= Solution
Use the <Minkowski metric> with signature $(-,+,+,+)$. The source point is the intersection of the worldline with the observation event's past <light cone>: $R^\mu R_\mu=0$ and $R^0>0$. In components this is $c(t-t_r)=|x-y(t_r)|$ with $t_r<t$. For a timelike worldline the retarded solution is unique in the usual complete-source setting.
Writing $R=|x-y(t_r)|$, $n=(x-y)/R$ and $u^\mu=\gamma(c,v)$ gives $R\cdot u=-\gamma cR(1-n\cdot v/c)$. Thus the <Liénard–Wiechert potentials> are $\phi=q\mu_0c^2/[4\pi R(1-n\cdot v/c)]$ and $A=q\mu_0v/[4\pi R(1-n\cdot v/c)]$. At leading order in $1/r$ and first order in $v/c$, set $R\simeq r,n\simeq\hat r$ and expand the denominator:
$$
\phi\simeq\frac{q\mu_0c}{4\pi r}(c+\hat r\cdot v(t_r)),\qquad A\simeq\frac{q\mu_0}{4\pi r}v(t_r).
$$
Now use $t_-=t-r/c$ and let $a=\dot v(t_-)$. Derivatives of $1/r$ and $\hat r$ give near-field terms of order $r^{-2}$; the radiation terms come from $\partial_t t_-=1$ and $\nabla t_-=-\hat r/c$. The <electric field> and <magnetic field> are therefore
$$
\boxed{E=\frac{q\mu_0}{4\pi r}[\hat r(\hat r\cdot a)-a],\qquad B=-\frac{q\mu_0}{4\pi cr}\hat r\times a=\frac1c\hat r\times E.}
$$
They are transverse to the observation direction. The <Poynting vector> is
$$
\boxed{\frac1{\mu_0}E\times B=\frac{q^2\mu_0}{16\pi^2cr^2}\hat r\,|\hat r\times a|^2.}
$$
Integrating over a sphere uses $\int\sin^2\theta\,d\Omega=8\pi/3$, yielding the nonrelativistic <Larmor formula> $P=q^2\mu_0|a|^2/(6\pi c)$. For <simple harmonic motion>, $|a|^2=A^2\omega^4\cos^2(\omega t_-)$, whose period average is $A^2\omega^4/2$. Multiplication by the period $2\pi/\omega$ gives
$$
\boxed{E_{\rm period}=q^2\mu_0A^2\omega^3/(6c).}
$$
This is the leading radiation energy in the nonrelativistic and far-field approximation.
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