The uniform-force model: explicit global motion of a label under a constant declared field, with uniqueness, dimensions and frame covariance

The map $F_i(t,x,v)=m_i\mathbf{g}$ is constant on $D=\mathbb{R}\times {\mathbb{R}}^{3N}\times {\mathbb{R}}^{3N}$, hence continuous there; and for every pair of states $(t,z),(t,z^{\mathrm{\prime }})$ one has $\parallel F(t,z)-F(t,z^{\mathrm{\prime }}){\parallel }_2=0\le 0\cdot \parallel z-z^{\mathrm{\prime }}{\parallel }_2$, so the local-Lipschitz requirement of clause (iii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance holds on every compact cylinder with constant $L=0$, in particular with a constant not depending on time; this proves claim 1.

Differentiate the displayed curve: by [L2] applied to each label's three components, ${\dot{x}}_i(t)=v_{0,i}+\mathbf{g}t$ and ${\ddot{x}}_i(t)=\mathbf{g}$ for every $t\in \mathbb{R}$; hence $m_i{\ddot{x}}_i(t)=m_i\mathbf{g}=F_i(t,x(t),\dot{x}(t))$ for every $i$, and $x(t)\in Q={\mathbb{R}}^{3N}$, $t\in I_0=\mathbb{R}$; the initial values $x(0)=x_0$ and $\dot{x}(0)=v_0$ are immediate. By the equivalence of clause (c) of The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, $x$ is a motion of the model on $\mathbb{R}$.

Dimensions: $[\mathbf{g}]=\mathrm{\ell }{\tau }^{-2}$, so $[\frac{1}{2}\mathbf{g}{t}^2]=\mathrm{\ell }{\tau }^{-2}\cdot {\tau }^2=\mathrm{\ell }$, matching the position dimension, and $[m_i\mathbf{g}]=\mu \mathrm{\ell }{\tau }^{-2}$, matching the force dimension of clause (iii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance; $[\dot{x}]=\mathrm{\ell }{\tau }^{-1}$ and $[\ddot{x}]=\mathrm{\ell }{\tau }^{-2}$ by the derivative rule of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework; this proves claim 3.

The datum $(0,x_0,v_0)$ has exactly one maximal motion by Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder clause 4, and every motion with this datum is a restriction of it; since the motion of step 1.2 is defined on all of $\mathbb{R}=I_0$, it is itself maximal (any extension would have to agree with it where both are defined, so no strictly larger interval is possible), and it is the unique maximal motion; this proves claim 2, including existence on the whole time window. The alternatives of Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain are not needed here, but the agreement is consistent with its clause 3: global existence follows from the explicit formula without a compact-trapping assumption; for stationary data ($\mathbf{g}=0$ and $v_0=0$) the state is in fact compactly trapped.

Covariance: let a second inertial frame be related to the chosen one by $t^{\mathrm{\prime }}=t+b$, $x^{\mathrm{\prime }}=Rx+vt+a_0$ with $R\in SO(3)$, and let $\eta$ be the description of the motion of step 1.2 in that frame; by [L3], ${\eta }_i^{\mathrm{\prime }\mathrm{\prime }}(t^{\mathrm{\prime }})=R{\ddot{x}}_i(t^{\mathrm{\prime }}-b)=R\mathbf{g}$ for every $i$, so the second frame sees the same equation of motion with the constant vector $R\mathbf{g}$ in place of $\mathbf{g}$; hence the class of uniform-force models is covariant under changes of inertial frame, with the parameter transforming as a spatial vector under the rotation and unaffected by the translation, boost or time shift; this proves claim 4.