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There are many answers on the site discussing motion of electrons in an electric potential field, See Why is voltage described as potential energy per charge?

but also mass tends to move toward a lower gravitational potential. I assume then that things in general tend to move toward a lower potential when the potential is defined so that it represents potential energy. However sometimes "potential" is used for a field that another field can be derived from usually by taking a gradient.

See What is a basic physics general definition of a 'potential'? for a discussion of the term "potential" in physics.

My question is:

Why physically do things in general tend to move toward a lower potential value in a potential field?

Do they in general? Is there a physical law? Is there a basic physical explanation? I know there are mathematical explanations. Does the potential have to be defined as potential energy for this to hold true? Also does this have to hold true if one field can be derived from another potential field. Etc. ..

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    $\begingroup$ Generally, a force is minus the gradient of the potential. If a body is not in an extremal point of a potential it will experience a force that will be directed towards a minimal point. $\endgroup$ – proton Jun 23 '18 at 17:59
  • $\begingroup$ This is still basically a mathematical argument: Why will it " ..experience a force that will be directed towards a minimal point. " $\endgroup$ – user45664 Jun 23 '18 at 18:04
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    $\begingroup$ Related? physics.stackexchange.com/q/255353 $\endgroup$ – Farcher Jun 23 '18 at 18:14
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    $\begingroup$ Possible duplicate of Why does a system try to minimize potential energy? $\endgroup$ – John Rennie Jun 23 '18 at 18:27
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    $\begingroup$ There isn't a physical law because potential and potential energy are quite literally just antiderivatives. Forces are physical and energy is just math built on top of force (yes I know in upper level courses, it's energy which becomes fundamental and maybe even physical according to many). Anyways, just as you can create an antiderivative function $\int_a^x f(x) dx$ in single variable calculus, you can create a potential energy (antiderivative) function in physics from the force in exactly the same way. There isn't law which requires us to use this mathematics. We could forget about energy $\endgroup$ – DWade64 Jun 23 '18 at 23:59
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To be frank, it is simply a definition. There is no physical input, we could easily define it so that objects tend to move towards maximums. We could define the potential such that the force it models is $F=\nabla V$.

I tend to think that the reason we've chosen to model things as moving towards minimums is so that we can pictorially think of a ball residing on the potential curve, sliding down because of a uniform gravitational acceleration. It helps build intuition for which points are stable/unstable equilibria.

To reiterate, as far as I know there is absolutely no physical reason for objects moving towards minima of potentials. It is all definition so that we can model dynamics. In just the same way that Newton's Second Law $F=ma$ is a definition for force. We need to define things before we start modelling anything, and there will always be an ambiguity when you do this.

Sorry that I can't give you the answer you want!

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    $\begingroup$ I agree that potential energy $U$ could have been defined so that the force $\vec{F}$ points uphill the energy function. And just to add another reason for the minus sign. You use that minus sign in the definition of $U$ so that total energy $E$ is $K+U$. Without that minus sign, energy would be defined as $K-U$. But this is just an aesthetic motivation to your more physical motivation $\endgroup$ – DWade64 Jun 23 '18 at 23:47

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