Harmonic forcing gives an exact error criterion for reduction of order

Use the adopted equation $m(a-\tau \dot{a})=f$ with positive constants $m>0$ and $\tau >0$, of units $\mathrm{k}\mathrm{g}$ and $\mathrm{s}$, respectively, and the laboratory conventions of the bounded-future branch theorem.

A phasor coefficient at frequency $\omega \in \mathbb{R}$ is a complex vector $g_0=g_R+i{g}_I\in {\mathbb{C}}^3$, with $g_R,g_I\in {\mathbb{R}}^3$, representing a real map $g:\mathbb{R}\to {\mathbb{R}}^3$ by the real-part representation

$g(t)=\mathrm{Re}\text{\hspace{0.17em}⁣}(g_0\exp (-i\omega t))=g_R\cos (\omega t)+g_I\sin (\omega t)\mathrm{.}$

At $\omega =0$ this representation determines only $g_R$. Adopt the canonical static convention $g_0=G+i0$ for the constant function $g(t)\equiv G\in {\mathbb{R}}^3$; in particular the static force below has $v=0$ and $f_0=u+i0$. Static acceleration phasors use the same convention.

Let $\omega \in \mathbb{R}$ and $f_0=u+iv\in {\mathbb{C}}^3$, with $u,v\in {\mathbb{R}}^3$. For any $z=u+iv\in {\mathbb{C}}^3$, define $\mathrm{\mid }z\mathrm{\mid }:=({\sum }_{j=0}^2\mathrm{\mid }{u}_j{\mathrm{\mid }}^2+{\sum }_{j=0}^2\mathrm{\mid }{v}_j{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, the Euclidean norm of its six real coordinates. The force amplitude $f_0$ has units N, $\omega$ has units ${\mathrm{s}}^{-1}$, and acceleration phasors have units $\mathrm{m}{\mathrm{s}}^{-2}$. Prescribe the real force

$f(t):=\mathrm{Re}\text{\hspace{0.17em}⁣}(f_0\exp (-i\omega t)),$

where real part is taken componentwise. The bounded-future acceleration has complex phasor amplitude

$a_{\ast ,0}=\frac{f_0}{m(1+i\omega \tau )}\mathrm{.}$

The first-order reduction-of-order acceleration $a_{\mathrm{R}\mathrm{O}}=(f+\tau f^{\mathrm{\prime }})\mathrm{/}m$ has phasor amplitude $a_{\mathrm{R}\mathrm{O},0}=f_0(1-i\omega \tau )\mathrm{/}m$. For $f_0\mathrm{\ne }0$, the absolute amplitude error relative to the force acceleration scale $\mathrm{\mid }{f}_0\mathrm{\mid }\mathrm{/}m$ is

$\frac{\mathrm{\mid }{a}_{\ast ,0}-a_{\mathrm{R}\mathrm{O},0}\mathrm{\mid }}{\mathrm{\mid }{f}_0\mathrm{\mid }\mathrm{/}m}=\frac{(\omega \tau {)}^2}{\sqrt{1+(\omega \tau {)}^2}}\mathrm{.}$

For any chosen tolerance $0<ϵ<1$, this relative amplitude error is at most $ϵ$ exactly when

$\mathrm{\mid }\omega \tau \mathrm{\mid }\le \sqrt{\frac{{ϵ}^2+\sqrt{{ϵ}^4+4{ϵ}^2}}{2}}\mathrm{.}$

Hence small error at a small prescribed tolerance requires small $\mathrm{\mid }\omega \tau \mathrm{\mid }$; no universal numerical cutoff or experimental frequency response is asserted.

Runaway and bounded-future branches of the Abraham–Lorentz equation

First-order reduction of order with an explicit residual and error bound

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Real and imaginary parts, complex conjugation, and modulus

$\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$

The derivatives of sine and cosine are cosine and minus sine