potential_force_explanation

Alright, I present my most convincing and least circular case for: potential, force, energy, and all that jaz.

This will all be in the context of classical mechanics; it generalizes, but in ways that would needlessly complicate the explanation.

First, let me establish some analogies that will help motivate the conventions that are used below.

Work done on an object by a force quantifies the resulting change in the object's kinetic energy.
So if a force does positive work on an object, it has increased its kinetic energy (sped it up),
and likewise if it does negative work, it has slowed it down.

The potential energy of a system quantifies how much work must be done on the system to change it to some reference state.
The reference state is arbitrary, but there is usually some conventional choice that makes the most sense/convenience for the problem being solved.
FOr example, in the case of a free-falling object: the reference state is conventionally chosen as the object being on the ground, and the potential energy is the work that must be done by gravity to drop the object to the ground.
In general, a system will be a collection of objects with their initial and current kinetic energies, forces that do work on those objects, and the current time.
So the kinetic energy of an object in the reference state will be equal to its initial kinetic energy, plus the work required to do on it to get it there (I.E> the initial potential energy).


Now, there are certain special systems, called conservative, where the potential energy depends only on the positions of the objects in the system.
When a force is called conservative, what is implicitly meant is that any system involving only that force is a conservative system.

Let's formulate this more symbolically, and consider a conservative system with only one object and only one static force f(x).

Let x' be the initial position of the object, U(x) be the potential energy (only as a function of position), and K denote kinetic energy, which is equal to mv(t)^2/2.

From the discussion above, we have that:
K(t)+U(x) = K(0)+U(x')

Let E = K(0)+U(x'), which is conventionally called the total energy of the system.

Then you can rewrite this as:
K(t)+U(x) = E

This gives another way to define conservative systems: they are systems where the kinetic energy of any object at any reference state depends only on the object's initial state, and not the path taken to get there. That is, the total energy E remains constant under state changes (conserved).

We can differentiate this equation with regards to time to get more useful information:

-dU/dx dx/dt = dK/dt
= d(mv(t)^2)/2
= m v(t) dv(t)/dt
= dx/dt f(x)
f(x) = -dU/dx
U(xi) = int_xi^x' f(x)dx

Notice that there is an arbitrary, but very well-established, choice being made here:
if potential energy had instead been defined as the work done to change an object to its current state, then the total energy equation would instead become:
K(t)-U(x) = E,

and the result of the differentiation would then give:
f(x) = dU/dx


Now you may reasonably protest against a couple parts of this explanation:

Why focus on system constrained like this, anyway?
It just happens that most of the classical forces are radially symmetric and not time-dependent, so their magnitudes only depend on an object's distance from the source.
Because of this, conservative systems end up being a pretty broad and useful category.

What is this vague energy thing you keep bringing up, anyway?
On a fundamental level, the energy in a system is the average of the velocities of its components.
In a closed system, the fundamental law of conservation of momentum tells us that its energy does not change, because particles can only change their velocity by bumping into another particle and proportionally changing its velocity.
Building up the rest of mechanics from this framework is called statistical mechanics and it is very complicated.