Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator

Let $N\ge 1$ particle labels with fixed masses $m_1,\dots ,m_N>0$ be given as in Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses, and fix one inertial frame of the model with spatial coordinates $x_i\in {\mathbb{R}}^3$ for each label $i$. Identify the space of all position tuples with ${\mathbb{R}}^{3N}=({\mathbb{R}}^3{)}^N$ and write its points as $x=(x_1,\dots ,x_N)$; each component $x_i$ has dimension $\mathrm{\ell }$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

(a) Unconstrained branch. The unconstrained configuration domain is an open set $Q\subseteq {\mathbb{R}}^{3N}$, chosen as part of the model. A tuple $x\in Q$ is an admissible configuration: an assignment of one position to each label at one time. A singular-force exclusion is the removal of the collision set

$\mathrm{\Delta }:=\{x\in {\mathbb{R}}^{3N}:\text{ there are }i\mathrm{\ne }j\text{ with }{x}_i=x_j\}$

from the domain, which is done when the model's force law is not defined at coincident positions: $\mathrm{\Delta }$ is closed (it is the union of the finitely many closed sets $\{x_i=x_j\}$, each the zero set of the continuous map $x\mapsto x_i-x_j$), so $Q:={\mathbb{R}}^{3N}\setminus \mathrm{\Delta }$ is open and is a legitimate configuration domain. Whether the exclusion is part of the model is a modelling decision recorded with the force law; the framework itself only requires $Q$ to be open and stated.

(b) Holonomic constraint branch. Let $O\subseteq {\mathbb{R}}^{3N}$ be an open ambient set, let $k\in \mathbb{N}$ with $1\le k\le 3N-1$, and let

$G=(G_1,\dots ,G_k):O⟶{\mathbb{R}}^k$

be a smooth map, each component carrying a declared dimension and unit fixed by the model (a component can always be rescaled by a nonzero constant without changing its zero set, and the rescaling is part of the declaration). Assume:

1. Nonempty level set: $G^{-1}(0)\mathrm{\ne }\mathrm{\varnothing }$;
2. Rank condition: the differential $DG(x):{\mathbb{R}}^{3N}\to {\mathbb{R}}^k$ is surjective, equivalently has rank $k$, at every $x\in G^{-1}(0)$.

Then $0$ is a regular value of $G$ in the sense of Regular and critical points and values, and the constrained configuration space is $Q:=G^{-1}(0)$. By A regular level set is an embedded submanifold — applied to the smooth map $G$ from the open subset $O$ of ${\mathbb{R}}^{3N}$ to ${\mathbb{R}}^k$, whose regular value $0$ has nonempty fibre — $Q$ is an embedded submanifold of ${\mathbb{R}}^{3N}$ of dimension $3N-k$ (Embedded submanifolds and slice charts). The degrees of freedom of this branch of the model is the number $\dim Q=3N-k\mathrm{.}$ In the sense of Constrained configurations, the mass matrix and the mass metric, $Q$ together with the inclusion $X:Q\hookrightarrow {\mathbb{R}}^{3N}$ is then a particle configuration, and the mass metric of that definition is available with the masses $m_i$; the configuration is constrained to $Q$ for all times in this branch.

(c) Time-dependent holonomic constraints. If the constraint map depends on time,

$G:\mathbb{R}\times O⟶{\mathbb{R}}^k,(t,x)⟼G(t,x),$

with $G$ smooth and $D_{x}G(t_0,x)$ of rank $k$ at every $x$ of the nonempty set $G(t_0,\cdot {)}^{-1}(0)$, then for each fixed $t_0\in \mathbb{R}$ the instantaneous configuration space $Q_{t_0}:=G(t_0,\cdot {)}^{-1}(0)$ is an embedded submanifold of dimension $3N-k$ by the same theorem applied to $G(t_0,\cdot )$. The family $\{Q_{t_0}\}$ varies with $t_0$; a curve $t\mapsto q(t)$ with $G(t,q(t))=0$ is not thereby a curve in one fixed manifold, and the kinematic treatment of such constraints (velocities, the reaction forces that maintain them and the distinction between rheonomic and scleronomic constraints) belongs to the framework's constraint development and is not part of this definition.

(d) Mass operator. In the ambient coordinates define the mass operator $\mathsf{M}:{\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N}$ by block multiplication,

$\mathsf{M}(x_1,\dots ,x_N):=(m_1{x}_1,\dots ,m_N{x}_N);$

its matrix in the standard basis is block diagonal with blocks $m_i{I}_3$, the mass matrix of Constrained configurations, the mass matrix and the mass metric. Its dimension is $\mu$: it multiplies positions of dimension $\mathrm{\ell }$ by masses of dimension $\mu$. On the constrained branch, for the inclusion $X:Q\hookrightarrow {\mathbb{R}}^{3N}$ and tangent vectors $h,z\in T_{q}Q$, the mass metric of that item is ${\mathcal{G}}_q(h,z):=⟨d{X}_q(h),\mathsf{M}d{X}_q(z){⟩}_{3N}={\sum }_{i=1}^N{m}_i{h}_i\cdot z_i,$ where $h_i,z_i\in {\mathbb{R}}^3$ are the blocks of $d{X}_q(h),d{X}_q(z)$ under the canonical Euclidean tangent identification specified in that supplier. It restricts the mass-weighted ambient bilinear form to $T_{q}Q$, rather than the unweighted Euclidean inner product.

(e) Velocity constraints are a different object. A velocity-map constraint on the open ambient set $O$ is a pair of smooth maps $A:\mathbb{R}\times O\to {\mathbb{R}}^{k^{\mathrm{\prime }}\times 3N},b:\mathbb{R}\times O\to {\mathbb{R}}^{k^{\mathrm{\prime }}},1\le k^{\mathrm{\prime }}\le 3N,$ with $A(t,x)$ of constant rank $k^{\mathrm{\prime }}$ everywhere in the declared domain, imposing $A(t,x(t))\dot{x}(t)+b(t,x(t))=0$ on a $C^1$ curve $x:J\to O$, where $J\subseteq \mathbb{R}$ is a nondegenerate open interval, the parameter $t$ is frame time, and $\dot{x}(t):=x^{\mathrm{\prime }}(t)$ is the componentwise derivative in the convention of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework. Equivalently $A(t,x)$ is a linear map ${\mathbb{R}}^{3N}\to {\mathbb{R}}^{k^{\mathrm{\prime }}}$. Each row's output unit is declared, and its corresponding component of $b$ has that same unit. The allowed velocities at $(t,x)$ form the affine set $\{v:A(t,x)v+b(t,x)=0\}$; its differences form $\ker A(t,x)$.

The law is locally holonomic-equivalent at $(t_0,x_0)$ if on some open neighbourhood $V\subseteq \mathbb{R}\times O$ of that point there are a $C^2$ map $g:V\to {\mathbb{R}}^{k^{\mathrm{\prime }}}$ with $D_{x}g$ of rank $k^{\mathrm{\prime }}$ and a $C^2$ matrix map $C:V\to {\mathbb{R}}^{k^{\mathrm{\prime }}\times k^{\mathrm{\prime }}}$ with $\det C\mathrm{\ne }0$ such that $A=C{D}_{x}g,b=C{\mathrm{\partial }}_{t}g\text{on }V\mathrm{.}$ Thus the velocity law is locally the derivative of the regular positional constraints $g(t,x)=\text{constant}$. It is locally nonholonomic at the specified point when no such neighbourhood and pair $(g,C)$ exists; a nonholonomic model in this local sense declares a point (or a region of points) with this failure. For a homogeneous time-independent law this rules out such a regular-level-set representation throughout a neighbourhood. It need not rule out an individual embedded submanifold $S\subseteq O$ with $T_{x}S=\ker A(x)$ for every $x\in S$, using the ambient tangent identification of Constrained configurations, the mass matrix and the mass metric; for an affine time-dependent law the criterion also includes the drift $b$. Absence of a single global pair alone does not establish local nonintegrability. The rank of $A$ counts instantaneous velocity restrictions, not the dimension of a positional configuration manifold; accessibility requires its own definition. Rank changes require a separate model.

• The definition is a stage, not a law. Nothing here says how $x$ varies with time; the equation of motion, the force law and the reaction forces of the constrained branch are adopted later in the framework, each with its own hypotheses. The dimension count of clause (b) is a statement about $G^{-1}(0)$ as a set, proved from the established regular-level theorem; it presupposes exactly the nonemptiness and rank hypotheses displayed there.

• Choice of coordinates. The definition uses the redundant particle positions $x\in {\mathbb{R}}^{3N}$ as coordinates, not a set of generalized coordinates of $Q$; describing a constrained configuration by $3N-k$ coordinates requires charts of the embedded submanifold, and the transformation between such charts is part of the differential geometry of $Q$. The distinction between the redundant description and a chart is not erased by the count $\dim Q$.

• Dimensions of constraint components. The dimension of a constraint component is part of the model's declaration, exactly as the unit of a physical quantity is: a constraint expressed in metres and the same constraint multiplied by a second quantity have different components and the same zero set. The rescaling freedom is recorded so that a model can choose convenient units, and it changes no physical content of the constraint.