Question: So I got the mathematical aspects down of what it means for a vector field to not be conservative, but I'm trying to make sense of the physical intuition. Why are so many vector fields found in nature conservative? What can be said about vector fields found in nature that are not conservative, for example, an induced electric field ([Conservative nature of electric field](https://physics.stackexchange.com/questions/612484/conservative-nature-of-electric-field)). This is a pretty open ended question. Thanks!

Answer: 
A vector fields is conservative if it is the gradient of a scalar field. In static cases we can use the the scalar Coulomb and the Newton potentials. The force fields are then conservative. In the more general case they are not. The Coulomb and 'magnetic' vector potential form a Lorentz vector. For gravity you have to use General Relativity, which I guess does not lead to a conservative force either.
Your description gives the impression that most of the fields in nature are conservative which is not true and quite the opposite in classical applications and open systems. Force fields and systems where for example friction or drag is found are not overall conservative. Many confuse the law of conservation of energy with these two definitions which is irrelevant.

In general a force field in a system that does not produce any work that consumes energy which is expelled out of the system (i.e. energy transformed from one form to the other for example electric power to heat leaving the system) is considered a conservative acting field. Other characteristics of conservative fields are that the overall work done in a closed loop path is path independent which is another way to say that the overall energy consumption, work done in the system by the field is zero. For example a gravitational field without air all the potential energy is transformed to kinetic and back, independent of the path trajectory chosen going from point A to point B and back to point A again inside the field. Therefore, no work is done that consumes energy thus no energy leaves the system and therefore a gravitational field is a conservative field. However, the situation in the system totally changes when you add air drag and friction where work done consumes energy in the form of heat expelled out of the system.

Another helpful characteristic-criterion of a conservative field is that the field force in most cases acts along to the motion vector whereas in an non-conservative field the force is in many cases perpendicular to the motion vector. But these are not general rules and there are exceptions like for example, friction which is not perpendicular to the motion of an object but still is a non-conservative force.

So in the case of an electric charge moving linearly inside a magneto-static field the electromagnetic Lorentz force is always perpendicular to the linear charge motion and therefore we say that the interacting magnetic field does zero work on the linear direction of the charge's motion however the charge will now move on a spiral trajectory changing therefore its linear velocity therefore the magnetic field in this case is non-conservative since it reduces the energy of the moving electric charge.

An electrostatic field on the other hand when acting on an stationary electric charge moved on a closed loop trajectory as in the case of a gravitational field, will in overall not change the energy of the moving charge, overall there is no energy consumed by the electrostatic field on the electric charge, therefore a conservative field.

An electric field generated by a changing magnetic field caused by a moving magnet, in this case the electrostatic field projected is [irrotational](https://en.wikipedia.org/wiki/Conservative_vector_field) (i.e. has a vanishing curl, vortex structure) which is another characteristic of conservative fields. See below the electrostatic field projected by moving magnet:

[![projected electrostatic field by a moving magnet](https://i.stack.imgur.com/lkhE7.png)](https://i.stack.imgur.com/lkhE7.png)

However when this conservative electrostatic field generated by the moving magnet is projected on a conductive loop or a solenoid or circuit only the field vectors which coincide with the loop wire will be induced inside the wire therefore creating a rotational field instead and thus a non-conservative electric field.

The list can go forever and the concept of conservative and non-conservative fields is one of the most complex subjects.

As a general conclusion a field is characterized as conservative or non-conservative always in relation with the system applied to.