A regular holonomic constraint, its virtual displacements and the ideal-reaction law

Let $n\ge 1$ and $k\ge 1$, let $O\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be open and nonempty, let $G:O\to {\mathbb{R}}^k$ be of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms), and write $G_q(t,q)$ for the Jacobian matrix of $q\mapsto G(t,q)$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case), a $k\times n$ matrix. The constraint

$G(t,q)=0$

is regular when

$G_q(t,q)\text{ has rank }k\text{ for every }(t,q)\in O,$

equivalently when $G_q$ is surjective at every point (Submersions and immersions between Euclidean open sets). Under this hypothesis each fibre $G(t,\cdot {)}^{-1}(0)$ is a regular level set and every point of it is a regular point in the sense of Regular and critical points, regular and critical values, and level sets.

Constraint set and admissible data. Fix a symmetric positive definite matrix $M\in M_n(\mathbb{R})$, meaning $M^{\mathsf{T}}=M$ and $u^{\mathsf{T}}Mu>0$ for every $0\mathrm{\ne }u\in {\mathbb{R}}^n$, and a continuous map $f:O\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, the given force. A map $q:I\to {\mathbb{R}}^n$ on an open interval $I\subseteq \mathbb{R}$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) is a constrained curve when

$(t,q(t))\in O,G(t,q(t))=0(t\in I)\mathrm{.}$

Virtual displacements and permitted velocities. For $(t,q)\in O$ the virtual displacements at $(t,q)$ are the elements of the kernel

$V(t,q):=\ker G_q(t,q)=\{\delta \in {\mathbb{R}}^n:G_q(t,q)\delta =0\}\subseteq {\mathbb{R}}^n,$

the tangent space to the level set of $G(t,\cdot )$ at $q$ in the sense of The tangent space to a regular level set; a permitted velocity at $(t,q)$ is a vector $v\in {\mathbb{R}}^n$ with

$G_t(t,q)+G_q(t,q)v=0.$

For a $C^1$ constrained curve the differentiated constraint forces $G_t(t,q(t))+G_q(t,q(t))q^{\mathrm{\prime }}(t)=0$, so its velocity is automatically permitted; the permitted condition is stated separately because it is the correct initial condition for the first-order system below.

Ideal-constraint motions. A $C^2$ constrained curve $q:I\to {\mathbb{R}}^n$ is an ideal-constraint motion of the data $(G,M,f)$ when

1. $q^{\mathrm{\prime }}(t)$ is a permitted velocity at $(t,q(t))$ for every $t\in I$, and
2. there is a continuous map $\lambda :I\to {\mathbb{R}}^k$ with

$M{q}^{\mathrm{\prime }\mathrm{\prime }}(t)=f(t,q(t),q^{\mathrm{\prime }}(t))+G_q(t,q(t){)}^{\mathsf{T}}\lambda (t)(t\in I)\mathrm{.}$

The vector field $R(t):=M{q}^{\mathrm{\prime }\mathrm{\prime }}(t)-f(t,q(t),q^{\mathrm{\prime }}(t))$ is the reaction and $R=G_q^{\mathsf{T}}\lambda$ is its multiplier form. Since $G_q(t,q{)}^{\mathsf{T}}\lambda \cdot \delta =\lambda \cdot G_q(t,q)\delta =0$ for every $\delta \in V(t,q)$, every ideal-constraint motion satisfies the d'Alembert condition

$(M{q}^{\mathrm{\prime }\mathrm{\prime }}(t)-f(t,q(t),q^{\mathrm{\prime }}(t)))\cdot \delta =0\text{for every }\delta \in V(t,q(t)),$

i.e. the reaction performs no virtual work; the converse implication, that every curve satisfying the d'Alembert condition is an ideal-constraint motion with a unique multiplier for its given reaction, is the content of Reactions annihilating the virtual displacements are transpose multiples, with a positive definite multiplier matrix and is not assumed here. Determination of the reaction and acceleration additionally uses the twice-differentiated constraint, as in The ideal holonomic system in a possibly time-dependent chart: explicit acceleration, constraint preservation, local existence and reaction power under its stated regularity hypotheses.

Scope. The definition is chartwise and time-dependent: the constraint may depend on $t$ (rheonomic case), its velocity condition is the fixed-time linearisation recorded above, and no relation between the variational (vakonomic) treatment of the constraint and the present d'Alembert law is asserted. The construction and long-time behaviour of solutions is the subject of The ideal holonomic system in a possibly time-dependent chart: explicit acceleration, constraint preservation, local existence and reaction power. For a holonomic constraint written in a chart, the multiplier $\lambda$ is not a measured quantity: only the reaction $G_q^{\mathsf{T}}\lambda$ enters the equations of motion.

• The definition covers the classical examples: a planar pendulum is $n=2$, $k=1$, $G(q)=\mathrm{\mid }q{\mathrm{\mid }}^2-l^2$ with $l>0$ on the chart $\{(x,y):y\mathrm{\ne }0\}$ or on ${\mathbb{R}}^2\setminus \{0\}$, where $G_q=2q^{\mathsf{T}}\mathrm{\ne }0$; a rolling disk in the plane with configuration $(x,\theta )$ is $G(q)=x-a\theta$, so that in that planar special case the constraint is holonomic, while general three-dimensional rolling is not reducible to a single level set. These examples are worked out on the companion page.

• The word ideal refers to the d'Alembert condition and not to any assertion that a real contact or joint enforces it; frictionless joints, no-slip rolling with sufficient static friction, and similar mechanisms are physical idealisations modelled by this law, and the empirical qualifications of such models belong to the consuming physics items, not here.