differentiable with the expected derivatives (clause 1 of the same lemma, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[L3]

Cross-product algebra (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors): the cross product is bilinear, so $a \times (u + v) = a \times u + a \times v$ and $a \times (λ u) = λ (a \times u)$ ; it is alternating, so $u \times u = 0$ and $u \times v = - (v \times u)$ . These identities distribute the force sums in step 1.1 and cancel each term in step 3.1.

[L4]

Reindexing (Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule clause 2): the ordered pairs $(i, j)$ with $i \neq j$ are the disjoint union of the pairs ${(i, j), (j, i)}$ over $i < j$ , so $\sum_{i} \sum_{j \neq i} a_{i j} = \sum_{i < j} (a_{i j} + a_{j i})$ for vectors $a_{i j}$ ; finite sums are linear (Laws of finite sums and finite products, [L3]).

[L5]

A continuous real-valued function on an interval with zero derivative at every interior point is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant); the interval $J$ has nonempty interior (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Proof

technique · direct; reindex the internal pair sums using antisymmetry, differentiate the angular momentum by the product rule, and apply the centrality and vanishing-torque hypotheses

1.1givenL1L3L4

Internal torque identity. Using [L4] and the antisymmetry of the pair maps, $\sum_{i} (x_{i} - o) \times \sum_{j \neq i} F_{i j} = \sum_{i < j} [(x_{i} - o) \times F_{i j} + (x_{j} - o) \times F_{j i}] = \sum_{i < j} (x_{i} - x_{j}) \times F_{i j} = I,$ because $F_{j i} = - F_{i j}$ and the cross product is bilinear by [L3]; the first equality also shows $I$ does not depend on $o$ . Adding $T_{o}^{e x t} = \sum_{i} (x_{i} - o) \times F_{i}^{e x t}$ and substituting the presentation $F_{i} = F_{i}^{e x t} + \sum_{j \neq i} F_{i j}$ gives $T_{o} = T_{o}^{e x t} + I$ , completing clause 1.

2.1step 1.1givenL1L2L3algebra

Balance. Each summand $(x_{i} - o) \times p_{i}$ of $J_{o}$ is a cross product of curves differentiable on $J$ (since $x_{i}$ is $C^{2}$ and $o$ is $C^{1}$ ), so by [L2] $J_{o}$ is differentiable on $J$ with $J_{o}^{'} = \sum_{i} (w_{i} - o^{'}) \times p_{i} + \sum_{i} (x_{i} - o) \times p_{i}^{'} .$ In the first sum, $\sum_{i} w_{i} \times p_{i} = \sum_{i} m_{i} w_{i} \times w_{i} = 0$ by alternation [L3], so it equals $- o^{'} \times P$ ; in the second sum, substitute $p_{i}^{'} = m_{i} {\ddot{x}}_{i} = F_{i} (t, x, w)$ from the equation of motion, which gives $\sum_{i} (x_{i} - o) \times F_{i} = T_{o} = T_{o}^{e x t} + I$ by clause 1. Therefore $J_{o}^{'} = T_{o}^{e x t} + I - o^{'} \times P$ , which is clause 2.

3.1step 2.1givenL3algebra

Centrality. Under the hypothesis of clause 3, for each $i < j$ $(x_{i} - x_{j}) \times F_{i j} = λ_{i j} (x_{i} - x_{j}) \times (x_{i} - x_{j}) = 0$ by alternation and bilinearity [L3]; hence $I = 0$ and step 2.1 gives $J_{o}^{'} = T_{o}^{e x t} - o^{'} \times P$ . If $o$ is constant then $o^{'} = 0$ and $J_{o}^{'} = T_{o}^{e x t}$ .

4.1step 2.1step 3.1L1L5∎

Conservation. Let $o$ be constant and $T_{o} \equiv 0$ on $J$ . Then $o^{'} = 0$ and step 2.1 gives $J_{o}^{'} = T_{o}^{e x t} + I = T_{o} = 0$ at every interior point of $J$ . $J_{o}$ is continuous on $J$ , being a finite sum of products of continuous curves; [L5] gives that $J_{o}$ is constant on $J$ . In the particular case, step 3.1 gives $T_{o} = T_{o}^{e x t} + I = T_{o}^{e x t} \equiv 0$ when the evaluated pair forces are central and the external torque vanishes; this includes a model with no external forces.

Remarks

The moving-origin correction. For a fixed origin the balance reads $J_{o}^{'} = T_{o}$ ; for a moving reference curve the extra term $- o^{'} \times P$ appears. The exact conservation condition is $T_{o}^{e x t} + I = o^{'} \times P$ along the motion. When the pair torque vanishes and the external torque is also zero, this reduces to $o^{'} \times P = 0$ ; the companion page checks the correction on explicit data.
Centrality is sufficient, not necessary. Clause 3 is one sufficient condition for $I \equiv 0$ ; clause 4 records that the conservation conclusion holds whenever the total torque about the fixed reference point vanishes, which can happen without centrality by cancellation among the pair torques. The companion counterexample makes this explicit.
No claim about nature. The pair structure, antisymmetry and centrality are properties a model may or may not declare; this item is a conditional deduction from them, and the framework's Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clause (i) records that an arbitrary force map need not admit such a presentation.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Power of the total force and the work-energy theorem: $K^{'} = P$ and $K (t_{1}) - K (t_{0}) = W ([t_{0}, t_{1}])$ along a motion

Physical scope

The work-energy identity of the force branch: along every motion the time derivative of the kinetic energy equals the power of the total force, and the change of kinetic energy over a time interval equals the work of the total force. The result is a conditional deduction from the adopted equation of motion and uses no potential, no pair structure and no conservation hypothesis; it asserts no frame-independent value and no measured quantity.

Statement

Adopt the force model and one inertial frame of Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame: masses $m_{1}, \dots, m_{N} > 0$ , an open configuration domain $Q$ , the force map $F$ with the standing hypotheses of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iii), and the quantities $w_{i}$ , $K = \frac{1}{2} \sum_{i} m_{i} ∥ w_{i} ∥_{2}^{2}, P = \sum_{i} ⟨ F_{i} (t, x, w), w_{i} ⟩, W ([t_{0}, t_{1}]) = \int_{t_{0}}^{t_{1}} P (t) d t$ along a curve family. Let $x : J \to Q$ be a motion of the model on an interval $J$ with nonempty interior. Then:

$K$ is of class $C^{1}$ on $J$ and $K^{'} (t) = P (t) (t \in J) :$ the rate of change of the kinetic energy equals the power of the total force;
for all $t_{0} < t_{1}$ in $J$ , $K (t_{1}) - K (t_{0}) = \int_{t_{0}}^{t_{1}} P (t) d t = W ([t_{0}, t_{1}]) :$ the change of the kinetic energy equals the work of the total force.

No potential, no pair decomposition and no conservation hypothesis is used: the identity is a consequence of the equation of motion alone, in the fixed frame. The power and the work are frame-dependent (see clause 4 of the frame-transformation item of this page); the mechanical energy statement of the potential branch is the energy balance of this page.

Facts & Assumptions

Given: A force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with a motion $x : J \to Q$ on an interval $J$ with nonempty interior, and the quantities $K$ , $P$ , $W$ of Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame.

[L1]

Notation and equation of motion (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model): $w_{i} = x_{i}^{'}$ is $C^{1}$ because $x_{i}$ is $C^{2}$ ; $p_{i} = m_{i} w_{i}$ ; and $m_{i} {\ddot{x}}_{i} (t) = F_{i} (t, x (t), w (t))$ for all $i$ , $t \in J$ . The Euclidean inner product satisfies $∥ u ∥_{2}^{2} = ⟨ u, u ⟩$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ) and finite sums are linear (Laws of finite sums and finite products).

[L2]

Derivative of the squared norm (Differentiation of sums, scalar multiples, inner products, cross products and squared norms of differentiable curves in $R^{3}$ clause 4): for a differentiable curve $u$ , the function $t \mapsto ∥ u (t) ∥_{2}^{2}$ is differentiable with derivative $2 ⟨ u^{'} (t), u (t) ⟩$ .

[L3]

Newton–Leibniz (Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative): if $G : [a, b] \to R$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , and $f$ is Riemann integrable with $f = G^{'}$ on $(a, b)$ , then $\int_{a}^{b} f = G (b) - G (a)$ . The interval $J$ has nonempty interior (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Proof

technique · direct; differentiate the kinetic energy using the equation of motion and integrate the resulting identity

1.1givenL1L2algebra

For each $i$ , $w_{i}$ is differentiable with derivative $w_{i}^{'} = {\ddot{x}}_{i}$ ; [L2] gives $\frac{d}{d t} ∥ w_{i} (t) ∥_{2}^{2} = 2 ⟨ {\ddot{x}}_{i} (t), w_{i} (t) ⟩ .$ Multiplying by $\frac{1}{2} m_{i}$ and summing over $i$ with the linearity of finite sums, $K^{'} (t) = \frac{1}{2} \sum_{i} m_{i} \cdot 2 ⟨ {\ddot{x}}_{i} (t), w_{i} (t) ⟩ = \sum_{i} ⟨ m_{i} {\ddot{x}}_{i} (t), w_{i} (t) ⟩ = \sum_{i} ⟨ F_{i} (t, x (t), w (t)), w_{i} (t) ⟩ = P (t),$ where the third equality is the equation of motion of [L1]. Each term is continuous in $t$ (the $F_{i}$ and $w_{i}$ are continuous), so $K^{'}$ is continuous and $K$ is $C^{1}$ ; this is clause 1.

2.1step 1.1L1L3∎

Clause 2: fix $t_{0} < t_{1}$ in $J$ . The function $K$ is continuous on $[t_{0}, t_{1}]$ and differentiable on $(t_{0}, t_{1})$ with $K^{'} = P$ by step 1.1, and $P$ is continuous on $[t_{0}, t_{1}]$ , hence Riemann integrable there; [L3] gives $\int_{t_{0}}^{t_{1}} P = K (t_{1}) - K (t_{0}),$ which is the first equality of clause 2, and the second equality is the definition of $W ([t_{0}, t_{1}])$ .

Remarks

No structure beyond the equation of motion. The power of the total force is the sum of the powers of whatever contributions the model declares, but the identity does not require the sum to split over pairs, and it holds for velocity-dependent and time-dependent forces alike.
Work depends on the frame and on the path. For the boost velocity $v$ in Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy clause 4, the power gains $⟨ R Σ, v ⟩$ , and the work gains the integral of this term. For a motion with the antisymmetric pair presentation, the momentum balance replaces that integral by $⟨ R (P (t_{1}) - P (t_{0})), v ⟩$ ; without those hypotheses the force-integral form applies. The work-energy identity holds in each frame separately.
Not yet an energy conservation statement. Clause 2 relates the kinetic energy to the work of the total force; a conserved mechanical energy requires a potential, which is the subject of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets and Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets

Physical scope

The potential branch of the force model: a scalar potential representing the force map through the Euclidean gradient, with or without declared residual forces; the potential and mechanical energy along a curve family; the non-uniqueness of the potential under addition of a function of time; the distinction between closed and exact time- and velocity-independent force fields, with local potentials on star-shaped subdomains and the failure of global exactness on a punctured domain; and equipotential level sets. The item defines the branch and records cited mathematical facts; it asserts no conservation law, no measured value and no global potential for an arbitrary closed field.

Definition

Adopt the force model and one inertial frame of Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, with configuration domain $Q$ , time window $I_{0}$ and force map $F$ . Dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework and gradients are Euclidean (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case); points of $Q$ are written $x = (x_{1}, \dots, x_{N})$ with $x_{i} \in R^{3}$ .

(a) Potential representation of the force map. A model presents its force map through a potential when it declares a function $U : I_{0} \times Q ⟶ R$ of class $C^{2}$ ( $C^{k}$ Euclidean maps and diffeomorphisms), of dimension $μ ℓ^{2} τ^{- 2}$ (read in joules), with $F_{i} (t, x, v) = - \nabla_{x_{i}} U (t, x) (i = 1, \dots, N)$ for every $(t, x, v)$ of the state domain; here $\nabla_{x_{i}} U$ is the gradient of $u \mapsto U (t, x_{1}, \dots, x_{i - 1}, u, x_{i + 1}, \dots, x_{N})$ at $u = x_{i}$ . More generally, a model may declare residual forces $d_{i} : I_{0} \times Q \times (R^{3})^{N} \to R^{3}$ of the standing regularity of clause (iii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance and present $F_{i} (t, x, v) = - \nabla_{x_{i}} U (t, x) + d_{i} (t, x, v);$ the pair $(U, d)$ is then the model's presentation and $F_{i}^{p} : = - \nabla_{x_{i}} U$ is its potential part. The scalar $U$ is a potential energy for the model's potential part; the framework's earlier admission of potentials in Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clause (iii) is the special case $d \equiv 0$ . A model that presents its whole force map this way with $d \equiv 0$ is a global potential model (the term "conservative model" is used only in this sense).

(b) Potential and mechanical energy along a curve family. For a curve family $x : J \to Q$ of the model (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame clause (a)) define the potential energy along the family and the mechanical energy by $U_{x} (t) : = U (t, x (t)), E (t) : = K (t) + U_{x} (t) (t \in J),$ with $K$ the kinetic energy of the definition. The potential part contributes to the force field but not directly to $K$ ; $E$ is defined only along the potential branch of the model, and along a motion it is the subject of the energy balance of this page.

(c) Normalization of the potential. Let $c : I_{0} \to R$ be differentiable and put $\tilde{U} : = U + c (t)$ , retaining $C^{2}$ regularity when a potential presentation of the standing class is required. The spatial gradient is unchanged, so this presents the same potential part with the same residual forces. The full converse is componentwise: if $\tilde{U}$ and $U$ are $C^{2}$ and have the same spatial gradient, then on each connected component $C$ of the open set $Q$ there is a $C^{2}$ function $c_{C} : I_{0} \to R$ with $\tilde{U} (t, x) - U (t, x) = c_{C} (t) (x \in C) .$ Here a connected component is a maximal connected subset of $Q$ . For fixed $t$ , the difference is constant on every ball contained in $Q$ : along any segment in the ball its derivative is the vanishing spatial gradient paired with the segment direction. The difference is consequently locally constant, and on a connected component it is constant, since a value's level set and its complement are both relatively open. Evaluating at any fixed point of $C$ gives the $C^{2}$ regularity of $c_{C}$ . Conversely, independent such functions on the components give a $C^{2}$ potential with unchanged spatial gradient; components of an open Euclidean set are open, because each sufficiently small ball about one of their points is connected and lies in that component. A single $c (t)$ describes every ambiguity precisely when the component functions agree, in particular when $Q$ is connected. No agreement follows from openness alone: on two disjoint balls the potentials $0$ and the function equal to $0$ on one ball and $1$ on the other have the same force. Every continuous motion on an interval remains in one component, so its energy shifts by that component's $c_{C} (t)$ and its energy rate by $c_{C}^{'} (t)$ . The representative is part of the model's declaration, and every energy statement names it.

(d) Closed and exact force fields. A time- and velocity-independent configuration-space force field is a $C^{1}$ map $φ = (φ_{1}, \dots, φ_{N}) : Q \to (R^{3})^{N}$ , where $Q \subseteq R^{3 N}$ is the open configuration domain and each $φ_{i} (x)$ is the force on label $i$ at configuration $x$ ; its components have force dimension. It represents a force map of the form $F_{i} (t, x, v) = φ_{i} (x)$ on the model's state domain. This configuration-space field is not in general an external single-label force: to identify its $i$ th component with an external map of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clause (i), one must further require that it depend on $x_{i}$ alone, as in the declared signature $F_{i}^{e x t} (t, x_{i}, v_{i})$ . Such a field is closed when $\partial_{k} φ_{j} = \partial_{j} φ_{k}$ for all coordinate indices $j, k \in {0, \dots, 3 N - 1}$ at every point of $Q$ , using the flattened coordinate indices of Exact and closed C1 vector fields, and exact with potential energy $U$ when $φ = - \nabla U$ for a $C^{2}$ function $U : Q \to R$ ; equivalently, the field $- φ$ is exact in the sense of Exact and closed C1 vector fields with potential $U$ . The following mathematical facts are used, with their proofs at the cited items:

every exact field is closed, by equality of mixed partial derivatives (Clairaut--Schwarz theorem for continuous second partial derivatives, applied to the $C^{2}$ potential);
on a star-shaped open domain every closed $C^{1}$ field is exact, by the Poincaré lemma (Poincare's lemma on a star-shaped domain: every closed C1 field is exact);
beyond item 2 this item asserts no implication from closedness to global exactness: on a general open domain a closed field need not have a potential energy, and the criterion used by the framework is exactness as defined above.

Consequently the existence of a global potential energy on the model's configuration domain is a property of the field on that domain, and a model whose potential part is a merely closed field on a non-star-shaped domain is not thereby a global potential model. The distinction is sharp: the companion examples page carries a model on an annular domain whose force field is smooth and closed, has nonzero circulation around a closed loop, and therefore admits no global potential and no global conserved energy of the form $K + U$ , while item 2 provides a local potential and a locally conserved energy on every star-shaped subpart of that domain.

(e) Local potentials. Let $V \subseteq Q$ be open and star-shaped with respect to one of its points (Star-shaped open subsets of Euclidean space) and let $φ$ be closed on $V$ . Then item 2 of (d) provides a $C^{2}$ potential $U_{V} : V \to R$ with $φ ∣_{V} = - \nabla U_{V}$ : a local potential of the field on $V$ . On the restricted configuration domain $V$ the model with force $- \nabla U_{V}$ is a global potential model, so the mechanical energy $K + U_{V}$ is available along curve families in $V$ ; this is how a potential is recovered from a merely closed field on a sufficiently small star-shaped piece of the domain, with no claim of a global representative.

(f) Equipotential level sets. For this clause assume additionally that $U : Q \to R$ is smooth, meaning $C^{k}$ for every $k \in N$ ( $C^{k}$ Euclidean maps and diffeomorphisms); the potential presentations of clause (a) still require only $C^{2}$ . Let $c \in R$ and suppose $U^{- 1} (c) \neq \emptyset$ and $\nabla U (x) \neq 0$ for every $x \in U^{- 1} (c)$ ; equivalently $c$ is a regular value of $U$ and every point of the level set is a regular point (Regular and critical points and values). Then $U^{- 1} (c)$ is an embedded submanifold of $Q$ of dimension $3 N - 1$ (A regular level set is an embedded submanifold, Embedded submanifolds and slice charts). No such conclusion is made at a level where $\nabla U$ vanishes somewhere on the level set, where the level set may have singular points.

(g) Status. The potential, its normalization, the residual forces and the declaration of a potential presentation are model data; the closed/exact statements of (d)-(f) are cited mathematics about the declared field. This item asserts no conservation law, no force law of nature and no measured value.

Remarks

The potential is not the force. Only the spatial gradient of $U$ enters the force map; changing $U$ by $c (t)$ changes no acceleration and no trajectory, but it changes the numerical value of $E$ and its balance. This is why every energy statement records its representative.
Residual forces are ordinary model data. In a presentation with $d \neq 0$ the split into potential part and residual forces is a modelling declaration; two presentations with the same total force and different splits produce the same motions and the same total power, while the potential-energy balance distributes that power differently between $\partial_{t} U$ and the residual power (Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form).
The obstruction is not smoothness. Fact 3 of (d) leaves closedness without a global conclusion on general domains; on a star-shaped subdomain the radial integral of the Poincaré lemma supplies the potential, and its value depends on the chosen centre. The companion counterexample records a smooth closed model field whose obstruction is its circulation around the removed axis, not any failure of regularity.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form

Physical scope

The mechanical energy balance of a model presented through a potential with declared residual forces: the derivative of the mechanical energy equals the partial time derivative of the potential plus the power of the residual forces; a time-independent potential with no residual forces conserves the mechanical energy along every motion; and the integrated form expresses the work of the total force through the change of the potential and the residual work. The result is a conditional deduction from the adopted potential presentation, with the normalization of the potential carried along; it asserts no measured value and no conservation outside the stated hypotheses.

Statement

Adopt the force model, one inertial frame, and the potential presentation of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (a): a $C^{2}$ potential $U : I_{0} \times Q ⟶ R$ of dimension $μ ℓ^{2} τ^{- 2}$ and residual forces $d_{i}$ of the standing regularity, with $F_{i} (t, x, v) = - \nabla_{x_{i}} U (t, x) + d_{i} (t, x, v) (i = 1, \dots, N)$ on the state domain. Let $x : J \to Q$ be a motion of the model on an interval $J$ with nonempty interior, write $d_{i} (t) : = d_{i} (t, x (t), w (t))$ for the evaluated residual forces and $E (t) : = K (t) + U (t, x (t))$ for the mechanical energy of the definition (clause (b)). Then:

$E$ is differentiable on $J$ and $E^{'} (t) = \partial_{t} U (t, x (t)) + \sum_{i} ⟨ d_{i} (t), w_{i} (t) ⟩ (t \in J) :$ the mechanical energy changes at the rate of the explicit time dependence of the potential plus the power of the residual forces;
if $\partial_{t} U \equiv 0$ on $I_{0} \times Q$ and each evaluated residual force $d_{i}$ vanishes identically along this motion, then $E$ is constant on $J$ . If, more strongly, every residual-force map $d_{i}$ vanishes on its entire state domain, the model is a global potential model with the declared representative $U$ ;
for all $t_{0} < t_{1}$ in $J$ , $K (t_{1}) - K (t_{0}) = U (t_{0}, x (t_{0})) - U (t_{1}, x (t_{1})) + \int_{t_{0}}^{t_{1}} [\partial_{t} U (t, x (t)) + \sum_{i} ⟨ d_{i} (t), w_{i} (t) ⟩] d t,$ equivalently $W ([t_{0}, t_{1}]) = U (t_{0}, x (t_{0})) - U (t_{1}, x (t_{1})) + \int_{t_{0}}^{t_{1}} [\partial_{t} U + \sum_{i} ⟨ d_{i}, w_{i} ⟩] d t$ : the work of the total force is the decrease of the potential energy plus the work of the explicit time dependence and of the residual forces.

The representative $U$ is the one declared by the model; replacing $U$ by $U + c (t)$ changes $E$ by $c (t)$ and $E^{'}$ by $c^{'} (t)$ , as recorded in Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c). The statement is read in the fixed frame; no empirical content is claimed.

Facts & Assumptions

Given: The potential presentation of the model, one inertial frame, a motion $x : J \to Q$ with velocities $w_{i} = x_{i}^{'}$ , evaluated residual forces $d_{i}$ , and the mechanical energy $E = K + U (t, x (t))$ .

[L1]

Notation (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clauses (a)-(b), Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame): $K$ , $W$ and $P$ are the kinetic energy, work and power of the definition; $Q$ is nonempty and open; $U$ is $C^{2}$ hence $C^{1}$ ; $t \mapsto U (t, x (t))$ is defined on $J$ ; the equation of motion is $m_{i} {\ddot{x}}_{i} = F_{i} (t, x, w)$ with $F_{i} = - \nabla_{x_{i}} U + d_{i}$ , so that $m_{i} {\ddot{x}}_{i} (t) = - \nabla_{x_{i}} U (t, x (t)) + d_{i} (t)$ with $d_{i} (t) = d_{i} (t, x (t), w (t))$ .

[L2]

Mathematical supplier (Work-energy balance $\frac{d}{d t} (K + U (t, x (t))) = \partial_{t} U (t, x (t)) + \sum_{i} ⟨ d_{i} (t), x_{i}^{'} (t) ⟩$ for forces $m_{i} x_{i}^{''} = - \nabla_{x_{i}} U + d_{i}$ ): under the displayed hypotheses with $U$ of class $C^{1}$ on the open $I_{0} \times Q$ and maps $d_{i}$ along the curve, if $m_{i} x_{i}^{''} (t) = - \nabla_{x_{i}} U (t, x (t)) + d_{i} (t)$ on $I$ , then $K^{'} = \sum_{i} ⟨ m_{i} x_{i}^{''}, x_{i}^{'} ⟩$ ; $\frac{d}{d t} U (t, x (t)) = \partial_{t} U (t, x (t)) + \sum_{i} ⟨ \nabla_{x_{i}} U (t, x (t)), x_{i}^{'} (t) ⟩$ ; $\frac{d}{d t} (K + U (t, x (t))) = \partial_{t} U (t, x (t)) + \sum_{i} ⟨ d_{i} (t), x_{i}^{'} (t) ⟩$ ; and if $d_{i} \equiv 0$ for all $i$ and $\partial_{t} U \equiv 0$ then $t \mapsto K + U (t, x (t))$ is constant on $I$ .

[L3]

Work-energy theorem (Power of the total force and the work-energy theorem: $K^{'} = P$ and $K (t_{1}) - K (t_{0}) = W ([t_{0}, t_{1}])$ along a motion): $K (t_{1}) - K (t_{0}) = W ([t_{0}, t_{1}]) = \int_{t_{0}}^{t_{1}} P$ for $t_{0} < t_{1}$ in $J$ , and $P = \sum_{i} ⟨ F_{i} (t, x, w), w_{i} ⟩$ . The Euclidean inner product is bilinear (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ) and finite sums are linear (Laws of finite sums and finite products).

[L4]

Newton–Leibniz (Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative): if $G$ is continuous on $[a, b]$ and differentiable on $(a, b)$ with integrable derivative $G^{'}$ , then $\int_{a}^{b} G^{'} = G (b) - G (a)$ ; a real function on an interval with zero derivative at every interior point is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant); the interval $J$ has nonempty interior (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

Proof

technique · direct; instantiate the mathematical work-energy balance for the potential presentation and integrate the identity to obtain the work form

1.1givenL1L2algebra

The hypotheses of [L2] hold: $I : = J$ is an interval with nonempty interior, $Q$ is nonempty and open, $U$ is $C^{1}$ on $I_{0} \times Q$ (indeed $C^{2}$ ), $x$ is twice differentiable, and the equation of motion together with the presentation gives $m_{i} {\ddot{x}}_{i} (t) = - \nabla_{x_{i}} U (t, x (t)) + d_{i} (t)$ with the evaluated residual forces of the statement. Applying [L2] gives that $E = K + U (t, x (t))$ is differentiable on $J$ with $E^{'} (t) = \partial_{t} U (t, x (t)) + \sum_{i} ⟨ d_{i} (t), w_{i} (t) ⟩,$ which is clause 1.

2.1step 1.1L2

Clause 2: if $\partial_{t} U \equiv 0$ and $d_{i} \equiv 0$ , then the right side of step 1.1 vanishes at every $t \in J$ ; the final clause of [L2] gives directly that $E$ is constant on $J$ .

2.2step 1.1L1L2L3algebra

Clause 3: fix $t_{0} < t_{1}$ in $J$ . By [L3], $K (t_{1}) - K (t_{0}) = \int_{t_{0}}^{t_{1}} P$ , and on $J$ $P = \sum_{i} ⟨ - \nabla_{x_{i}} U (t, x (t)) + d_{i} (t), w_{i} (t) ⟩ = - \sum_{i} ⟨ \nabla_{x_{i}} U (t, x (t)), w_{i} (t) ⟩ + \sum_{i} ⟨ d_{i} (t), w_{i} (t) ⟩ .$ By the chain-rule clause of [L2], $\sum_{i} ⟨ \nabla_{x_{i}} U (t, x (t)), w_{i} (t) ⟩ = \frac{d}{d t} U (t, x (t)) - \partial_{t} U (t, x (t))$ , so $P = \partial_{t} U (t, x (t)) + \sum_{i} ⟨ d_{i} (t), w_{i} (t) ⟩ - \frac{d}{d t} U (t, x (t)) .$

3.1step 2.2L1L3L4algebra∎

The function $t \mapsto U (t, x (t))$ is $C^{1}$ on $J$ (composition of the $C^{1}$ map $U$ with the $C^{1}$ curve $t \mapsto (t, x (t))$ ), so [L4] gives $\int_{t_{0}}^{t_{1}} \frac{d}{d t} U (t, x (t)) d t = U (t_{1}, x (t_{1})) - U (t_{0}, x (t_{0}))$ . Integrating the identity of step 2.2 over $[t_{0}, t_{1}]$ and substituting this value, $K (t_{1}) - K (t_{0}) = \int_{t_{0}}^{t_{1}} P = U (t_{0}, x (t_{0})) - U (t_{1}, x (t_{1})) + \int_{t_{0}}^{t_{1}} [\partial_{t} U (t, x (t)) + \sum_{i} ⟨ d_{i} (t), w_{i} (t) ⟩] d t,$ which is the first form of clause 3; the second form is the definition of $W ([t_{0}, t_{1}])$ by [L1] and [L3].

Remarks

Three distinct contributions. The balance separates the explicit time dependence of the potential ( $\partial_{t} U$ ), the power of the residual forces, and the potential difference; a model with a time-dependent potential or with residual forces is driven, and its mechanical energy is not conserved in general. The companion worked example computes all three terms on explicit data.
Conservation is conditional and representation-dependent. Clause 2 requires the representative $U$ to be time-independent and the residual forces to vanish; a potential $U + c (t)$ with time-dependent $c$ turns a conserved model into a presentation with $E^{'} = c^{'} (t)$ , as clause (c) of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets records. Energy statements therefore name the representative.
Frame dependence of conservation. Under the scalar transfer of Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy clause 5, with its boost velocity $v$ , $\tilde{E} = E + ⟨ v, R P ⟩ + \frac{1}{2} M ∥ v ∥_{2}^{2}$ . Thus if both $E$ and $P$ are constant, then $\tilde{E}$ is constant. If $P$ varies, constancy of $E$ alone does not imply constancy of $\tilde{E}$ ; the transferred potential can also have explicit time dependence.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy

Physical scope

The transformation laws of the model's mechanical quantities under a change of inertial frame: the centre of mass and centre-of-mass velocity transform affinely, the total momentum gains M times the boost velocity, the kinetic energy gains a cross term and the boost kinetic energy, the angular momentum and torque about corresponding C1 reference curves acquire the stated rotation and boost terms, and power and work transform according to the transferred force map of the force postulate, with the momentum-increment work form requiring a motion with the antisymmetric pair presentation; for a globally potential model whose potential is transferred by the scalar rule, the mechanical energy gains the same kinetic boost terms. The result is a conditional deduction from the adopted frame, worldline and force laws; it asserts no invariance of these quantities and no measured value.

Statement

Adopt the force model and the notation of Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, and let $F = (o, u, (e_{1}, e_{2}, e_{3})), F^{'} = (o^{'}, u^{'}, (e_{1}^{'}, e_{2}^{'}, e_{3}^{'}))$ be two inertial frames with coordinates $(t, x)$ and $(t^{'}, x^{'})$ related by $t^{'} = t + b, x^{'} = R x + v t + a_{0}, b \in R, a_{0}, v \in R^{3}, R \in S O (3)$ (Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes clause 1, The orthogonal group $O (3)$ and the rotation group $S O (3)$ ). Let $x : J \to Q$ be a curve family of the model in the first frame and let $η_{i} (t^{'}) : = R x_{i} (t^{'} - b) + v (t^{'} - b) + a_{0} (t^{'} \in J + b)$ be the description of the same worldlines in the second frame. Write $σ_{i} (t) : = F_{i} (t, x (t), w (t))$ and $Σ (t) : = \sum_{i} σ_{i} (t)$ for the evaluated total forces in the first frame, and read the second frame's quantities from the description $η$ with the transferred force map $F_{*}$ of clause (vi) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: ${\tilde{σ}}_{i} (t^{'}) : = F_{* i} (t^{'}, η (t^{'}), η^{'} (t^{'}))$ , $\tilde{P} : = \sum_{i} m_{i} η_{i}^{'}$ , $\tilde{X} : = M^{- 1} \sum_{i} m_{i} η_{i}$ , $\tilde{K} : = \frac{1}{2} \sum_{i} m_{i} ∥ η_{i}^{'} ∥_{2}^{2}$ , ${\tilde{J}}_{\tilde{o}} : = \sum_{i} (η_{i} - \tilde{o}) \times m_{i} η_{i}^{'}$ and ${\tilde{T}}_{\tilde{o}} : = \sum_{i} (η_{i} - \tilde{o}) \times {\tilde{σ}}_{i}$ for any $C^{1}$ reference curve $\tilde{o} : J + b \to R^{3}$ (clause 3 uses the curve corresponding to $o$ ), $\tilde{P} : = \sum_{i} ⟨ {\tilde{σ}}_{i}, η_{i}^{'} ⟩$ and $\tilde{W} ([t_{0}^{'}, t_{1}^{'}]) : = \int_{t_{0}^{'}}^{t_{1}^{'}} \tilde{P} (t^{'}) d t^{'}$ for $t_{0}^{'} \leq t_{1}^{'}$ in $J + b$ . Then, with all first-frame quantities read at the corresponding time $t^{'} - b$ :

Centre of mass and momentum. $\tilde{X} (t^{'}) = R X (t^{'} - b) + v (t^{'} - b) + a_{0}, {\tilde{X}}^{'} (t^{'}) = R V (t^{'} - b) + v, \tilde{P} (t^{'}) = R P (t^{'} - b) + M v;$
Kinetic energy. $\tilde{K} (t^{'}) = K (t^{'} - b) + ⟨ v, R P (t^{'} - b) ⟩ + \frac{1}{2} M ∥ v ∥_{2}^{2};$
Angular momentum and torque. For every $C^{1}$ reference curve $o : J \to R^{3}$ of the definition, let $\tilde{o} (t^{'}) : = R o (t^{'} - b) + v (t^{'} - b) + a_{0}$ and $X_{o} (t) : = M^{- 1} \sum_{i} m_{i} (x_{i} (t) - o (t)) = X (t) - o (t)$ . Then ${\tilde{J}}_{\tilde{o}} (t^{'}) = R J_{o} (t^{'} - b) + (R X_{o} (t^{'} - b)) \times (M v), {\tilde{T}}_{\tilde{o}} (t^{'}) = R T_{o} (t^{'} - b),$ where $T_{o} (t) : = \sum_{i} (x_{i} (t) - o (t)) \times σ_{i} (t)$ is the evaluated total torque of the first frame about $o (t)$ ;
Power and work. $\tilde{P} (t^{'}) = P (t^{'} - b) + ⟨ R Σ (t^{'} - b), v ⟩,$ and for every curve family and $t_{0} < t_{1}$ in $J$ $\tilde{W} ([t_{0} + b, t_{1} + b]) = W ([t_{0}, t_{1}]) + \int_{t_{0}}^{t_{1}} ⟨ R Σ (t), v ⟩ d t .$ If in addition $x$ is a motion and the model invokes the pair presentation of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch with the antisymmetry clause, then $Σ = P^{'}$ along the motion by Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces, and $\tilde{W} ([t_{0} + b, t_{1} + b]) = W ([t_{0}, t_{1}]) + ⟨ R (P (t_{1}) - P (t_{0})), v ⟩;$
Potential and mechanical energy. If the model is a global potential model of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (a) with potential $U : I_{0} \times Q \to R$ of class $C^{2}$ and the second frame's model declares the scalar transfer $U_{*} (t^{'}, y) : = U (t^{'} - b, R^{T} (y_{1} - v (t^{'} - b) - a_{0}), \dots, R^{T} (y_{N} - v (t^{'} - b) - a_{0})),$ then $U_{*}$ is $C^{2}$ on its open transferred domain, $F_{* i} (t^{'}, y, u) = - \nabla_{y_{i}} U_{*} (t^{'}, y)$ at corresponding data, and the mechanical energies of clause (b) of that definition satisfy $\tilde{E} (t^{'}) : = \tilde{K} (t^{'}) + U_{*} (t^{'}, η (t^{'})) = E (t^{'} - b) + ⟨ v, R P (t^{'} - b) ⟩ + \frac{1}{2} M ∥ v ∥_{2}^{2} .$

Whether the second frame's model uses the scalar transfer of clause 5 is a modelling declaration; a model whose potential is not transferred by this rule (for instance a background potential that selects a frame) is a different model in the second frame, and no transformation law of $E$ is asserted for it. No empirical content is claimed.

Facts & Assumptions

Given: The force model and one curve family $x : J \to Q$ in the first inertial frame, a second inertial frame with the transformation parameters $(b, a_{0}, v, R)$ , and the description $η$ of the worldlines in the second frame; for clause 3, also a $C^{1}$ reference curve $o : J \to R^{3}$ .

[L1]

Kinematics (Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes clause 2): each $η_{i}$ is of class $C^{2}$ and $η_{i}^{'} (t^{'}) = R w_{i} (t^{'} - b) + v, η_{i}^{''} (t^{'}) = R {\ddot{x}}_{i} (t^{'} - b) (t^{'} \in J + b) .$

[L2]

Notions (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame): $w_{i} = x_{i}^{'}$ , $p_{i} = m_{i} w_{i}$ , $P = \sum_{i} p_{i}$ , $X = M^{- 1} \sum_{i} m_{i} x_{i}$ , $V = X^{'}$ , $K = \frac{1}{2} \sum_{i} m_{i} ∥ w_{i} ∥_{2}^{2}$ , $P = \sum_{i} ⟨ σ_{i}, w_{i} ⟩$ , $J_{o} = \sum_{i} (x_{i} - o (t)) \times p_{i}$ and $T_{o} = \sum_{i} (x_{i} - o (t)) \times σ_{i}$ for a $C^{1}$ reference curve $o$ , and the work $W$ is the integral of $P$ ; the corresponding tilde-quantities of the statement are the same construction in the second frame from $η$ and ${\tilde{σ}}_{i} = F_{* i} (t^{'}, η, η^{'})$ . For a motion, $m_{i} {\ddot{x}}_{i} = σ_{i}$ and the transferred description satisfies $m_{i} η_{i}^{''} = {\tilde{σ}}_{i}$ , with ${\tilde{σ}}_{i} (t^{'}) = R σ_{i} (t^{'} - b)$ at corresponding data (clause (vi) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, which defines $F_{*}$ by $F_{*} (t^{'}, y, u) = R_{N} F (t^{'} - b, R_{N}^{T} (y - \overset{ˉ}{v} (t^{'} - b) - {\overset{ˉ}{a}}_{0}), R_{N}^{T} (u - \overset{ˉ}{v}))$ , where $R_{N} : = diag (R, \dots, R)$ and $\overset{ˉ}{v} : = (v, \dots, v)$ , ${\overset{ˉ}{a}}_{0} : = (a_{0}, \dots, a_{0})$ are the repeated $N$ -tuples; for a global potential model $σ_{i} = - \nabla_{x_{i}} U$ and $E = K + U (t, x (t))$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clauses (a)-(b)).

[L3]

Orthogonality consequences: for $R \in S O (3)$ one has $R^{T} R = I_{3}$ ( $S O (3)$ is a subgroup of the general linear group, with $R^{- 1} = R^{T}$ ), hence $⟨ R u, R z ⟩ = u^{T} R^{T} R z = u^{T} z = ⟨ u, z ⟩, ∥ R u ∥_{2} = ∥ u ∥_{2}$ for all $u, z \in R^{3}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, The transpose $A^{T}$ of a matrix); moreover $R (u \times z) = (R u) \times (R z),$ because $R [u]_{\times} z = R [u]_{\times} R^{T} R z = [R u]_{\times} R z$ by clause 3 and clause 1 of The skew-matrix/cross-product correspondence $[ω]_{\times} z = ω \times z$ and its conjugation law $R [ω]_{\times} R^{T} = [R ω]_{\times}$ together with $R^{T} R = I_{3}$ (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors).

[L4]

Chain rule and gradients (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, A total derivative computes every directional derivative, and its matrix is the Jacobian, If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, $C^{k}$ Euclidean maps and diffeomorphisms, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case): the maps of the statement are compositions of $C^{1}$ maps with the affine transfer $(t^{'}, y) \mapsto (t^{'} - b, R_{N}^{T} (y - \overset{ˉ}{v} (t^{'} - b) - {\overset{ˉ}{a}}_{0}))$ , whose Jacobian in $y$ is the orthogonal block matrix $R_{N}^{T}$ ; finite sums, scalar multiples and integrals of differentiable curves behave as in The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Laws of finite sums and finite products and Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative.

[L5]

Balance supplier (Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces): for a motion of a model that invokes the external-plus-pair presentation with the antisymmetry clause, $\dot{P} = \sum_{i} F_{i}^{e x t}$ , and also $\sum_{i} F_{i} = \sum_{i} F_{i}^{e x t}$ because the internal pairs cancel.

Proof

technique · direct; compute each quantity of the second frame from the kinematic laws and the orthogonality of $R$, and integrate the power identity for the work law

1.1givenL1L2L4algebra

Centre of mass. By the definition of $\tilde{X}$ and the kinematic law, $\tilde{X} (t^{'}) = M^{- 1} \sum_{i} m_{i} (R x_{i} (t^{'} - b) + v (t^{'} - b) + a_{0}) = R (M^{- 1} \sum_{i} m_{i} x_{i} (t^{'} - b)) + v (t^{'} - b) + a_{0} = R X (t^{'} - b) + v (t^{'} - b) + a_{0},$ where linearity of $R$ and the finite sum are used ([L2], [L4]); its $t^{'}$ -derivative is $R V (t^{'} - b) + v$ by the chain rule applied to $t^{'} \mapsto X (t^{'} - b)$ with $V = X^{'}$ .

1.2L3

Orthogonality. For $R \in S O (3)$ and $u, z \in R^{3}$ , $⟨ R u, R z ⟩ = ⟨ u, z ⟩$ and $∥ R u ∥_{2} = ∥ u ∥_{2}$ because $⟨ R u, R z ⟩ = (R u)^{T} (R z) = u^{T} R^{T} R z = u^{T} z$ with $R^{T} R = I_{3}$ ; and $R (u \times z) = (R u) \times (R z)$ because $R (u \times z) = R [u]_{\times} z = R [u]_{\times} R^{T} R z = [R u]_{\times} (R z)$ by the skew-matrix correspondence and $R^{T} R = I_{3}$ . These two consequences are used freely below.

1.3givenL2L4algebra

Potential transfer. Let the model be a global potential model with $σ_{i} = - \nabla_{x_{i}} U$ and let $U_{*}$ be the scalar transfer of the statement. $U_{*}$ is $C^{2}$ , being the composition of the $C^{2}$ map $U$ with the affine transfer $y \mapsto R_{N}^{T} (y - \overset{ˉ}{v} (t^{'} - b) - {\overset{ˉ}{a}}_{0})$ and the time shift ([L4]). Its spatial gradient is computed by the chain rule: at corresponding data $(t^{'}, y)$ with $x = R_{N}^{T} (y - \overset{ˉ}{v} (t^{'} - b) - {\overset{ˉ}{a}}_{0})$ , $\nabla_{y_{i}} U_{*} (t^{'}, y) = R \nabla_{x_{i}} U (t, x),$ because the Jacobian of $y_{i} \mapsto R^{T} (y_{i} - v (t^{'} - b) - a_{0})$ is $R^{T}$ and $(R^{T})^{T} = R$ (The transpose $A^{T}$ of a matrix, A total derivative computes every directional derivative, and its matrix is the Jacobian). Hence $- \nabla_{y_{i}} U_{*} (t^{'}, η (t^{'})) = R (- \nabla_{x_{i}} U (t, x (t))) = R σ_{i} (t) = F_{* i} (t^{'}, η (t^{'}), η^{'} (t^{'}))$ at corresponding times, so the transferred model presents its force map through $U_{*}$ .

2.1step 1.1step 1.2L1L2algebra

Momentum and kinetic energy. By [L1], $\tilde{P} = \sum_{i} m_{i} (R w_{i} + v) = R \sum_{i} m_{i} w_{i} + M v = R P (t^{'} - b) + M v .$ For the kinetic energy, expand with bilinearity of the inner product and step 1.2: $∥ η_{i}^{'} ∥_{2}^{2} = ∥ R w_{i} + v ∥_{2}^{2} = ∥ w_{i} ∥_{2}^{2} + 2 ⟨ R w_{i}, v ⟩ + ∥ v ∥_{2}^{2},$ so, summing with the masses, $\tilde{K} = \frac{1}{2} \sum_{i} m_{i} ∥ w_{i} ∥_{2}^{2} + ⟨ R \sum_{i} m_{i} w_{i}, v ⟩ + \frac{1}{2} M ∥ v ∥_{2}^{2} = K (t^{'} - b) + ⟨ R P (t^{'} - b), v ⟩ + \frac{1}{2} M ∥ v ∥_{2}^{2},$ which with step 1.1 is clauses 1 and 2.

2.2step 1.2L1L2algebra

Angular momentum and torque. At corresponding times $t = t^{'} - b$ , the definitions give $η_{i} (t^{'}) - \tilde{o} (t^{'}) = R (x_{i} (t) - o (t))$ . By the cross-product identity of step 1.2, ${\tilde{J}}_{\tilde{o}} = \sum_{i} (R (x_{i} - o)) \times m_{i} (R w_{i} + v) = \sum_{i} m_{i} R ((x_{i} - o) \times w_{i}) + \sum_{i} (R (x_{i} - o)) \times m_{i} v .$ The first sum is $R \sum_{i} (x_{i} - o) \times m_{i} w_{i} = R J_{o} (t^{'} - b)$ ; in the second, $\sum_{i} (R (x_{i} - o)) \times m_{i} v = R [\sum_{i} m_{i} (x_{i} - o) \times R^{T} v] = R [(M X_{o}) \times R^{T} v] = (R M X_{o}) \times v = (R X_{o}) \times (M v),$ using $v = R (R^{T} v)$ , the cross-product identity, the definition of $X_{o}$ and bilinearity. Hence ${\tilde{J}}_{\tilde{o}} = R J_{o} + (R X_{o}) \times (M v)$ . For the torque, the transfer rule gives ${\tilde{σ}}_{i} (t^{'}) = R σ_{i} (t^{'} - b)$ at corresponding data, so ${\tilde{T}}_{\tilde{o}} = \sum_{i} (R (x_{i} - o)) \times R σ_{i} = \sum_{i} R ((x_{i} - o) \times σ_{i}) = R T_{o} (t^{'} - b),$ again by step 1.2; this is clause 3.

2.3step 1.2L1L2algebra

Power. By the transfer rule and [L1], $\tilde{P} = \sum_{i} ⟨ R σ_{i}, R w_{i} + v ⟩ = \sum_{i} ⟨ σ_{i}, w_{i} ⟩ + ⟨ R \sum_{i} σ_{i}, v ⟩ = P (t^{'} - b) + ⟨ R Σ (t^{'} - b), v ⟩,$ using step 1.2 in the first equality of the middle expression.

3.1step 2.3L2L4L5algebra

Work. The force map is continuous and the curve family is $C^{2}$ , so $P$ and $⟨ R Σ, v ⟩$ are continuous on $J$ and these Riemann integrals exist. Integrating the power identity of step 2.3 over $[t_{0} + b, t_{1} + b]$ and substituting $t = t^{'} - b$ gives $\tilde{W} ([t_{0} + b, t_{1} + b]) = \int_{t_{0}}^{t_{1}} P (t) d t + \int_{t_{0}}^{t_{1}} ⟨ R Σ (t), v ⟩ d t = W ([t_{0}, t_{1}]) + \int_{t_{0}}^{t_{1}} ⟨ R Σ (t), v ⟩ d t,$ the general work law in clause 4. If $x$ is a motion of a model with the antisymmetric pair presentation, then [L5] and Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces give $Σ = P^{'}$ on $J$ ; since $P$ is $C^{1}$ by that balance theorem, the Newton–Leibniz theorem applied componentwise yields $\int_{t_{0}}^{t_{1}} ⟨ R Σ (t), v ⟩ d t = ⟨ R (P (t_{1}) - P (t_{0})), v ⟩,$ which is the conditional specialization in clause 4.

4.1step 2.1step 1.3L2algebra∎

Mechanical energy. With the transfer of step 1.3, $U_{*} (t^{'}, η (t^{'})) = U (t^{'} - b, x (t^{'} - b)),$ by the definitions of $U_{*}$ and $η$ ; combining with clause 2 of step 2.1, $\tilde{E} = \tilde{K} + U_{*} (t^{'}, η (t^{'})) = K (t^{'} - b) + ⟨ R P (t^{'} - b), v ⟩ + \frac{1}{2} M ∥ v ∥_{2}^{2} + U (t^{'} - b, x (t^{'} - b)) = E (t^{'} - b) + ⟨ v, R P (t^{'} - b) ⟩ + \frac{1}{2} M ∥ v ∥_{2}^{2},$ which is clause 5.

Remarks

Conservation under a frame change has explicit conditions. If $P$ is constant, then $\tilde{P} = R P + M v$ is constant; if both $P$ and $K$ are constant, then the kinetic-energy law makes $\tilde{K}$ constant. Under the scalar potential transfer, constancy of both $E$ and $P$ makes $\tilde{E}$ constant. The angular-momentum law includes the time-dependent term $(R X_{o} (t^{'} - b)) \times (M v)$ , so constancy of $J_{o}$ alone does not generally make ${\tilde{J}}_{\tilde{o}}$ constant; it does so exactly when that added term is constant.
The transferred model is a declaration. Clause (vi) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance fixes how a force law is read in another inertial frame; clause 5 adds the analogous scalar transfer for a potential and is asserted only when the model declares it. A background potential that does not transfer this way selects a frame, consistent with Status audit and model boundaries of the force page: open systems, frame selection, breakdown and empirical scope.

RemarkRemark: AI-adaptedProof: Not applicableOpen item page →

Status audit and scope of the balance page: conditional conservation, open systems, frames, potential normalization and empirical boundaries

Physical scope

A scope and status audit of the balance page: which objects are defined, which statements are conditional deductions from the force and interaction posts, which hypotheses the stated conservation conclusions use, and which claims are deliberately not made; the frame dependence of the conserved quantities, the open-system boundary, the normalization of a potential, the local-versus-global potential distinction and the empirical status of model parameters. Bookkeeping and scope only; it asserts no law, no measurement and no conservation beyond the items it cites.

Status audit of this page

This remark records the status of the notions of the balance page. It is bookkeeping: it asserts no law and no empirical claim, and every mathematical statement it repeats is proved in the item cited.

Adopted assumptions. The force branch is the postulate Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: the force map with its standing continuity and state-local-Lipschitz hypotheses, the equation of motion $m_{i} {\ddot{x}}_{i} = F_{i} (t, x, \dot{x})$ in every inertial frame, the additive composition of declared force contributions and the transformation rule of a force law under a change of inertial frame. The restricted interaction models are the postulate Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch: the external-plus-pair presentation, the optional antisymmetry and centrality clauses, the potential branch and the gravitational branch. The linear drag model is the postulate The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient.

Defined objects. The total mass, total momentum, centre of mass, relative positions, angular momentum about a reference curve, total and external torques, internal torque sum, kinetic energy, power and work are defined in Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame; potential presentations, potential and mechanical energy, the closed/exact distinction for time- and velocity-independent configuration-space force fields and equipotential level sets are defined in Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets. Motions and maximal motions are The associated first-order system, motions, maximal motions and the Newtonian flow of a force model.

Conditional deductions. The splitting of the kinetic sum and $P = M V$ are 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). The momentum balance and the centre-of-mass equation, under the antisymmetric external-plus-pair presentation, are Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces. The angular-momentum balance about fixed and moving reference points, with the centrality case, is Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case. The power identity and the work-energy theorem are Power of the total force and the work-energy theorem: $K^{'} = P$ and $K (t_{1}) - K (t_{0}) = W ([t_{0}, t_{1}])$ along a motion. The mechanical energy balance and conservation for a static potential without residual forces are Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form. The transformation laws of all these quantities under a change of inertial frame are Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy.

Computed examples. The companion page checks the laws on explicit data: the two-body Hooke model with an exact circular relative motion (momentum, angular momentum including the moving-origin correction, kinetic splitting, energy conservation and the kinetic boost law), a driven one-dimensional model with a time-dependent potential and a residual force (the balance, the integrated work form and the normalization ambiguity), and the two counterexamples recorded below.

Deliberately not claimed on this page. The conservation conclusions stated here are conditional: momentum is conserved under the antisymmetric pair presentation and vanishing total external force; angular momentum is conserved about a fixed reference point when the total torque vanishes, with centrality only one sufficient condition (the companion counterexample exhibits non-central pair forces whose torque sum cancels identically). For mechanical energy, the exact rate is the right-hand side of clause 1 of Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form, so it is constant along a motion whenever that combined rate vanishes there. A time-independent potential representative with residual forces vanishing along the motion is a sufficient case; the stronger condition that the residual-force maps vanish throughout their state domains makes the whole model a global potential model. Neither condition is claimed as necessary for every motion on which the energy happens to be constant. No theorem of this page asserts that any interaction of nature has the required structure. No constraint reactions, no impulsive forces, no variable-mass models and no measurements appear.

Hypotheses and counterexamples

The hypotheses are used, not decorative. Antisymmetry is exactly what cancels the internal forces in Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces; centrality is a sufficient condition in clause 3 of Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case, and a model with non-central pairs can still conserve angular momentum when the pair torques cancel. The companion counterexample makes the two situations concrete.
Closed is not exact. The potential branch requires a global potential; a merely closed field on a non-star-shaped domain need not have one, and the companion counterexample exhibits this on an annular domain excluding an axis, with nonzero circulation around a loop. On every star-shaped subdomain a local potential and a local mechanical energy exist (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (e)).
Energy can be removed by residual forces. A residual force such as the linear drag term of The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient has power $⟨ - c v, v ⟩ = - c ∥ v ∥_{2}^{2} \leq 0$ along the model's motions, by bilinearity of the Euclidean inner product (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ); the balance Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form then reports a strictly negative residual contribution whenever $v \neq 0$ and $c > 0$ . This is a statement about the declared model, not about nature.

Frames, open systems and empirical boundaries

Conservation is frame-relative. Under a change of inertial frame the values of $P$ , $K$ , $J_{o}$ and $E$ change by the laws of Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy. Constancy of $P$ makes $\tilde{P}$ constant; constancy of both $P$ and $K$ makes $\tilde{K}$ constant; under scalar potential transfer, constancy of both $E$ and $P$ makes $\tilde{E}$ constant. For angular momentum, constancy of $J_{o}$ transfers precisely when the boost term $(R X_{o}) \times (M v)$ is constant. The displayed terms need not be constant, and no conservation law of this page asserts a frame-independent numerical value.
Open systems. The pair structure, antisymmetry and centrality are properties of the declared labels; for a proper subsystem the forces to omitted labels enter as external forces that need satisfy no such property. The subsystem's momentum, angular momentum and energy may be exchanged with the environment; zero net force, zero total torque about a fixed reference point, or zero mechanical-energy rate can give the respective conservation conclusions under the cited hypotheses. This continues the boundary recorded in Status audit and model boundaries of the force page: open systems, frame selection, breakdown and empirical scope; the balance theorems are statements about the specified system, not about an arbitrary collection of bodies.
Empirical status. Masses, force parameters, potentials, residual-force coefficients and the choice of potential representative are model data; a numerical value taken from measurement is an empirical premise with its conditions and uncertainty, and the pages of this pair record no observation. Agreement of a computed motion with a reported observation would test the combined model, apparatus and calibration assumptions, and would be evidence about them, not a proof of the model. Finite-time breakdown of a maximal motion remains possible by Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain and does not affect the balance identities, which hold on every interval on which the motion is defined.

5 · Examples, experiments and counterexamples

None yet.

Momentum, work and energy balances

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

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

Definition

Remarks

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)

Statement

Facts & Assumptions

Proof

Remarks

Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces

Statement

Facts & Assumptions

Proof

Remarks

Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case

Statement

Facts & Assumptions

Proof

Remarks

Power of the total force and the work-energy theorem: K′=PK′=P and K(t1)−K(t0)=W([t0,t1])K(t1​)−K(t0​)=W([t0​,t1​]) along a motion

Statement

Facts & Assumptions

Proof

Remarks

Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets

Definition

Remarks

Mechanical energy balance for a potential presentation: E′=∂tUE′=∂t​U plus the residual power, conservation for a static potential with no residual forces, and the integrated work form

Statement

Facts & Assumptions

Proof

Remarks

Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy

Statement

Facts & Assumptions

Proof

Remarks

Status audit and scope of the balance page: conditional conservation, open systems, frames, potential normalization and empirical boundaries

Status audit of this page

Hypotheses and counterexamples

Frames, open systems and empirical boundaries

5 · Examples, experiments and counterexamples

Sources

Power of the total force and the work-energy theorem: $K^{'} = P$ and $K (t_{1}) - K (t_{0}) = W ([t_{0}, t_{1}])$ along a motion

Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form