Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch

Adopt, whenever a model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance invokes them, the following restricted interaction assumptions. Throughout, the data $I_0$, $Q$, $m_1,\dots ,m_N$ and the force map $F$ are those of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, and a state writes $z=(x,v)$ with $x=(x_1,\dots ,x_N)\in Q$ and $v=(v_1,\dots ,v_N)\in {\mathbb{R}}^{3N}$.

(i) External-plus-pair decomposition. The model's force map is presented as

$F_i(t,z)=F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}(t,x_i,v_i)+{\sum }_{j\mathrm{\ne }i}{F}_{ij}(t,x_i,x_j,v_i,v_j)(i=1,\dots ,N),$

where each $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}$ is a single-label external force map and each $F_{ij}$ (for $i\mathrm{\ne }j$, an ordered pair) is a pair force map, all defined on their stated open domains and satisfying the standing continuity and state-local-Lipschitz hypotheses of clause (iii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with respect to the arguments they actually depend on. This is a restriction of the model class: an arbitrary force map $F$ of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance need not admit such a decomposition, and the decomposition is part of the model's declaration, not a theorem. It is relative to the declared set of labels: for a subsystem, a pair force to a label outside the declared set appears as an external force, and no antisymmetry or pairing property of the full system restricts that external force.

(ii) Optional pair assumptions. In addition, a model may adopt, separately or together, the following assumptions about a pair force $F_{ij}$:

1. Antisymmetry (weak form of the third law). $F_{ij}(t,x_i,x_j,v_i,v_j)=-F_{ji}(t,x_j,x_i,v_j,v_i)$ for all arguments in the domain.
2. Centrality. $F_{ij}(t,x_i,x_j,v_i,v_j)$ is parallel to $x_i-x_j$, i.e. lies in the real line spanned by $x_i-x_j$, whenever $x_i\mathrm{\ne }{x}_j$.

The strong form of the third law means antisymmetry together with centrality; centrality alone does not imply antisymmetry.

Each assumption is a further restriction of the model class and is adopted only where invoked. Both are sufficient hypotheses used by later balance statements, and neither is a consequence of the equation of motion: a velocity-dependent pair force, or an interaction mediated by a field omitted from the declared labels, need satisfy neither.

(iii) Potential model. A model may present its force map through a scalar potential $U:I_0\times Q\to \mathbb{R}$ of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms), of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$, read in joules (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework), through the rule

$F_i(t,z)=-{\mathrm{\nabla }}_{x_i}U(t,x)(i=1,\dots ,N),$

the gradient being the Euclidean gradient in the position variable of each label (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). This is a restriction of the force class: only force maps that are the negative gradient of such a potential belong to it. The same force map arises from every potential of the form $U(t,x)+c(t)$ with a differentiable function $c$ of $t$ alone whenever the declared $C^2$ regularity is retained. More generally, for each connected component $C$ of $Q$ (a maximal connected subset), two $C^2$ potentials for the same force differ by an independently chosen $C^2$ function $c_C(t)$ on $I_0\times C$. A single time function suffices when $Q$ is connected; openness alone does not require the constants on different components to agree. The chosen normalization is part of any later energy statement and is recorded with it, and no energy statement is made on this page.

(iv) Pair and external potentials are examples, not the general form. When $U$ has the form

$U(t,x)={\sum }_i{u}_i(t,x_i)+{\sum }_{i<j}{u}_{ij}(t,x_i,x_j)$

with $C^2$ scalar functions $u_i$, $u_{ij}$ on the respective open time-position domains, each with energy dimension, the decomposition (i) holds with $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}(t,x_i,v_i)=-{\mathrm{\nabla }}_{x_i}{u}_i(t,x_i)$ and, for every $i<j$, both ordered-pair maps given by

$F_{ij}(t,x_i,x_j,v_i,v_j)=-{\mathrm{\nabla }}_{x_i}{u}_{ij}(t,x_i,x_j),F_{ji}(t,x_j,x_i,v_j,v_i)=-{\mathrm{\nabla }}_{x_j}{u}_{ij}(t,x_i,x_j)\mathrm{.}$

Here the two gradients differentiate the first and second position arguments, respectively, of the same declared $u_{ij}$; no $u_{ji}$ is required. These maps are independent of velocities. Differentiating the finite sum for $U$ with respect to $x_i$ collects exactly the external term and one term for each $j\mathrm{\ne }i$, giving (i). Antisymmetry is not asserted for a general pair potential. Potentials of the still more special form $u_{ij}=u_{ij}(t,\mid x_i-x_j\mid )$ are the central pair potentials; these are examples, and no generality is claimed for any of these subclasses. Forces with nontrivial velocity dependence, such as the drag model The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient with $c>0$, lie outside every potential model of this clause; when $c=0$ that model is a constant force and has a potential.

(v) Newtonian gravitational branch. The model may adopt the pair potential

$U_{ij}(x_i,x_j)=-\frac{G_{\mathrm{N}}{m}_i^g{m}_j^g}{\mid x_i-x_j\mid }$

on the collision-excluded configuration domain ($x_i\mathrm{\ne }{x}_j$ for all $i\mathrm{\ne }j$), where $G_{\mathrm{N}}>0$ is a constant of dimension ${\mathrm{\ell }}^3{\mu }^{-1}{\tau }^{-2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework) and the numbers $m_1^g,\dots ,m_N^g>0$ are gravitational coupling parameters of the labels, primitive model data distinct from the inertial masses $m_i$ of the equation of motion. By clause (iii) this potential supplies the attractive inverse-square pair force $F_{ij}=-G_{\mathrm{N}}{m}_i^g{m}_j^g(x_i-x_j)\mathrm{/}\mid x_i-x_j{\mid }^3$, the content of the elementary differentiation ${\mathrm{\nabla }}_x(1\mathrm{/}\mid x-y\mid )=-(x-y)\mathrm{/}\mid x-y{\mid }^3$; a concrete instance of that differentiation is verified on the companion page. The adopted model satisfies the antisymmetry and centrality options of (ii) where they are invoked.

(vi) The mass identification is an empirical assumption. The identification

$m_i^g=m_i(i=1,\dots ,N)$

of gravitational couplings with inertial masses is not part of clauses (i)–(v) and is not a consequence of the equation of motion. It is an additional empirical assumption, adopted only when a model states it, and it is the subject of the weak equivalence principle; the sources record its empirical support and its historical status, and this page records no measurement. Likewise, the inverse-square form in (v) is an adopted law, not a consequence of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance.

(vii) Status. Clauses (i)–(v) are adopted modelling assumptions that restrict the force maps of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance; they add no dynamics beyond that postulate, and every theorem of this page remains conditional on these adopted hypotheses. The balances of momentum, angular momentum and energy, when asserted later, are theorems with their own hypotheses on this structure and are not made here.

• Why the restrictions are separated. Antisymmetry, centrality and the existence of a potential are stated as separate restrictions. A model's declared assumptions must be jointly consistent with its force maps and decomposition; no logical independence or realizability of every combination is asserted. Keeping the hypotheses explicit prevents the later balance theorems from silently assuming more than their proofs use.

• Open subsystems. The pair form is declared for the labels of the model. If a physical system is described by a proper subset of its bodies, interactions with the remaining bodies are external forces in the sense of clause (i), and the pair properties of the full system say nothing about them.

• No inverse-square from the second law. The gravitational potential and the inverse-square force are additional adopted laws; the second law alone determines no force law, and this page proves no property of nature from the form of the equation of motion.

• Dimensions of the gravitational couplings. Assign each independent gravitational coupling the mass dimension $[m_i^g]=\mu$, read in kilograms. Then the potential in clause (v) has dimension

  $[G_{\mathrm{N}}{m}_i^g{m}_j^g\mathrm{/}\mathrm{\mid }{x}_i-x_j\mathrm{\mid }]=({\mathrm{\ell }}^3{\mu }^{-1}{\tau }^{-2}){\mu }^2{\mathrm{\ell }}^{-1}=\mu {\mathrm{\ell }}^2{\tau }^{-2}\mathrm{.}$

  as required for energy. If a model additionally adopts $m_i^g=m_i$, this assignment is dimensionally consistent with the inertial mass dimension $[m_i]=\mu$; dimensional agreement does not establish that empirical identification.