Constrained configurations, the mass matrix and the mass metric

Let $N\ge 1$; the ambient dimension is $3N$. Identify ${\mathbb{R}}^{3N}$ with the $N$-fold product $({\mathbb{R}}^3{)}^N$ and write a point as $x=(x_1,\dots ,x_N)$ with $x_i\in {\mathbb{R}}^3$, explicitly $x_i(k):=x(3(i-1)+k)$ for $1\le i\le N$ and $0\le k<3$; the Euclidean inner product on ${\mathbb{R}}^{3N}$ is

$⟨x,y{⟩}_{3N}={\sum }_{i=1}^N⟨x_i,y_i{⟩}_2,$

with $⟨\cdot ,\cdot {⟩}_2$ the inner product of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$.

Let $d\ge 0$ and let $Q$ be a smooth $d$-dimensional manifold (Smooth manifolds and their smooth charts) and let

$X:Q⟶{\mathbb{R}}^{3N}$

be a smooth map. The manifold $Q$ is the configuration space and $X$ is the configuration map; the pair $(Q,X)$ is called a particle configuration when $Q$ is an embedded submanifold of ${\mathbb{R}}^{3N}$ and $X$ is the inclusion (Embedded submanifolds and slice charts), and a parametrized configuration otherwise. For $q\in Q$ the intrinsic differential is $d{X}_q:T_{q}Q\to T_{X(q)}{\mathbb{R}}^{3N}$ as in The differential of a smooth map (Derivations at a point and the tangent space for the tangent space). For $a\in {\mathbb{R}}^{3N}$ and the coordinate function ${\pi }_k(x):=x(k)$, define ${\iota }_a:T_a{\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N}$ by ${\iota }_a(w)(k):=w([{\pi }_k])$ for $0\le k<3N$. This is the standard linear isomorphism: in the identity chart the coordinate derivations form a basis by Coordinate derivations form a basis of the tangent space, and evaluation on coordinate germs sends that basis to the standard unit vectors. Throughout this item and its mass-metric formulas, the notation $d{X}_q(u)$ in Euclidean expressions abbreviates ${\iota }_{X(q)}(d{X}_q(u))$. In particular its components are $u([{\pi }_k\circ X])$, so this convention retains the usual Jacobian and inclusion formulas.

Let $m_1,\dots ,m_N>0$ be masses. The mass matrix is the linear map $M:{\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N}$ acting on each block by $M(x_1,\dots ,x_N)=(m_1{x}_1,\dots ,m_N{x}_N)$; its matrix in the standard basis is block diagonal with blocks $m_i{I}_3$ (Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes).

The mass metric of $(Q,X,m_1,\dots ,m_N)$ is the family of maps

$G_q:T_{q}Q\times T_{q}Q⟶\mathbb{R},G_q(u,v):=⟨d{X}_q(u),Md{X}_q(v){⟩}_{3N},$

for $q\in Q$ and $u,v\in T_{q}Q$. The kinetic energy of a tangent vector $v\in T_{q}Q$ is

$T_q(v):=\frac{1}{2}{G}_q(v,v)\mathrm{.}$

Each $G_q$ is a symmetric bilinear form on the real vector space $T_{q}Q$ in the sense of Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms; that the family depends smoothly on $q$ means that in every smooth local chart of $Q$ the coefficient functions $q\mapsto G_q({\mathrm{\partial }}_i,{\mathrm{\partial }}_j)$ are smooth, and is stated in the smoothness remark below.

• Smooth dependence. In any smooth chart $\psi :W\to \psi (W)\subseteq {\mathbb{R}}^d$ on $Q$, write $X_{\psi }:=X\circ {\psi }^{-1}$ and let $J_X$ be its $3N\times d$ Jacobian. The coordinate matrix of $G$ is $J_X^{\mathsf{T}}M{J}_X$. Because $X_{\psi }$ is $C^{\mathrm{\infty }}$, each entry of its Jacobian is $C^{\mathrm{\infty }}$ by the multi-index definition of $C^k$ ($C^r$ and smooth maps between smooth manifolds, $C^k$ Euclidean maps and diffeomorphisms, $C^k$ maps and multi-index derivative notation in Euclidean space); finite sums and products preserve smoothness ($C^k$ Euclidean maps are closed under componentwise algebra and composition). Hence the coefficients of $G$ are smooth in every chart, as stated in the Definition.

• In a local chart with coordinate vector fields ${\mathrm{\partial }}_1,\dots ,{\mathrm{\partial }}_d$ the matrix of $d{X}_q$ is the $3N\times d$ Jacobian $X_q$ with entries ${\mathrm{\partial }}_j{X}^k(q)$, and the matrix of $G_q$ is $X_q^{\mathsf{T}}M{X}_q$; this finite computation is The mass metric is symmetric, positive semidefinite, and positive definite exactly when the configuration differential is injective. When $X$ is the inclusion of an embedded $Q$, the differential is the inclusion $T_{q}Q\hookrightarrow {\mathbb{R}}^{3N}$ and the formula reads $G_q(u,v)=⟨u,Mv{⟩}_{3N}$.

• The mass metric is a mathematical construction from the data $(Q,X,m_i)$: no physical law, no identification of $m_i$ with any measured inertial or gravitational mass, and no assertion that the motion of any physical system minimizes anything is made here.

• The kinetic energy $T_q$ is not a function on $Q$ but a function on tangent vectors depending on the base point; this dependence is the reason the mass metric is recorded as a family indexed by $q$.