### Basics

Potential energy is any energy based in the stress (term used loosely) involved in an interaction between particles (or larger structures). Under certain circumstances, it can be turned into any other kind of energy, and created from almost any kind of energy.

For example, if you stretch a spring, you have stressed the mechanical structure. In order to bend the spring you pushed spring out of its relaxed state, putting energy into it. If you release the spring, it will recoil. In terms of energy, the potential energy of the spring became kinetic energy of the spring's motion.

Suppose you pick up a weight. The energy you gave that weight by picking it up has become its potential energy. If you allow the weight to fall, that energy will become kinetic energy. If you put it on a table, the energy will remain potential.

### Potential and Kinetic

One of the useful things about this connection between kinetic and potential energies is that the total energy is conserved. That is to say, the sum is always the same. Unfortunately, real macroscopic systems have an additional term (Thermal energy) which gradually soaks up the energy over time (see the second law of thermodynamics). Fortunately, we can often arrange for this rate of dissipation to be low. When it is, one is essentially swapping potential and kinetic energies back and forth. Another problem is isolating the system so energy doesn't leave altogether, or come in from outside. If you can prevent that, then much becomes possible: once you have found out the total energy of this lossless isolated system, it won't change. Therefore, if you find out the momentary value of either the potential or kinetic energy, then you can calculate the momentary value of the other simply by taking the difference from the total.

Upon experimentation, one would find that the potential energy of springs in general follows the following rule:

Spring potential = ½ k * x^{2}

where x is the distance you have pulled (or pushed) it from its equilibrium position, and k is a constant determined by the spring. The ½ was put in so that the value of this k would be the same value as for Hooke's Law

Similarly, if one undertook experiments on a human scale, one would discover that the potential energy due to gravity obeys a simple rule:

gravity potential = mgh

where m is the mass of the object, g is the local gravitational acceleration, and h is the height elevated. This is a local approximation, since g does not change much on a human scale. However, the true ^{1} gravitational potential (which you might discover if you were to launch something into orbit) is

gravity potential = - G*m_{1}*m_{2} / r

where G is the Universal Gravitational Constant, the m's are the masses of the two objects in question, and r is the distance between them ^{2}. Over small distances (not small r, but small differences in r), this actually looks a lot like the formula above, with

g = ½ G*m_{1} / r_{0}^{2}

with m_{1} being the mass that *doesn't* show up in the local gravity formula, and r_{0} being a typical radius for the system.

### Potential and Force

As you no doubt noticed in the first two examples, holding the weight up and holding the spring stretched required force. A general rule is that objects will exert forces to reduce their potential energy. Here is a quick derivation:

One of the fundamental equations of energy is

Energy = Force ⋅ Distance

(That is a dot product, btw) By taking the gradient of both sides (in respect to position, here manifesting itself in the 'Distance' variable), and assuming that all the force was pushing against potential energy, we get

Force pushing against potential = ∇ PE

Applying Newton's third law, we get

Force due to potential = - ∇ PE

So the more sharply the potential drops, the harder it pushes. And it pushes in the 'downhill' direction toward lower potential energy.

Note that by applying this to the spring potential, we recover Hooke's Law. Similarly, applying this to the local gravitational potential we recover the constant force of gravity... and applying this to the general gravitational potential we recover the inverse square law. See? This all hangs together.

An interesting feature of using the gradient is that it allows all sort of coordinate systems. These are used in Lagrangian Dymanics and Hamiltonian Dynamics.

For example, for a rigid pendulum, the potential of the pendulum is a simple function of the angle of the pendulum: - mgL cos(θ), with θ = 0 at the bottom.

Another case is in many body problems: one can calculate the force on each body by taking the gradient of the total potential in respect to the positions (and whatever else would affect the potential) of all the bodies.

There are many many kinds of potential energy beyond the two that arise in an introductory physics course (i.e. gravitation and simple harmonic oscillating potentials). Electrostatic potentials look a lot like gravity, with charge instead of mass. Magnetic potentials depend not only on position, but orientation. That is why magnets held near each other try to turn to face the same direction. In the quantum dynamics of surfaces, there are all kinds of outlandish combined potentials from all of the many forces acting on a particle. Nonetheless, these forces can be combined into one coherent potential, greatly simplifying the model of the system.

### Potential and other things

Note that since the only part of the potential which has any impact upon the observable universe is its derivative, the *value in absolute terms* of the potential is arbitrary (so long as the relative values remain the same). There are several conventions as to where to put zero, each useful in a different situation:

- Put zero at the lowest value possible. This has the advantage that the potential is always positive, which will help minimize sign errors. Not good if the potential involves a singularity pointing downwards.
- Put zero at a separation of infinity. This is good for singularities, either up or down. Get used to this for astrophysics or electromagnetism.
- Put zero at the value that most of the system has. If you are dealing with an externally imposed potential which is mostly flat, with a bump up or down here and there... put zero where it will do you the most good: where most of your particles are.

Requiring that a theory produce the same predictions no matter what constant you add to the potential is called

Gauge symmetry.

Of course, since the potential energy gives one the force on an object, it is very useful for predicting behavior. It comprises half of the Lagrangian and the Hamiltonian, which are central figures in both classical mechanics and quantum mechanics.

^{1}Of course, if you were to dig down into the earth, you would find that this immediately ceases to apply. Instead, the potential becomes the spring potential centered on the center of the Earth, with k set so that the force at the surface produces the usual acceleration of gravity (read the Potential and Force section to get to that part). This abrupt change is due to more and more mass being further from the center than you are, pulling you up instead of down. It is up to you to choose the arbitrary constants for these two potentials so that they don't cause a discontinuity.

^{2} What if there are more than two objects, you ask? Then calculate this for each pair of objects and add up the sum. This holds generally true for other kinds of systems. The total potential is simply the sum of all individual potentials.