Ideal scleronomic holonomic constraints: the substituted Lagrange equations are equivalent to the constrained model equation

Adopt the force-branch data of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: $N_p\ge 1$ labelled particles with masses $m_1,\dots ,m_{N_p}>0$, an open time window $I_0$, an open configuration domain $Q\subseteq {\mathbb{R}}^{3N_p}$ and a force map $F$ with the standing regularity; adopt a regular holonomic-constraint datum $(O,k,g_c)$ of Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction, where $g_c:=G$ is the constraint map named $G$ in that definition, and the ideal-reaction law of Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits. Assume the datum is scleronomic with time-independent locus $\mathcal{C}\subseteq {\mathbb{R}}^{3N_p}$:

$\mathrm{\Sigma }=\{(t,x)\in O:x\in \mathcal{C}\}\mathrm{.}$

Let $(U,d)$ be a potential presentation of $F$ in the sense of clause (a) of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, so that $F_i(t,x,v)=-{\mathrm{\nabla }}_{x_i}U(t,x)+d_i(t,x,v)$. Let $r:=3N_p-k$, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^r$ be open and nonempty, and let $X:\mathrm{\Omega }\to Q$ be a parametrization of the constrained configurations in the sense of clause (d) of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces: $X$ is of class $C^3$, $DX(q)$ is injective for every $q\in \mathrm{\Omega }$, $X(\mathrm{\Omega })=\mathcal{C}\cap Q$ and $(t,X(q))\in O$ for all $(t,q)\in I_0\times \mathrm{\Omega }$. Injectivity of $X$ is not assumed: the image condition requires coverage of the constrained configurations, while multiple coordinate representatives are permitted. Define the kinetic matrix, substituted potential, substituted Lagrangian and residual generalized force

$\mathsf{G}(q):=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q),\tilde{U}(t,q):=U(t,X(q)),\tilde{L}(t,q,v):=\frac{1}{2}{v}^{\mathsf{T}}\mathsf{G}(q)v-\tilde{U}(t,q),$

$Q^{\mathrm{r}\mathrm{e}\mathrm{s}}(t,q,v):=DX(q{)}^{\mathsf{T}}d(t,X(q),DX(q)v)\mathrm{.}$

Let $J\subseteq I_0$ be an interval with nonempty interior, let $q:J\to \mathrm{\Omega }$ be a $C^2$ curve and put $x:=X\circ q$. Then the following are equivalent:

1. $x$ is a constrained motion of the model with an ideal reaction field: there is a continuous $R:J\to {\mathbb{R}}^{3N_p}$ with $\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}(t)=F(t,x(t),x^{\mathrm{\prime }}(t))+R(t)$ and $R(t)\cdot \delta =0$ for every $t\in J$ and every virtual displacement $\delta \in \ker D_x{g}_c(t,x(t))$;
2. $\mathrm{\Lambda }(\tilde{L})(q)(t)=Q^{\mathrm{r}\mathrm{e}\mathrm{s}}(t,q(t),q^{\mathrm{\prime }}(t))$ for every $t\in J$, i.e. the declared Lagrange equations of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces hold along $q$.

If these hold, then:

3. the reaction is unique: $R(t)=\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}(t)-F(t,x(t),x^{\mathrm{\prime }}(t))$ for every $t\in J$, and there is a unique continuous $\lambda :J\to {\mathbb{R}}^k$ with $R(t)=D_x{g}_c(t,x(t){)}^{\mathsf{T}}\lambda (t)$; if the declared constraint components have dimensions ${\gamma }_{\alpha }$, then $[{\lambda }_{\alpha }]=\mu {\mathrm{\ell }}^2{\tau }^{-2}\mathrm{/}{\gamma }_{\alpha }$;
4. the virtual displacements at $x(t)$ are exactly the image of $DX(q(t))$, $\ker D_x{g}_c(t,x(t))=DX(q(t))\left({\mathbb{R}}^r\right)$;
5. the statement is conditional on the adopted force branch, the presentation, the scleronomic regular constraint datum and the ideal-reaction law. No solution existence, no uniqueness of the coordinate curve for given data and no empirical content are asserted.

• What global coverage means. The equivalence uses $X(\mathrm{\Omega })=\mathcal{C}\cap Q$, so every constrained configuration occurs as $X(q)$. The proof is pointwise and does not require one-to-one coordinates: coverings and other multiple-representation surjective immersions are permitted. For a map whose image covers only part of the locus, the theorem applies after restricting the constraint datum to that part.

• Scleronomy is used twice. Time-independence of the locus gives $(g_c{)}_t=0$ on $\mathrm{\Sigma }$ (so parametrized curves have permitted velocity) and lets the kinetic energy be read as $\frac{1}{2}{v}^{\mathsf{T}}\mathsf{G}(q)v$ without mixed $X_t$ terms. The rheonomic case has additional velocity-proportional terms in $\tilde{L}$ and its own treatment through the moving-chart and constraint pages; no statement about it is made here.