About fundamental physics (static fields, forces, and energy) Disclaimer, so that you can provide a more informative answers: I have a degree in mathematics but I am quite ignorant in physics.
I was reflecting about the ability of a charged particle (any fundamental charge, but let's restrict ourselves to an electric charge) of exerting a force without expenditure of energy.
I mean, let's imagine a charged particle in motion with respect to an observer's frame of reference. It radiates electro-magnetic waves which carry energy and momenta away from it. Once those quantities are depleted, the particle is reduced to be at rest wrt such reference frame.
This makes sense.
Now imagine a particle at rest wrt the observer's frame. It has nothing to lose (except for its mass, which is finite anyhow). No energy, no momenta (again, aprt from mass-bound energy).
Still, it produces a static field which can exert a force (indefinitely) that can accelerate objects, thus making them gain energy.
At the prensent state of my knowledge, I find it strange to understand. May you provide a simple explanation?
EDIT: To rephrase my question, I find it unsettling that an entity can exert a force upon other entities without expenditure of other quantities (energy of whatever).
 A: Electromagnetic theory obeys law of local conservation of energy. This means that we can define energy as a quantity distributed in space, and its amount in some observed region changes solely by transport through boundaries of that region.
When one particle's field acts and accelerates another (second) particle, energy is being sucked away from the space near the second particle and is being transformed into kinetic energy of that particle. So, the kinetic energy that appeared "from nothing" was actually already there in the space around the second particle, it just had a different form.
There are formulae for energy density in space due to EM interaction. In macroscopic theory (the particles have to have their charge distributed in some non-zero volume), the energy density is given by squares of electric and magnetic field:
$$
\frac{dE}{dV} = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2.
$$
A: Here is a possible, although somewhat simplistic, explanation.
Let's use an electron as a charged particle to make the explanation more concrete.
The energy of the electrostatic field around an electron is generated at the time it is separated from a neutral atom and is taken away. This energy is equal to the work that had to be done by someone to move the electron against the attraction force from the positively charged atom (positive ion) left behind.
NOTE: In fact, a similar field will be created around the positive ion and those two fields could be treated as one, but, if the distance between the particles is significant, the fact that the fields are shared may not be obvious from examining the fields in their immediate neighborhoods. 
This electrostatic field can perform work, say, by accelerating some charges, but any work performed by the field will reduce its energy and its capacity to do work is limited by its initial energy (or the work done to create it).
For instance, if the positive ion left behind is allowed to fly all the way toward the electron, the electrostatic field will perform exactly the same work (by accelerating the ion) someone has originally performed to separate the charges and create the field. 
Once the electron and the ion reunite (with a little bang - dissipating the kinetic energy acquired by the ion), there won't be any electrostatic field or electrostatic energy left.   
So, if we understand the origin of the electrostatic field, we should not be surprised by it ability to perform some limited work.  
