Is a free particle one on which there's no NET force or one on which there's no force at all? I'm getting different definitions from different sources. Some claim that free particles have no forces acting on them at all (i.e. even if a particle has forces acting on it such that they cancel, it's not free). Other sources explicitly state that there is no net force (like this). Other sources, when I google about this, state that free particles are "free from external influence" - which is kind of vague and can be interpreted either way.
Can someone resolve the confusion?
 A: Forces often come from force fields like gravitational or electromagnetic. These differ in different parts of space, so they may cancel in one point but not in another.
Now, suppose a particle is at a point where all forces cancel (and there's a non-empty set of forces being in superposition). Let's now perturb its position. In general, the forces now won't cancel, and the particle will experience acceleration. Suppose that the forces are all directed into the initial point where no net force acted on the particle. Then this initial point would be the point of stable equilibrium. Now, if the particle has too small kinetic energy, it will be bound in the potential well, unable to escape it.
I think it's fair to say that in the case described above a particle, even at the point of equilibrium, where all forces cancel, is not free (it can't escape arbitrarily far given arbitrarily long time). So, I think the case where there's no net force, but the set of forces is non-empty shouldn't be included in the definition of a free particle.
A: The following answer assumes we are dealing with an inertial frame of reference.
It really depends on what is meant by a "free" particle. The dictionary definition of "free" as an adjective is "not under the control or in the power of another". If a particle is not subjected to a net force then what we do know is that the particle either remains at rest or in uniform linear motion according to Newton's first and second law. That doesn't mean the particle is not subjected to any forces. Just that it's not subjected to a net force. Per the definition of "free" the particle is not under the control of the forces.
A dictionary definition of "influence" is "the capacity to have an effect on the character, development, or behavior of someone or something". So here again, if there are forces acting on a particle but they do not affect the character of the particle, then by this definition those forces are not influencing the particle. That can only be the case where the net force of those external forces is zero.
Hope this helps.
