The associated first-order system, motions, maximal motions and the Newtonian flow of a force model

Let a force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance be given: time window $I_0\subseteq \mathbb{R}$ nondegenerate and open, configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ open (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator clause (a)), fixed masses $m_1,\dots ,m_N>0$ (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)), and a force map $F$ with the standing continuity and state-local-Lipschitz hypotheses of clause (iii) of that postulate.

(a) State domain and inverse-mass operator. The state domain of the model is

$D:=I_0\times Q\times ({\mathbb{R}}^3{)}^N\subseteq \mathbb{R}\times {\mathbb{R}}^{3N}\times {\mathbb{R}}^{3N},$

an open subset of $\mathbb{R}\times {\mathbb{R}}^{6N}$; a point $(t,x,v)\in D$ is a time together with the state $z=(x,v)$ of positions $x\in Q$ and coordinate velocities $v\in ({\mathbb{R}}^3{)}^N$ of the labels (Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The inverse-mass operator is the linear map

${\mathsf{M}}^{-1}:{\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N},{\mathsf{M}}^{-1}(u_1,\dots ,u_N):=(\frac{u_1}{m_1},\dots ,\frac{u_N}{m_N}),$

the inverse of the mass operator $\mathsf{M}$ of clause (d) of Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator.

(b) Associated first-order system. The associated first-order field of the model is

$\mathcal{F}:D⟶{\mathbb{R}}^{3N}\times {\mathbb{R}}^{3N},\mathcal{F}(t,x,v):=(v,{\mathsf{M}}^{-1}F(t,x,v)),$

and the associated first-order system is $z^{\mathrm{\prime }}=\mathcal{F}(t,z)$ on the open domain $D$, in the sense of First-order systems, initial value problems, and solutions on intervals; here $z=(x,v)\in {\mathbb{R}}^{6N}$. The map $\mathcal{F}$ satisfies the Picard–Lindelöf hypotheses of Picard-Lindelöf local existence and uniqueness for first-order systems: it is continuous on $D$, and it is locally Lipschitz in the state variable $z$ locally uniformly in time, because its first component $z\mapsto v$ is linear and its second is the composition of the force map $F$ — which is locally Lipschitz in $(x,v)$ locally uniformly in time by Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iii) — with the linear map ${\mathsf{M}}^{-1}$.

(c) Motions. Let $J\subseteq \mathbb{R}$ be an interval with nonempty interior, endpoints read with one-sided derivatives as in First-order systems, initial value problems, and solutions on intervals. A motion of the model on $J$ is a curve $x:J\to Q$ of class $C^2$ such that $t\in I_0$ for every $t\in J$ and

$m_i{\ddot{x}}_i(t)=F_i(t,x(t),\dot{x}(t))(i=1,\dots ,N,t\in J)\mathrm{.}$

Equivalently: the state curve $z=(x,\dot{x}):J\to Q\times {\mathbb{R}}^{3N}$ is a solution of the associated first-order system on $J$. Indeed, if $x$ is a motion then $z=(x,x^{\mathrm{\prime }})$ is $C^1$ with $z^{\mathrm{\prime }}=(x^{\mathrm{\prime }},x^{\mathrm{\prime }\mathrm{\prime }})=(x^{\mathrm{\prime }},{\mathsf{M}}^{-1}F(t,x,x^{\mathrm{\prime }}))=\mathcal{F}(t,z)$ and $(t,z(t))\in D$; conversely, if $z=(x,v)$ is a solution with $(t,z(t))\in D$ for all $t$, then $x^{\mathrm{\prime }}=v$ and $v^{\mathrm{\prime }}={\mathsf{M}}^{-1}F(t,x,v)$, so $v$ is $C^1$, $x$ is $C^2$, and multiplying the second equation by the masses returns the displayed second-order equation. Motions of the labels of the model are exactly these curves by Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iv).

(d) Initial data and maximal motion. An initial datum of the model is a triple $(t_0,x_0,v_0)$ with $t_0\in I_0$, $x_0\in Q$ and $v_0\in ({\mathbb{R}}^3{)}^N$; write $z_0:=(x_0,v_0)$. A maximal motion with this datum is a motion $x:J_{\max }\to Q$ with $x(t_0)=x_0$ and $\dot{x}(t_0)=v_0$ that has no extension to a motion of the model on a strictly larger interval. By the Picard–Lindelöf hypotheses verified in (b) and the maximal-solution theorem Every Picard–Lindelöf initial value problem has one maximal solution on an open interval (with Extensions, maximal solutions, maximal intervals, and global solutions): every initial datum has exactly one maximal motion, its maximal interval $J_{\max }$ is an open interval containing $t_0$ and contained in $I_0$, and every motion with the same datum is the restriction of the maximal one. The restriction statement transfers from the associated first-order system to motions through the equivalence in (c).

(e) Newtonian flow. The Newtonian flow of the model is the map

$\mathrm{\Phi }:\mathrm{\Omega }⟶Q\times {\mathbb{R}}^{3N},\mathrm{\Omega }:=\{(t,t_0,z_0):t_0\in I_0,z_0\in Q\times {\mathbb{R}}^{3N},t\in J_{\max }(t_0,z_0)\},$

defined by $\mathrm{\Phi }(t;t_0,z_0):=z(t)$, where $t\mapsto z(t)=(x(t),\dot{x}(t))$ is the state curve of the maximal motion with datum $(t_0,z_0)$. By (d) the assignment is well defined: the maximal motion is unique, so $\mathrm{\Phi }$ is a function on $\mathrm{\Omega }$ and not merely a relation. For fixed $t_0$ and $z_0$ the curve $t\mapsto \mathrm{\Phi }(t;t_0,z_0)$ is defined exactly on the maximal interval $J_{\max }(t_0,z_0)$.

• The flow is a solution family, not a completeness claim. Whether $J_{\max }(t_0,z_0)$ is all of $I_0$ is a property of the particular force map and datum, not part of this definition; the criterion for a finite endpoint is the subject of Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain.

• Regularity of the flow is a separate question. This definition asserts no continuity, differentiability or measurability of $\mathrm{\Phi }$ in any variable; such statements would be separate results with their own hypotheses on the differentiability of $F$.

• Frame covariance. A motion in the sense of (c) is defined with respect to one fixed inertial frame; the transformation of the description between inertial frames and of the force map is clause (vi) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, and under $t^{\mathrm{\prime }}=t+b$ the maximal interval becomes $J_{\max }+b$. Maximality is preserved on the transferred domain: any proper extension there would pull back by the inverse affine transformation to a proper extension of the original motion.