Constant-magnetic-field slow test-orbit comparison

Put $B_z=\sqrt{ϵ}c{b}_0\mathrm{/}(\alpha L)$ and supply the real one-form $A=(B_z\mathrm{/}2)(x^{1}d{x}^2-x^{2}d{x}^1)$ on all of Minkowski space. Its coefficients are smooth and have SI units tesla metres, and direct differentiation gives $dA=B_{z}d{x}^1\wedge d{x}^2=F$, with $(dA{)}_{0i}=0$ and $(dA{)}_{12}=B_z$. Thus the local-potential hypothesis of [F6] is met on every finite curve segment, and the action model gives its proper-time Lorentz law throughout the Example's displayed range, independently of the general small-cutoff comparison in [F1]. In flat coordinates this law has $U^0=\gamma c$, $U^i=\gamma w^i$, and $\mathrm{\nabla }=\mathrm{\partial }$. With $E=0$ and $F_{ij}={ϵ}_{ijk}{B}_k$, its spatial components reduce to $d(\gamma m\mathbf{w})\mathrm{/}dt=q\mathbf{w}\times \mathbf{B}$. Its dot product with $\mathbf{w}$ is zero, so $\mathrm{\mid }\mathbf{w}\mathrm{\mid }$ and $\gamma =(1-\mathrm{\mid }\mathbf{w}{\mathrm{\mid }}^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}=(1-ϵ\mathrm{\mid }{\mathbf{W}}_0{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}$ are constant. Since ${\mathbf{W}}_0$ is transverse to ${\mathbf{e}}_z$, the motion stays in that transverse plane. Define $J\mathbf{v}:=\mathbf{v}\times b_0{\mathbf{e}}_z$ there; $J$ is skew-symmetric and $J^2=-b_0^{2}I$. Identify the transverse plane with the real Euclidean normed space ${\mathbb{R}}^2$ using the ordered coordinate basis $({\mathbf{e}}_x,{\mathbf{e}}_y)$; it is Banach by [F3], and $\mathrm{\mid }Jv\mathrm{\mid }=\mathrm{\mid }{b}_0\mathrm{\mid }\mathrm{\mid }v\mathrm{\mid }$ proves that $J$ is bounded with $\mathrm{\parallel }J{\mathrm{\parallel }}_{2\to 2}=\mathrm{\mid }{b}_0\mathrm{\mid }$. Define $E(t):={\sum }_{n=0}^{\mathrm{\infty }}{t}^n{J}^n\mathrm{/}n!$ by [F3], and write $e^{tJ}:=E(t)$. Rescaling the exact force equation gives ${\mathbf{W}}^{\mathrm{\prime }}=J\mathbf{W}\mathrm{/}\gamma$ and ${\mathbf{X}}^{\mathrm{\prime }}=\mathbf{W}$. The formula $\mathbf{W}(S)=E(S\mathrm{/}\gamma ){\mathbf{W}}_0$ has the matched initial velocity and satisfies the velocity equation by the derivative law in [F3]; the Newton–Lorentz comparison is ${\mathbf{W}}_N(S)=E(S){\mathbf{W}}_0$. For either constant generator $aJ$ with $a={\gamma }^{-1}$ or $a=1$, any solution with this initial value is unique: differentiating $E(-aS)\mathbf{W}(S)$ gives zero, so [F4] makes it constant and multiplication by $E(aS)$ recovers the displayed formula.

By [F3], $Q(t):=E(t{)}^{T}E(t)$ is smooth and $Q^{\mathrm{\prime }}(t)=E(t{)}^T(J^T+J)E(t)=0$, with $Q(0)=I$. Clause 2 of [F4], applied to its four entries as a real vector on any compact interval, gives $Q(t)=I$, so every $E(t)$ preserves the Euclidean norm. For fixed $S>0$, put $r={\gamma }^{-1}$ and $P(s):=E(rs)E(S-s)$ on $[0,S]$. The semigroup and derivative laws in [F3] imply that these factors commute with $J$ and $P^{\mathrm{\prime }}(s)=(r-1)JE(rs)E(S-s)$. Its derivative is continuous, so the componentwise integral identity in [F4] gives $E(rS)-E(S)={\int }_0^S(r-1)JE(rs)E(S-s)ds$. Apply this identity to ${\mathbf{W}}_0$ and then [F5]. Both exponential factors preserve norm, so $\mathrm{\mid }\mathbf{W}-{\mathbf{W}}_N\mathrm{\mid }\le S\mathrm{\mid }{b}_0\mathrm{\mid }\mathrm{\mid }1-{\gamma }^{-1}\mathrm{\mid }\mathrm{\mid }{\mathbf{W}}_0\mathrm{\mid }$. At $S=0$ the difference and integral vanish directly. The exact identity $1-{\gamma }^{-1}=ϵ\mathrm{\mid }{\mathbf{W}}_0{\mathrm{\mid }}^2\mathrm{/}(1+\sqrt{1-ϵ\mathrm{\mid }{\mathbf{W}}_0{\mathrm{\mid }}^2})\le ϵ\mathrm{\mid }{\mathbf{W}}_0{\mathrm{\mid }}^2$ proves the velocity bound throughout the displayed parameter range, including ${\mathbf{W}}_0=0$.

Define $J^{-1}=-J\mathrm{/}{b}_0^2$ and the positions explicitly by $\mathbf{X}(S)=\mathbf{X}(0)+\gamma J^{-1}(E(S\mathrm{/}\gamma )-I){\mathbf{W}}_0$ and ${\mathbf{X}}_N(S)=\mathbf{X}(0)+J^{-1}(E(S)-I){\mathbf{W}}_0$. Their derivatives are the respective velocities by [F3], and their initial positions match. Clause 1 of [F4] and [F5] give $\mathrm{\mid }\mathbf{X}(S)-{\mathbf{X}}_N(S)\mathrm{\mid }\le {\int }_0^S\mathrm{\mid }\mathbf{W}(s)-{\mathbf{W}}_N(s)\mathrm{\mid }ds\le \mathrm{\mid }{b}_0\mathrm{\mid }ϵ\mathrm{\mid }{\mathbf{W}}_0{\mathrm{\mid }}^3{\int }_0^{S}sds=\frac{1}{2}\mathrm{\mid }{b}_0\mathrm{\mid }ϵ\mathrm{\mid }{\mathbf{W}}_0{\mathrm{\mid }}^3{S}^2$, with $S=0$ immediate. The smooth orthogonal flows exist for all real $S$ and give $\mathrm{\mid }\mathbf{W}\mathrm{\mid }=\mathrm{\mid }{\mathbf{W}}_N\mathrm{\mid }=\mathrm{\mid }{\mathbf{W}}_0\mathrm{\mid }$ and $\mathrm{\mid }\mathbf{X}-\mathbf{X}(0)\mathrm{\mid },\mathrm{\mid }{\mathbf{X}}_N-{\mathbf{X}}_N(0)\mathrm{\mid }\le S_{\ast }\mathrm{\mid }{\mathbf{W}}_0\mathrm{\mid }$ on the stated finite interval, hence common compact containment. Their constant speed is strictly below $c$ because $ϵ\mathrm{\mid }{\mathbf{W}}_0{\mathrm{\mid }}^2\le 1\mathrm{/}2$, and setting proper time $\tau =t\mathrm{/}\gamma$ reconstructs the future-directed proper-time test worldline satisfying the force law used in step 1.1. These direct calculations establish the displayed example throughout its stated range; for sufficiently small $ϵ$ they also lie within the general theorem's hypotheses [F1]. The metric and field remain prescribed inputs by [F2], with no coupled-background equation.