Reduced d'Alembert equation for an immersion: generalized forces and the quadratic correction

Let $n,m\ge 1$, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open and nonempty, and let $X:\mathrm{\Omega }\to {\mathbb{R}}^m$ be of class $C^3$ with $DX(q)$ injective for every $q\in \mathrm{\Omega }$; fix an immersion in the sense of Submersions and immersions between Euclidean open sets (this forces $n\le m$). Let $M\in M_m(\mathbb{R})$ be symmetric positive definite, meaning $M^{\mathsf{T}}=M$ and $u^{\mathsf{T}}Mu>0$ for every $0\mathrm{\ne }u\in {\mathbb{R}}^m$, let $D\subseteq \mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m$ be open and let $F:D\to {\mathbb{R}}^m$ be continuous, let $I\subseteq \mathbb{R}$ be an open interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and let $q:I\to \mathrm{\Omega }$ be a $C^2$ curve with $x(t):=X(q(t))$ and $(t,x(t),x^{\mathrm{\prime }}(t))\in D$ for all $t\in I$. Suppose there is a continuous $R:I\to {\mathbb{R}}^m$ with

$M{x}^{\mathrm{\prime }\mathrm{\prime }}(t)=F(t,x(t),x^{\mathrm{\prime }}(t))+R(t),R(t)\cdot DX(q(t))\delta =0\text{for all }\delta \in {\mathbb{R}}^n,t\in I\mathrm{.}$

Then, writing $DX(q(t){)}^{\mathsf{T}}$ for the transpose and $(X_{qq}[q^{\mathrm{\prime }},q^{\mathrm{\prime }}]{)}_k:={\sum }_{i,j}{\mathrm{\partial }}_i{\mathrm{\partial }}_j{X}_k(q)q_i^{\mathrm{\prime }}{q}_j^{\mathrm{\prime }}$, one has

$DX(q{)}^{\mathsf{T}}MDX(q)q^{\mathrm{\prime }\mathrm{\prime }}=DX(q{)}^{\mathsf{T}}F(t,x,x^{\mathrm{\prime }})-DX(q{)}^{\mathsf{T}}M{X}_{qq}[q^{\mathrm{\prime }},q^{\mathrm{\prime }}]$

on $I$. The vector $DX(q{)}^{\mathsf{T}}F(t,x,x^{\mathrm{\prime }})$ is the generalized force of the ambient force $F$ in the coordinates $q$, and $D{X}^{\mathsf{T}}M{X}_{qq}[q^{\mathrm{\prime }},q^{\mathrm{\prime }}]$ is the acceleration correction of the parametrisation; when $X$ is affine, the correction vanishes.

• The corollary is the mathematical content of the reduced d'Alembert principle: the tangent projection of the ambient equation is taken, the normal component $R$ drops out, and the applied force enters as the generalized force $D{X}^{\mathsf{T}}F$. The condition $R\cdot DX\delta =0$ is an explicit hypothesis on $R$; the corollary does not assert that any particular mechanism realises it, nor that an ambient constraint produces such an $R$.

• The matrix $D{X}^{\mathsf{T}}MDX$ is symmetric and positive definite on ${\mathbb{R}}^n$: for $0\mathrm{\ne }u\in {\mathbb{R}}^n$, injectivity gives $DXu\mathrm{\ne }0$ and $u^{\mathsf{T}}D{X}^{\mathsf{T}}MDXu=(DXu{)}^{\mathsf{T}}M(DXu)>0$; this is the reduced mass matrix used in The ideal planar pendulum: equation of motion, conserved energy, Lyapunov stability and the frequency $\sqrt{g\mathrm{/}l}$ ↗ and The planar rolling disk: the holonomic constraint $x-a\theta =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ and the acceleration $f\mathrm{/}(m+I\mathrm{/}{a}^2)$ ↗.