Energy paradox in classical electrodynamics? Consider two massive charged objects at rest with a large horizontal distance $d$ between them (object $1$: mass $m_1$, charge $q_1$ and object $2$: mass $m_2$, charge $q_2$).
I apply a constant vertical force $\vec{f_1}$ upwards to object $1$ so that it gains an acceleration $\vec{a_1}=\vec{f_1}/m_1$.
The total amount of power $P_1$ that object $1$ radiates is given by the Larmor formula:
$$P_1=\frac{2}{3}\frac{q_1^2 a_1^2}{4\pi\epsilon_0c^3}.\tag{1}$$
Now assume that object $2$ is constrained to move only in the vertical direction. If the horizontal distance $d$ between the objects is large then only the "radiative" part of the Lienard-Wiechert electric field due to object $1$ can do any work on object $2$. The vertical force $\vec{f_2}$ acting on object $2$ is given by:
$$\vec{f_2}=-\frac{q_1q_2}{4\pi\epsilon_0c^2d}\vec{a_1}.\tag{2}$$
The power received by object $2$, $P_2$, is given by:
$$P_2=\vec{f_2}\cdot\vec{v_2}.\tag{3}$$
The equation of motion of object $2$ is given by:
$$m_2 \frac{d\vec{v_2}}{dt}=\vec{f_2}.\tag{4}$$
As the vertical force $\vec{f_2}$ is constant and the object $2$ is initially at rest then integrating Eqn.(4) gives:
$$\vec{v_2}=\frac{t\vec{f_2}}{m_2}.\tag{5}$$
Substituting Eqn.(2) and Eqn.(5) into Eqn.(3) we find that the power $P_2$ received by object $2$ is given by
$$P_2=\Big(\frac{q_1q_2}{4\pi\epsilon_0c^2d}\Big)^2\frac{a_1^2t}{m_2}.\tag{6}$$
Finally, the ratio of the power received by object $2$, $P_2$, to the power emitted by object $1$, $P_1$, is given by
$$\frac{P_2}{P_1}=\frac{3}{2}\frac{q_2^2t}{4\pi\epsilon_0cd^2m_2}.\tag{7}$$
Thus eventually object $2$ receives more power than the total power emitted by object $1$.
What's gone wrong? :)
Additional calculation
I've put some numbers into the equations in such a way that the final velocities $v_1$, $v_2$ remain non-relativistic and only the radiative part of the Lienard-Weichert field from object $1$ to object $2$ is significant:
$$
\begin{eqnarray*}
a_1&=&1.0\times10^{10}\ \hbox{m/s^2}\\
q_1&=&1.0\ \hbox{C}\\
q_2&=&1.0\times10^{10}\ \hbox{C}\\
m_2&=&1.0\ \hbox{kg}\\
\Delta t&=&2.5\times10^{-5}\ \hbox{s}\\
d&=&7.5\times10^{6}\ \hbox{m}\\
c\Delta t/d&=&1.0\times10^{-3}\\
v_1/c&=&8.3\times10^{-4}\\
v_2/c&=&1.1\times10^{-7}\\
\end{eqnarray*}
$$
The energy $E_1$ radiated by object $1$ from $t=0$ to $t=\Delta t$ is:
$$E_1=5.5\times10^{-1}\ \hbox{Joules}$$
The energy $E_2$ received by object $2$ from $t=d/c$ to $t=d/c+\Delta t$ is:
$$E_2=5.5\times10^{2}\ \hbox{Joules}$$
Therefore object $2$ receives a thousand times more energy than was emitted by object $1$.
Unphysical assumption
The electrostatic energy of body 2 must be at least $q_2^2/4\pi\epsilon_0d=10^{23}$ Joules. This is greater than the rest mass energy of body 2 which equals $m_2c^2=10^{16}$ Joules. This seems to show that my argument is unphysical and therefore not a paradox.
 A: In answer to the last question -- many things, IMHO

*

*Be more careful about forces and accelerations, at high enough speeds constant force will not lead to further constant gains in speed. Also I would suggest not working with some fictitious forces. You are accelerating charges, so let your external force will be supplied by electric field. Less chance to end up with unphysical situation.


*You said that distance between objects is large, so why are you using what looks like electrostatic formula to compute the force on charge 2? The logical chain is $q_1$ generates electromagnetic field $\to$ electromagnetic field propagates $\to$ electromagnetic field acts on $q_2$


*What about the electromagnetic field generated by $q_2$ and how it acts on $q_1$?


*What about recoil due to electromagnetic field? It carries momentum, so it will create a 'drag'?


*Careful about 'eventually' statements. You have some unspecified force and you are using non-relativistic formulae. They will only work at low speeds. To get to 'eventually' your treatment has to be relativistic.
A proper way of doing this would be to write Lagrangian for two charges and an external constant electric field, add your constraints as Lagrange multipliers. Then derive coupled differential equations for the motion of two charges. Probably you will also have differential equations for the generated field. Then you solve all of these together. It may be possible to simplify things, but you will have to state carefully under which conditions such simplifications apply.
And use relativistic Lagrangians for the particles.
Forces will not be necessary in such treatment
A: Your calculations are correct, total Larmor's power radiated by body 1 is much lower than power of impressed radiation force due to 1 acquired by 2. That it, is it higher not right from the start of interaction (because impressed power is 0 when body 2 has 0 speed) but after some time. We can derive this time from your formulae to be
$$
t^* = \frac{\frac{2}{3}m_2 c^2}{F_C c}
$$
where $F_C$ is Coulomb force of charged body 2 on hypothetical charged body of the same charge separated by distance $d$. Thus $t^*$ is the time needed for the Coulomb force of the body 2 to impress energy $m_2c^2$ to hypothetical body of same charge moving with speed of light away from body 2.
In usual circumstances charge available on single body is very low and energy $m_2c^2$ is very high, so this time is extremely long. Your time $\Delta t$ can be very low because you assume extremely high value of $q_2$ and extremely low value for $m_2$. Maximum net charge we can concentrate on an isolated body of diameter 1m on Earth is probably not bigger than 1e-3 C. Even if this weighted only 1kg the time $t^*$ comes out around 10 billion years.  Even if we shorten the distance $d$ to 10m, the time is 26 days. So it is hard to observe this kind of effect. Further decrease of distance will be hard as the repelling force becomes very high.
Larmor's formula gives net EM energy flowing out of accelerated charge body provided there are no other charges anywhere near the region.
In your example, there is such a charged body: the body 2. This makes Larmor's formula derivation assumptions invalid. Total EM field is sum of fields of 1 and 2 and because body 2 is charged, total EM field in its vicinity is much higher than it would be were the body 2 absent. That's why the body 2 is able to acquire much more energy than Larmor's formula would suggest is available. Much more energy is available, because body 2 is charged and has its own strong electric and magnetic field. This fact is entirely ignored by Larmor's formula derivation.
Larmor's formula is pretty much useless when analyzing interaction of charged bodies and energy transfer between them. One must go back to analysis of  mutual EM forces as you have done.
