Cm Math Geometry

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open and let $F:U\to {\mathbb{R}}^n$ be a map. Consider the autonomous first-order system

$z^{\mathrm{\prime }}(t)=F(z(t)),$

read as the first-order system of First-order systems, initial value problems, and solutions on intervals with $D=\mathbb{R}\times U$ and right-hand side $(t,z)\mapsto F(z)$. A solution below means a solution of this system on an interval containing $0$; a solution whose interval is $[0,\mathrm{\infty })$ or contains it is said to be defined for all $t\ge 0$.

A solution $\tilde{z}:J\to U$ extends $z:I\to U$ when $I\subseteq J$ and $\tilde{z}{\mathrm{\mid }}_I=z$; the extension is proper if $I⊊J$. A solution is maximal when it has no proper extension (Extensions, maximal solutions, maximal intervals, and global solutions). All stability predicates below quantify over maximal solutions whose intervals contain $0$; their local restrictions remain solutions. This convention asserts neither existence nor uniqueness of maximal solutions for an arbitrary map $F$.

A point $z_{\ast }\in U$ is an equilibrium (also fixed point or zero) of $F$ when $F(z_{\ast })=0$; then the constant function $z(t)=z_{\ast }$ is a solution on every interval containing $0$.

Let $z_{\ast }\in U$ be an equilibrium.

1. $z_{\ast }$ is Lyapunov stable (or simply stable) when for every Euclidean ball $B_{\epsilon }(z_{\ast })$ with $\overline{B_{\epsilon }(z_{\ast })}\subseteq U$ (Open ball, closed ball and sphere in a metric space) there is $\delta \in (0,\epsilon ]$ such that every maximal solution $z$ with $\mathrm{\parallel }z(0)-z_{\ast }{\mathrm{\parallel }}_2<\delta$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$) is defined for all $t\ge 0$ and satisfies

$\mathrm{\parallel }z(t)-z_{\ast }{\mathrm{\parallel }}_2<\epsilon (t\ge 0)\mathrm{.}$

2. $z_{\ast }$ is asymptotically stable when it is Lyapunov stable and in addition there is ${\delta }_0>0$ such that every maximal solution $z$ with $\mathrm{\parallel }z(0)-z_{\ast }{\mathrm{\parallel }}_2<{\delta }_0$ is defined for all $t\ge 0$ and satisfies $\mathrm{\parallel }z(t)-z_{\ast }{\mathrm{\parallel }}_2\to 0$ as $t\to \mathrm{\infty }$.

An equilibrium which is not Lyapunov stable is unstable. An equilibrium is globally asymptotically stable when it is asymptotically stable and every maximal solution of the system is defined for all $t\ge 0$ and converges to it as $t\to \mathrm{\infty }$.

• The two clauses are not equivalent. The equation $z^{\mathrm{\prime }}=0$ on $\mathbb{R}$ has every point Lyapunov stable and no point asymptotically stable. Conversely, convergence of nearby solutions to $z_{\ast }$ does not by itself imply stability, which is why clause 2 requires stability explicitly; a standard planar witness is recorded in Problem 6.16 of the cited source. No claim about any physical system is made by either clause.

• The quantification in clause 1 is invariant under replacing Euclidean balls by arbitrary neighbourhoods of $z_{\ast }$: if the ball form holds then the neighbourhood form holds by choosing a ball inside the given neighbourhood, and conversely. Requiring $\overline{B_{\epsilon }(z_{\ast })}\subseteq U$ costs no generality, because $\epsilon$ is only a test radius and $U$ is open.

• Whether the definition is non-vacuous for a given $F$ depends on an existence theory for solutions. Under the continuity and local-Lipschitz hypotheses of Picard-Lindelof, every initial value problem has a unique maximal solution (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval), and clause 1 can be read as forbidding finite-time breakdown before any boundary of the ball is reached. No such hypothesis is built into this definition.

• Linear stability and orbital stability are defined separately in Invariant sets, orbital stability of an invariant set, and linear stability of an equilibrium; hyperbolicity, limit cycles and homoclinic trajectories in Hyperbolic equilibria, hyperbolic fixed points, periodic orbits, limit cycles and homoclinic trajectories. The stability of a limit cycle is orbital stability of its image, not Lyapunov stability of any of its points.

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open and let $F:U\to {\mathbb{R}}^n$ be a map; retain the convention of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system that a solution means a solution of $z^{\mathrm{\prime }}(t)=F(z(t))$, read as a system of First-order systems, initial value problems, and solutions on intervals, and that it is defined for all $t\ge 0$ when its interval contains $[0,\mathrm{\infty })$. A maximal solution means one with no proper extension, as defined there; the invariance and stability predicates below quantify over maximal solutions through time $0$, not their arbitrary local restrictions. No existence or uniqueness is asserted for an arbitrary $F$.

Invariant sets. A set $S\subseteq U$ is forward invariant (for this system) when every maximal solution $z$ with $z(0)\in S$ is defined for all $t\ge 0$ and satisfies $z(t)\in S$ for all $t\ge 0$.

Distance to a set. For a nonempty $S\subseteq U$ define

$\mathrm{dist}(z,S):=\inf \{\mathrm{\parallel }z-s{\mathrm{\parallel }}_2:s\in S\}(z\in {\mathbb{R}}^n),$

the infimum of a nonempty set of nonnegative reals that is bounded below, computed for the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Define the Euclidean closure of $S$ by

$\bar{S}:=\{z\in {\mathbb{R}}^n:\text{for every }\epsilon >0\text{ there is }s\in S\text{ with }\mathrm{\parallel }z-s{\mathrm{\parallel }}_2<\epsilon \}\mathrm{.}$

The map $z\mapsto \mathrm{dist}(z,S)$ is nonnegative and vanishes exactly on $\bar{S}$: by the defining property of the infimum, $\mathrm{dist}(z,S)=0$ exactly when every positive-radius Euclidean ball about $z$ meets $S$. When $S=\{z_{\ast }\}$ is a singleton, $\mathrm{dist}(z,S)=\mathrm{\parallel }z-z_{\ast }{\mathrm{\parallel }}_2$.

Orbital stability. Let $S\subseteq U$ be nonempty and forward invariant. Then $S$ is orbitally stable when for every $\epsilon >0$ there is $\delta >0$ such that every maximal solution $z$ with $\mathrm{dist}(z(0),S)<\delta$ is defined for all $t\ge 0$ and satisfies

$\mathrm{dist}(z(t),S)<\epsilon (t\ge 0)\mathrm{.}$

It is asymptotically orbitally stable when it is orbitally stable and there is ${\delta }_0>0$ such that every maximal solution with $\mathrm{dist}(z(0),S)<{\delta }_0$ is defined for all $t\ge 0$ and satisfies $\mathrm{dist}(z(t),S)\to 0$ as $t\to \mathrm{\infty }$.

Linear stability. Let $z_{\ast }\in U$ be an equilibrium of $F$ and suppose $F$ is totally differentiable at $z_{\ast }$ in the sense of The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, with total derivative $DF(z_{\ast }):{\mathbb{R}}^n\to {\mathbb{R}}^n$; write $L:=DF(z_{\ast })$. The linearized system at $z_{\ast }$ is the autonomous linear system

$y^{\mathrm{\prime }}(t)=Ly(t)\mathrm{.}$

The equilibrium $z_{\ast }$ is linearly stable when every maximal solution $y$ of the linearized system (First-order systems, initial value problems, and solutions on intervals) is defined for all $t\ge 0$ and is bounded on $[0,\mathrm{\infty })$, that is, there is a real constant $C\ge 0$ with $\mathrm{\parallel }y(t){\mathrm{\parallel }}_2\le C$ for every $t\ge 0$. It is linearly asymptotically stable when every maximal solution of the linearized system is defined for all $t\ge 0$ and satisfies $\mathrm{\parallel }y(t){\mathrm{\parallel }}_2\to 0$ as $t\to \mathrm{\infty }$.

• Orbital stability of $S$ is a statement about the set $S$, not about any chosen point of it: it controls the distance of a whole solution to $S$ and does not assert that a solution starting near one point of $S$ stays near that point. Lyapunov stability of an equilibrium is the special case $S=\{z_{\ast }\}$; orbital stability of a limit cycle concerns its periodic image $S$ (Hyperbolic equilibria, hyperbolic fixed points, periodic orbits, limit cycles and homoclinic trajectories) when that image is forward invariant in the sense of the Definition. Under the arbitrary-$F$ hypotheses here, a periodic solution alone does not ensure that every maximal solution through its image remains there; forward invariance must be established separately.

• Linear stability is a property of the linear system alone. It is not equivalent to Lyapunov stability of $z_{\ast }$ for the nonlinear system; linearization transfers stability only under additional spectral hypotheses, which this definition does not supply.

• The constant $C$ in the definition of linear stability is not required to be uniform over a family of systems, and no rate of decay is required; linear asymptotic stability is a strictly stronger property than linear stability. Both are defined here for the concrete Euclidean finite-dimensional case only.

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open and let $F:U\to {\mathbb{R}}^n$ be a map; a solution means a solution of $z^{\mathrm{\prime }}(t)=F(z(t))$ as in Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system.

For a real linear map $A:{\mathbb{R}}^n\to {\mathbb{R}}^n$, define its complex-linear extension $A_{\mathbb{C}}:{\mathbb{C}}^n\to {\mathbb{C}}^n$ by $A_{\mathbb{C}}(x+iy):=Ax+iAy$ for $x,y\in {\mathbb{R}}^n$. In both spectral definitions below, an eigenvalue means an eigenvalue of $A_{\mathbb{C}}$ over $\mathbb{C}$ (Eigenvalues, eigenvectors, eigenspaces $E_{\lambda }(T)=\ker (T-\lambda I)$, and the spectrum ${\sigma }_F(T)$ of an endomorphism with field $F=\mathbb{C}$); $\mathrm{Re}\lambda$ and $\mathrm{\mid }\lambda \mathrm{\mid }$ are the real part and modulus of Real and imaginary parts, complex conjugation, and modulus.

Hyperbolic equilibrium. Let $z_{\ast }\in U$ be an equilibrium of $F$ and suppose $F$ is totally differentiable at $z_{\ast }$ (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder) with total derivative $DF(z_{\ast })$. Then $z_{\ast }$ is hyperbolic when no eigenvalue of the real matrix of $DF(z_{\ast })$ has real part $0$ (Eigenvalues, eigenvectors, eigenspaces $E_{\lambda }(T)=\ker (T-\lambda I)$, and the spectrum ${\sigma }_F(T)$ of an endomorphism).

Hyperbolic fixed point of a map. Let $V\subseteq {\mathbb{R}}^n$ be open, let $\phi :V\to V$ be a map and let $p\in V$ satisfy $\phi (p)=p$. Suppose $\phi$ is totally differentiable at $p$. Then $p$ is a hyperbolic fixed point of $\phi$ when no eigenvalue of $D\phi (p)$ has modulus $1$.

Periodic solutions and periodic orbits. A solution $z:\mathbb{R}\to U$ is $T$-periodic for $T>0$ when $z(t+T)=z(t)$ for every $t\in \mathbb{R}$, and periodic when it is $T$-periodic for some $T>0$. It is nonconstant when $z(t_1)\mathrm{\ne }z(t_2)$ for some $t_1,t_2$. The image of a periodic solution is $\{z(t):t\in \mathbb{R}\}$; a subset of $U$ is a periodic orbit when it is the image of a nonconstant periodic solution. A number $T>0$ is a least period of $z$ when $T$ is a period of $z$ and no $T^{\mathrm{\prime }}\in (0,T)$ is a period of $z$.

Limit cycle. A periodic orbit $S\subseteq U$ is a limit cycle when it is isolated: there is an open set $W\subseteq U$ with $S\subseteq W$ such that the image of every nonconstant periodic solution lying in $W$ equals $S$; equivalently, no other periodic orbit of the system is contained in $W$. Stability or attraction of $S$ is a separate property and is not part of the definition.

Homoclinic and heteroclinic trajectories. A solution $z$ defined on all of $\mathbb{R}$ is homoclinic to the equilibrium $z_{\ast }$ when $z(t)\to z_{\ast }$ as $t\to +\mathrm{\infty }$ and as $t\to -\mathrm{\infty }$ and $z$ is not the constant solution $z\equiv z_{\ast }$. It is heteroclinic from $z_{\ast }$ to $z_{\ast \ast }$ when $z_{\ast }\mathrm{\ne }{z}_{\ast \ast }$ are equilibria and $z(t)\to z_{\ast \ast }$ as $t\to +\mathrm{\infty }$ and $z(t)\to z_{\ast }$ as $t\to -\mathrm{\infty }$.

• Hyperbolicity is a spectral condition on the derivative at a single point. It does not by itself assert Lyapunov stability, instability, the existence of local stable or unstable manifolds, or any statement about the nonlinear system; each of those is a theorem about the system under stated hypotheses, not a definitional consequence.

• The qualifier "isolated" in the definition of a limit cycle is taken with respect to the periodic orbits of the same system on an open neighbourhood of the cycle's image. A given periodic orbit may be isolated in this sense while being orbitally unstable; conversely an orbitally stable periodic orbit in a system with a continuum of periodic orbits filling an annulus is not a limit cycle.

• Stability and attraction of an invariant set are defined in Invariant sets, orbital stability of an invariant set, and linear stability of an equilibrium; for a periodic orbit they are separate from the isolation condition used here, and neither is implied by it.

• For an equilibrium $z_{\ast }$, adopt the all-solutions convention when uniqueness is not assumed. For $a\in U$, let $\mathcal{M}(a)$ be the set of maximal solutions through $z(0)=a$, with maximality as in Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system. Define $W^{+}(z_{\ast })$ to consist of those $a$ for which $\mathcal{M}(a)\mathrm{\ne }\mathrm{\varnothing }$ and every $z:J\to U$ in $\mathcal{M}(a)$ has $[0,\mathrm{\infty })\subseteq J$ and $z(t)\to z_{\ast }$ as $t\to +\mathrm{\infty }$. Define $W^{-}(z_{\ast })$ by the same nonemptiness requirement, $(-\mathrm{\infty },0]\subseteq J$ for every such solution, and convergence as $t\to -\mathrm{\infty }$. With a unique maximal solution these reduce to the usual stable and unstable sets; without uniqueness they explicitly require convergence of all trajectories and differ from an existential convention. Calling either set a differentiable manifold requires a stable-manifold theorem with its own hypotheses; no such conclusion is drawn here.

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open, let $P>0$, and let $F:\mathbb{R}\times U\to {\mathbb{R}}^n$ be a map that is continuous and locally Lipschitz in the state variable, in the sense of the hypotheses of Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, and that is $P$-periodic:

$F(t+P,z)=F(t,z)(t\in \mathbb{R},z\in U)\mathrm{.}$

Consider the first-order system $x^{\mathrm{\prime }}(t)=F(t,x(t))$ of First-order systems, initial value problems, and solutions on intervals on the open domain $\mathbb{R}\times U$. For $z\in U$ let

$x_z:J_z⟶U$

be the unique maximal solution of the initial value problem $x^{\mathrm{\prime }}(t)=F(t,x(t))$, $x(0)=z$; here $J_z$ is the open maximal interval of existence containing $0$, and by Every Picard–Lindelöf initial value problem has one maximal solution on an open interval every solution of this initial value problem is a restriction of $x_z$ (Extensions, maximal solutions, maximal intervals, and global solutions, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

The domain of the stroboscopic map is

$D_P:=\{z\in U:[0,P]\subseteq J_z\},$

the set of initial states whose solution exists over one full period. The stroboscopic map (or time-$P$ map) of the system is the function

${\theta }_P:D_P⟶U,{\theta }_P(z):=x_z(P)\mathrm{.}$

This is a well-defined function: for $z\in D_P$ the interval $J_z$ contains $P$, so $x_z(P)$ is a point of $U$, and by the uniqueness clause of the maximal-solution theorem the value $x_z(P)$ does not depend on any choice, since $x_z$ is the only maximal solution through $(0,z)$.

• The definition depends on the chosen initial time $0$. Replacing the initial time by $t_0$ and defining ${\theta }_P^{t_0}(z):=x_{t_0,z}(t_0+P)$ on the corresponding domain gives the same family of maps up to the time-translation flow of the $P$-periodic system; only the initial time $0$ is used on this page.

• The domain $D_P$ is the set where the solution exists up to time $P$; it need not be invariant under ${\theta }_P$ or equal all of $U$. Openness of this domain is not proved here. Formal iterates ${\theta }_P^k$ are defined only on the subset of points for which each successive value remains in $D_P$; the precise statement is Iterates of the stroboscopic map, and equilibria as fixed points.

• If $F$ does not depend on $t$ (the autonomous case), then ${\theta }_P$ is the time-$P$ map of the autonomous system and every equilibrium $z_{\ast }$ of the autonomous right-hand side satisfies ${\theta }_P(z_{\ast })=z_{\ast }$. A fixed point $z\in D_P$ yields a $P$-periodic solution: repeat $x_z{\mathrm{\mid }}_{[0,P]}$ on every interval $[kP,(k+1)P]$, $k\in \mathbb{Z}$. Values at each joining point equal $z$, and the one-sided derivatives agree because $F(P,z)=F(0,z)$. Periodicity of $F$ makes each segment solve the same equation; hence the repeated curve is a solution on $\mathbb{R}$, and uniqueness identifies it with $x_z$. Conversely a $P$-periodic solution through $z$ gives ${\theta }_P(z)=z$. Constant solutions are included, so a fixed point alone does not imply a nonconstant periodic orbit.

• The map ${\theta }_P$ can be studied as a discrete-time map. Applying the hyperbolic-fixed-point definition of Hyperbolic equilibria, hyperbolic fixed points, periodic orbits, limit cycles and homoclinic trajectories requires a differentiable self-map of an open neighbourhood of the fixed point; such a restriction is not supplied here. No relation to a continuous-time stability criterion is asserted here.

Retain all notation and hypotheses of The stroboscopic (time-$P$) map of a $P$-periodic first-order system: $U\subseteq {\mathbb{R}}^n$ is open, $P>0$, and $F:\mathbb{R}\times U\to {\mathbb{R}}^n$ is continuous, locally Lipschitz in the state variable, and $P$-periodic.

1. If $z\in D_P$ and ${\theta }_P(z)\in D_P$, then $[0,2P]\subseteq J_z$ and

${\theta }_P({\theta }_P(z))=x_z(2P)\mathrm{.}$

2. If $z_{\ast }\in U$ satisfies $F(t,z_{\ast })=0$ for every $t\in \mathbb{R}$, then $z_{\ast }\in D_P$ and ${\theta }_P(z_{\ast })=z_{\ast }$.

• Clause 1 is the precise sense in which iterating a stroboscopic map follows the continuous solution across successive periods: each iterate requires the previous value to lie in $D_P$ again, and then the iterate equals the value of the same maximal solution at the corresponding multiple of $P$. The identity extends by induction to $k$-fold iterates as long as every intermediate value lies in $D_P$.

• No claim is made that $D_P$ is nonempty, open, or forward invariant, or that ${\theta }_P$ has periodic points. Clause 2 shows only that every equilibrium of the $P$-periodic vector field gives a fixed point of ${\theta }_P$ with domain membership.

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 $q$.

Let $N\ge 1$, let $Q$ be a smooth manifold, let $X:Q\to {\mathbb{R}}^{3N}$ be smooth, let $m_1,\dots ,m_N>0$ and let $G_q$ be the mass metric of Constrained configurations, the mass matrix and the mass metric.

1. For every $q\in Q$ the map $G_q$ is a symmetric bilinear form on $T_{q}Q$.
2. $G_q(u,u)={\sum }_{i=1}^N{m}_i\mathrm{\parallel }d{X}_q(u{)}_i{\mathrm{\parallel }}_2^2\ge 0$ for every $u\in T_{q}Q$, so $G_q$ is positive semidefinite.
3. $G_q$ is positive definite (Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form) if and only if $d{X}_q$ is injective. In particular, if $X$ is an immersion (Immersions, submersions, and constant-rank maps) then $G_q$ is positive definite for every $q$.
4. If $b_1,\dots ,b_n$ is an ordered basis of $T_{q}Q$ and $A$ is the matrix whose $j$-th column is the coordinate vector of $d{X}_q(b_j)$ in the standard basis of ${\mathbb{R}}^{3N}$, then the matrix of the bilinear form $G_q$ in this basis is $A^{\mathsf{T}}MA$, with $M$ the mass matrix.

A material region is a nonempty, bounded, open subset $B\subseteq {\mathbb{R}}^3$. This region predicate is choice-free and does not assume a density or measure.

For the following Lebesgue density and mass construction, assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)): Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume supplies the measure ${\lambda }_3$ on its Lebesgue sigma-algebra (Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, Measures on sigma-algebras). Since $B$ is open, it is Lebesgue measurable by the Borel-measurability theorem (Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable). Put ${\mathcal{L}}_B:=\{E\cap B:E\in \mathcal{L}({\mathbb{R}}^3)\}$ and ${\lambda }_B(C):={\lambda }_3(C)$ for $C\in {\mathcal{L}}_B$. The trace family ${\mathcal{L}}_B$ is a sigma-algebra on $B$ (Sigma-algebras); countable additivity of ${\lambda }_B$ follows by applying that of ${\lambda }_3$ to disjoint trace-measurable sets, so $(B,{\mathcal{L}}_B,{\lambda }_B)$ is a measure space (Measures on sigma-algebras). In the displays below, ${\int }_B\rho d{\lambda }_3$ means the Lebesgue integral of $\rho$ on this measure space (Integrable real and complex functions, and their integrals). A density on $B$ is a measurable function

$\rho :B⟶[0,\mathrm{\infty })$

measurable as a map from $(B,{\mathcal{L}}_B)$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Measurable spaces and measurable sets, A measurable function between measurable spaces, The Borel sigma-algebra of a topological space) and integrable on $(B,{\mathcal{L}}_B,{\lambda }_B)$; the mass of the pair $(B,\rho )$ is

$\mathrm{mass}(B,\rho ):={\int }_B\rho d{\lambda }_3\in [0,\mathrm{\infty })\mathrm{.}$

For this scalar function, bounded means that some $M<\mathrm{\infty }$ satisfies $0\le \rho (X)\le M$ for every $X\in B$. Continuous means that for every $X\in B$ and every $\epsilon >0$ there is $\delta >0$ such that $Y\in B$ and $\mathrm{\parallel }X-Y{\mathrm{\parallel }}_2<\delta$ imply $\mathrm{\mid }\rho (X)-\rho (Y)\mathrm{\mid }<\epsilon$. For $k\ge 1$, of class $C^k$ on $B$ means that every iterated coordinate partial derivative of order at most $k$ exists on $B$ and is continuous there; no extension to $\mathrm{\partial }B$ is required unless a consuming statement says so. A measured material region is a triple $(B,\rho ,m)$ with $m=\mathrm{mass}(B,\rho )$ recorded.

• Only measurability and integrability on the restricted Lebesgue measure space $(B,{\mathcal{L}}_B,{\lambda }_B)$, together with the displayed nonnegative codomain, are part of the density definition. A density is not required to be positive, bounded, continuous, or supported away from $\mathrm{\partial }B$, and the definition asserts no conservation law: the mass is a number attached to $(B,\rho )$.

• Boundary regularity is a declared hypothesis, not part of the word "region": when a statement needs it, it says that the boundary $\mathrm{\partial }B$ is a $C^1$ embedded surface (or a smooth/rectifiable hypersurface) and states the surface-measure conventions used. The local transport and balance items of this collection state their own boundary hypotheses; none is inherited from this definition.

• The unit of $\rho$ is mass per unit volume only after a unit system is fixed. The mathematical object here is a function and a number; no physical identification of the number $m$ with a measured mass is made, and no empirical claim is attached.

• The statement "$\rho$ is mass per volume" is the interpretation convention $m={\int }_B\rho d{\lambda }_3$ recorded above; it is a definition of the interpretation, not a physical law. In particular, if $\rho$ is a summable function with a discontinuity across a surface, the integral above is still the Lebesgue integral and no interfacial mass is implied unless it is separately declared.

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be a spatial set and let $I\subseteq \mathbb{R}$ be a time interval. No openness of $\mathrm{\Omega }$ or regularity of a traction map is assumed by this definition; consumers impose any needed additional hypotheses. Let $S^2:=\{n\in {\mathbb{R}}^3:\mathrm{\parallel }n{\mathrm{\parallel }}_2=1\}$ be the unit sphere of Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$.

A traction field is a map

$t:S^2⟶{\mathbb{R}}^3,$

read with the following convention: $t(n)$ is the force per unit area exerted by the material on the side towards which $n$ points upon the material on the other side, at the point of a surface whose unit normal is $n$. The dependence of $t$ on position and time is suppressed in this notation; a traction field on a set $\mathrm{\Omega }\times I$ is a map $t:\mathrm{\Omega }\times I\times S^2\to {\mathbb{R}}^3$, and all statements below apply to each fixed $(y,s)$.

For $v\in {\mathbb{R}}^3$ use one-based component labels $v_i:=v(i-1)$, and for $\sigma \in M_3(\mathbb{R})$ use ${\sigma }_{ij}:=\sigma (i-1,j-1)$, where $1\le i,j\le 3$; $e_i$ is the vector with value $1$ at $i-1$ and $0$ at the other two indices. Define the matrix-vector action $\sigma v\in {\mathbb{R}}^3$ by $(\sigma v)(k):={\sum }_{j<3}\sigma (k,j)v(j)$ for $0\le k<3$, equivalently $(\sigma v{)}_i={\sum }_{j=1}^3{\sigma }_{ij}{v}_j$. It is the rectangular product of Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes after identifying $v$ with the column matrix $C_v(j,0):=v(j)$ and the result with its sole column.