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I am struggling to find a concise formal definition of external and internal forces. From my understanding:

Internal: Forces acting upon the object that cause changes of form of mechanical energy, without changing the total mechanical energy possessed by the object

External: Forces acting upon the object that change its total mechanical energy.

Now from intuition, I arrive at the 'correct' categorisation of common forces as stated in books etc:

Internal: gravity, Coulomb (electric), magnetic
External: friction, drag, applied force

However I want to have a correct way of distinguishing between the two types. Is there a direct link between a conservative vector field and internal force (and therefore non-conservative vector fields and external)? But then how is that influenced by equipotentials for force fields?

A concise mathematical statement would be ideal if possible!
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