I know that defining potential for non-conservative forces is not possible and we can define potential and potential energy for conservative forces only. But can we define it for all conservative forces?

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    $\begingroup$ Yes, by definition. $\endgroup$ – Qmechanic Mar 14 at 0:30

A conservative vector field is, by definition, a vector field that can be written as the gradient of a function. Since conservative forces are vector fields, they all can be written as a gradient of a function (that function is the potential)

  • $\begingroup$ Is writing "conservative forces are vector fields" correct? Shouldn't you write they are directly related? $\endgroup$ – user8550821 Mar 14 at 1:57
  • $\begingroup$ @user8550821 But they are vector fields. What's the issue? $\endgroup$ – Aaron Stevens Mar 14 at 1:58
  • $\begingroup$ @Aaron Stevens Sorry If I am wrong I am a high school student, as per my understanding forces produce field. Can you elaborate where I am wrong? $\endgroup$ – user8550821 Mar 14 at 16:51
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    $\begingroup$ @user8550821 a "vector field" is a mathematical term describing a function that takes a point in space as an input and outputs a vector. So things like electric and magnetic fields are "vector fields", but other things like forces, fluid flow, etc. are also "vector fields" because they are described by vectors existing at points in space. Saying that a force is a vector field is not the same thing as saying something like the electric force is the same thing as the electric field. The terminology can be confusing I suppose. $\endgroup$ – Aaron Stevens Mar 14 at 16:58

The potential is the property of a conservative field to do work when moving a test particle. This test particle should be so small that the filed itself is (nearly) not influenced.

Test particles for gravitation are small masses, for the electrostatic filed they are small electric charges. For the magnetic field we need a magnetic monopole which does not exist or is not detected yet. But a magnetic monopole can be simulated, e.g. by a magnetic dipole made form very thin rubber, which has nearly no tension for small movements.

  • $\begingroup$ The potential is the property of a conservative field to do work when moving a test particle. Non-conservative forces also do work when moving a test particle. You should probably put more constraints on this property so as to not confuse novices. The statement is technically true, but it isn't specified enough to be special for conservative forces. $\endgroup$ – Aaron Stevens Mar 14 at 5:59

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