Ideal holonomic constraints determine the reaction: multiplier form, explicit acceleration, unique constrained motions and the reaction-power identity

Adopt a holonomic-constraint model of Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction with datum $(O,k,G)$, mass operator $\mathsf{M}$, force map $F$, and the ideal-reaction law of Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits. Suppose, in addition to the hypotheses of those items:

(h1) $G$ is of class $C^3$ on $O$, and $D_{x}G(t,x)$ has rank $k$ for every $(t,x)\in O$ (not only on the constraint locus);

(h2) the force map is of class $C^1$ on $I_0\times Q\times ({\mathbb{R}}^3{)}^N$ ($C^k$ Euclidean maps and diffeomorphisms).

Set $O^{\mathrm{\prime }}:=O\cap (I_0\times Q)$. This is an open nonempty set because $\mathrm{\Sigma }\subseteq O^{\mathrm{\prime }}$ and $\mathrm{\Sigma }$ is nonempty. The matrix-valued map $C:O\to M_k(\mathbb{R})$ below is defined on $O$, while $\mathrm{\Lambda }:O^{\mathrm{\prime }}\times ({\mathbb{R}}^3{)}^N\to {\mathbb{R}}^k$ and $a:O^{\mathrm{\prime }}\times ({\mathbb{R}}^3{)}^N\to {\mathbb{R}}^{3N}$ are defined on the force-map domain restricted to the constraint datum; this is the time-state domain of the first-order system. Every constrained motion in the statement has $(t,x(t))\in \mathrm{\Sigma }\subseteq O^{\mathrm{\prime }}$, so these functions are defined along it.

Let $x:J\to Q$ be a constrained motion of the model with reaction field $R$ satisfying the ideal-reaction law, on an interval $J\subseteq I_0$ with nonempty interior, and write $w=x^{\mathrm{\prime }}$ for its velocity. Then:

1. multiplier form. For every $t\in J$ there is a unique $\lambda (t)\in {\mathbb{R}}^k$ with $R(t)=D_{x}G(t,x(t){)}^{\mathsf{T}}\lambda (t)$; the map $\lambda :J\to {\mathbb{R}}^k$ so defined is continuous, and with

$C(t,x):=D_{x}G(t,x){\mathsf{M}}^{-1}{D}_{x}G(t,x{)}^{\mathsf{T}},\mathrm{\Lambda }(t,x,v):=-C(t,x{)}^{-1}[D_{x}G{\mathsf{M}}^{-1}F(t,x,v)+G_{tt}+2G_{tq}[v]+G_{qq}[v,v]],$

the terms being evaluated at $(t,x)$ or $(t,x,v)$, the $k\times k$ matrix $C(t,x)$ is symmetric positive definite and invertible and $\lambda (t)=\mathrm{\Lambda }(t,x(t),w(t))(t\in J):$ the multiplier, and hence the reaction, is determined by the model data and the motion, not a free or measured quantity.

2. determined dynamics. With $a(t,x,v):={\mathsf{M}}^{-1}(F(t,x,v)+D_{x}G(t,x{)}^{\mathsf{T}}\mathrm{\Lambda }(t,x,v))$, the tuple $(x,w)$ solves the first-order system $x^{\mathrm{\prime }}=v,v^{\mathrm{\prime }}=a(t,x,v)$ on $J$; conversely every solution $(x,v)$ of this system on an interval whose initial datum is consistent, with $R(t):=D_{x}G(t,x(t){)}^{\mathsf{T}}\mathrm{\Lambda }(t,x(t),v(t))$, is a constrained motion of the model satisfying the ideal-reaction law; and through every consistent datum $(t_0,x_0,v_0)$ there is a unique maximal solution of the system. Hence the holonomic-constraint model with ideal reactions has determined dynamics on its constraint locus.

3. preservation. Along every solution of clause 2 with consistent data, $G(t,x(t))=0,G_t(t,x(t))+D_{x}G(t,x(t))v(t)=0$ for every $t$ of its maximal interval.

4. reaction power. Along every ideal constrained motion as in clause 1, $R(t)\cdot w(t)=-\lambda (t)\cdot G_t(t,x(t))(t\in J),$ the pairings being the Euclidean inner products.

The statement is conditional on the adopted force branch and ideal-reaction law and on hypotheses (h1)-(h2); it asserts no empirical value, and the multiplier is a representation tied to the declared components of $G$.

• The multiplier is a representation. The unique $\lambda (t)$ is attached to the declared components $G_{\alpha }$; multiplying $G$ by an invertible matrix-valued function $\rho (t,x)$ leaves the reaction $R$ and the constraint locus unchanged but changes $\lambda$ to ${\rho }^{-\mathsf{T}}\lambda$. Only $R$ enters the equation of motion, so no statement of this page may be read as attributing an independent measured meaning to $\lambda$.

• Where the hypotheses enter. The rank hypothesis (h1) on all of $O$ is used to make $C$ invertible and $\mathrm{\Lambda }$ everywhere defined and $C^1$; the $C^1$ hypothesis on $F$ is used for the same purpose; the ideal-reaction law supplies the conormal membership of $R$, without which the motion is not determined. At a point of rank loss the multiplier matrix degenerates, the acceleration is not determined by the data, and the statement does not apply; no continuation across such a point is asserted.

• Consistency is an initial-data condition. A solution of the system through nonconsistent data still exists but does not satisfy $G=0$; the constrained motions are exactly the solutions with consistent data, as clause 2 records. Nothing here asserts existence of a constrained motion through data whose position is off the locus or whose velocity is not permitted.

• No energy statement. Clause 4 is the power identity of the reaction; the reading of its consequences for mechanical energy, including the vanishing of the reaction work for scleronomic constraints and the work done by rheonomic constraints, belongs to the rheonomic-power theorem of this page.