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

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 $x_i\in {\mathbb{R}}^3$.

(a) Potential representation of the force map. A model presents its force map through a potential when it declares a function $U:I_0\times Q⟶\mathbb{R}$ of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms), of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ (read in joules), with $F_i(t,x,v)=-{\mathrm{\nabla }}_{x_i}U(t,x)(i=1,\dots ,N)$ for every $(t,x,v)$ of the state domain; here ${\mathrm{\nabla }}_{x_i}U$ 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 $d_i:I_0\times Q\times ({\mathbb{R}}^3{)}^N\to {\mathbb{R}}^3$ 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 $F_i(t,x,v)=-{\mathrm{\nabla }}_{x_i}U(t,x)+d_i(t,x,v);$ the pair $(U,d)$ is then the model's presentation and $F_i^{\mathrm{p}}:=-{\mathrm{\nabla }}_{x_i}U$ 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 $c:I_0\to \mathbb{R}$ 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 $c_C:I_0\to \mathbb{R}$ with $\tilde{U}(t,x)-U(t,x)=c_C(t)(x\in C)\mathrm{.}$ 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 $c_C^{\mathrm{\prime }}(t)$. 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 $\phi =({\phi }_1,\dots ,{\phi }_N):Q\to ({\mathbb{R}}^3{)}^N$, where $Q\subseteq {\mathbb{R}}^{3N}$ is the open configuration domain and each ${\phi }_i(x)$ is the force on label $i$ at configuration $x$; its components have force dimension. It represents a force map of the form $F_i(t,x,v)={\phi }_i(x)$ 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 $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}(t,x_i,v_i)$. Such a field is closed when ${\mathrm{\partial }}_k{\phi }_j={\mathrm{\partial }}_j{\phi }_k$ for all coordinate indices $j,k\in \{0,\dots ,3N-1\}$ at every point of $Q$, using the flattened coordinate indices of Exact and closed C1 vector fields, and exact with potential energy $U$ when $\phi =-\mathrm{\nabla }U$ for a $C^2$ function $U:Q\to \mathbb{R}$; equivalently, the field $-\phi$ 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:

1. 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);
2. 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);
3. 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 $\phi$ be closed on $V$. Then item 2 of (d) provides a $C^2$ potential $U_V:V\to \mathbb{R}$ with $\phi {\mathrm{\mid }}_V=-\mathrm{\nabla }{U}_V$: a local potential of the field on $V$. On the restricted configuration domain $V$ the model with force $-\mathrm{\nabla }{U}_V$ 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 $U:Q\to \mathbb{R}$ is smooth, meaning $C^k$ for every $k\in \mathbb{N}$ ($C^k$ Euclidean maps and diffeomorphisms); the potential presentations of clause (a) still require only $C^2$. Let $c\in \mathbb{R}$ and suppose $U^{-1}(c)\mathrm{\ne }\mathrm{\varnothing }$ and $\mathrm{\nabla }U(x)\mathrm{\ne }0$ 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 $3N-1$ (A regular level set is an embedded submanifold, Embedded submanifolds and slice charts). No such conclusion is made at a level where $\mathrm{\nabla }U$ 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.

• 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 $d\mathrm{\ne }0$ 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 ${\mathrm{\partial }}_{t}U$ and the residual power (Mechanical energy balance for a potential presentation: $E^{\mathrm{\prime }}={\mathrm{\partial }}_{t}U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form).

• 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.