Gfyuki

I know that planets can't have infinite energy, due to the law of conservation of energy.

However, I'm confused because I see a contradiction and it would be great if someone could explain it.

Energy is defined as the capacity to do work. Work is defined as Force x Distance. Force is defined as Mass x Acceleration. Thus, if we accelerate a mass for some distance by using some force, we are doing work, and we must have had energy in order to do that work.

In orbit, planets change direction, which is a change in velocity, which is an acceleration. Planets have mass, and they are moving over a particular distance. Thus, work is being done to move the planets.

In an ideal world, planets continue to orbit forever. Thus, infinite work will be done on the planets as they orbit.

How can infinite work be done (or finite work over an infinite time period, if you'd like to think of it that way) with a finite amount of energy?

Where is the flaw in this argument?

Your definitions are incorrect. Force is rate of change of momentum and is a vector. More importantly, the work done by a force is not force x distance, it is the force resolved in the direction of the displacement x the magnitude of the displacement. This is more formally known as the scalar product of force and displacement.

In the case of a (circular orbit), the centripetal force supplied by gravity is at right angles to the displacement, so no work is done.

Power expended when moving in orbit $vec {F}.vec {v}=-nabla phi .frac {dvec {r}}{dt}=-frac {dphi}{dt}$ , $phi$ is gravitational potential. Hence the work of gravitational forces is $W=int {vec {F}.vec {v} dt}=-int {frac {dphi}{dt}dt}$. For a periodic motion, the integral $W$ over the period is zero. For hyperbolic motion, the integral $W$ over the entire time of motion is zero.

In an ideal world, even then it is impossible to supply infinite energy.

I think the flaw here is that work done is only a measure of the change in energy. Obviously the satellite responds to the force of gravity acting on it (which in turn implies that the satellite does not have infinite energy).

What you have to understand is that work done on an object is the dot product of the force and displacement vector. This is where the $costheta$ term becomes vital. Remember that $theta$ is the angle between the force and the displacement vector. The object makes displacement, true. But that is a horizontal displacement. The force cannot change the magnitude as such, because you have to apply a force in the direction of motion to cause the change in speed. Since the force acts perpendicularly, the object would have to move towards the planet.
Which it does, but it does not fall to the planet either. Because the direction of force changes here. And hence, there is a circular motion.

Because the satellite has that constant speed $v$ (I am assuming simplest idea of a circular orbit), it has a kinetic energy. Because the satellite is attracted by gravity, there is a corresponding potential energy.

Remember that work is force times displacement, not distance. Displacement is a vector, which means when a planet moves a full circle, its overall displacement is zero, resulting in a work of zero.