Potential Energy
A story in the Proxima Shared Universe by Bjorn Townsend

She drifted a meter and a half above the bulkhead that she was presently referencing as the "floor", her room pleasantly dark. The only light came in sparks of green and yellow, flickering like static fireflies, as her fibernet hub traded packets with the rest of the station's network. At moments like this she liked to meditate on the glowing patterns of light the hub made, trying to see a pattern in the traffic. Idly she toyed at working up a randomness algorithm which would model the chaos of network traffic, but decided eventually that there were too many variables for her to do it in her head, and it was more fun to try to get the pattern intuitively anyway. There was also the matter of school.

Sigh. "Computer: lights." Her voice was soft, full, and very British. An obedient beep, and the florescents sprang jarringly to life, illuminating her room. She'd done what she could with the place, with paint and computer-printed pictures and a half-finished Persian rug her friend Mojan was weaving for her. The rug was tacked loosely to the "floor", and occasionally waffled up and down in the air currents. Her favorite part was, of course, the window. Windows were rare on most habitats, but the station's designers knew all too well that humans needed space, and being cooped up on an asteroid only a few hundred meters in diameter, with nothing but rock and metal around, would drive anyone batty after a few years. Therefore, expensive windows were sprinkled around liberally, made of nanotech diamond so they wouldn't instantly shatter when a piece of ballistic rock hit them. The only problem was the fact that there was no way to make them radiation-opaque, so there were metal shutters to go over them when the sun flared. More expense, but everyone agreed the view was worth it.

She spared the view one last glance, sighed, grabbed her laptop, and kicked off a wall. The pressure-door slid aside for her and she drifted down the corridor, heading off to class.


"Now, you've established that true capitalism isn't viable in our present situation, given that we haven't got enough resources to survive the ensuing competition. Yet we still have a barter system, rather than taking the Socialist planned-economy route. We're not even going to talk about Communism, as we all know what happened to that idea." Doctor J looked back at his students, who were nodding and rolling their eyes; they'd heard his Soviet Union rant so many times they could recite it from memory. ("Blah blah blah Stalin blah blah blah blah Cold War blah blah American imperialism blah blah Afghanistan blah blah hypocrisy blah blah blah economic collapse...").

"So," he asked after a moment, "why aren't we Socialists? We all share the same challenges, living in the Asteroid Belt, which is a profoundly hostile environment, and having nothing to do with the home planet, having done our very best to tell Earth to leave us the hell alone. Once we've answered that question, we need to look more deeply at the present situation: since we don't live in a corporate capitalist society, why off Earth are United AeroSpace and Shinma Industries and the other corporates still around at all? And if they are still around, why aren't they running everything, exerting their considerable influence to create a situation of maximum benefit to themselves and their employees?"

The students, floating around Doctor J in a half-sphere pattern, their laptops projecting brightly-colored holographic displays, were silent for a moment, thinking. After a moment, one boy spoke.

"It seems clear to me, sir. We've looked at the economic summaries for the past ten years. The barter-prices have a very clear pattern; anyone who bothered to run the numbers would see it. Take titanium ore; despite fluctuations in supply and demand as new stations are set up and new ships are built or recycled, its price has remained comparatively consistent. There has been variation, but it hasn't been so great that it impacts individuals too drastically. Nobody's getting poor and nobody's getting rich. Someone is fixing the economy, probably stabilizing it so that market forces don't go critical and cause famine or societal revolt. Whoever is doing this realizes that market fluctuations could destroy Belt society, and is working to prevent that by controlling barter-prices. We are living in a Socialist economy, and we don't even know it."

Doctor J smiled. "Well done, Mister Kacirk. I take it the rest of you have arrived at similar conclusions?" A pause, and reluctant nods. Doctor J smiled. They suspected, but they hadn't bothered to do the work that young Jeffrey Kacirk had, so he uncovered a secret they hadn't learned. The other kids were kicking themselves for not digging deep enough. They would be more thorough in the future. It wouldn't cause a backlash against Jeffrey, of course; he just jumped a few notches in the social pecking order. No doubt the corporate computers at UAS would register a few more cracking attempts as the students tried to find pertinent documents. "If any of you can discover why the corporations are doing this secretly under the guise of a free market by next week, or at least come up with a workable theory, I'll personally fly you over to Perihelion City for dinner, with real chicken, not cloned stuff out of a vat."

There were predatory looks at this, and the holographic data displays blinked and shifted as students ran scripts that tried to sniff encrypted network traffic. Doctor J smiled faintly. He'd probably have ten reports in his inbox by night-shift. "That's all for today. Please send proposals for next week's topics and areas of study by 2200 hours tonight."

And to think these kids weren't even in their teens.


"Computer: lights off. Project flatscreen. Media, play Delerium, Track One, Karma. Volume setting five." The computer beeped compliance, the lights flicked off, a colorful data display clouded into existence before her, and the gentle laughter of children on a dusty African plain filled the room. That was the image that came to her, at least, as she listened to the opening of the album. She could only imagine what it must be like to feel dirt and sun and wind. She had lived in space for all her twelve years, and much as she loved the Belt, she wondered how the other eight billion lived. After the Exodus, of course, the Belt society had cut off contact with Earth, its members trying to lock away the stultifying lives they had led on the homeworld. She had heard enough horror stories of what people were like on Earth, and wanted nothing to do with them. But it sounded like a beautiful place, full of wonders that a child of delta-V could only guess at. Never having to match velocities; using wires to communicate rather than laser links; gravity everywhere, with no need to spin up the habitat; plants and green growing things ubiquitous. And there were the mysteries, the myths, the things that once made people afraid to go out at night, before the human monsters made the night an even more fearful place. They were supposed to be frightening, but she thought them wonderful. To think that there might be something still beyond human ken and impossible in human science, something so secret, gave her hope. And therein lay her sadness: in the Belt, no one had time to dream. Everyone was too busy surviving. She knew that dreams were the foundation of the Belt society, that its formation had been driven by the vision and hope of a small group of talented romantics.

She sighed, sadly and prettily, and summoned an email client. She typed:

From: "Rigel A. McCaffrey" (rigel@kitsune.monostation5.uas.com)
To: "Doctor Fahd Jondhpur" (drj@zero.monostation5.uas.com)
Cc:
Subject: Illusions

Doctor Jondhpur:

They fool us to keep us free.

One day, we will walk again on the surface of a world, whether it is Earth or some other distant planet. The romantics which founded the Belt society have a dream of a freewheeling world, full of chaos, with the butterfly's wings flapping all the time. They want us to live in an ever-changing place, full of possibility. They want us to remain accustomed to economic freedom, so that when we again have the resources to make our lives more than a matter of daily survival we will be better prepared for the shock of a free market economy. I don't know exactly what they have planned, but they'll need a new world to do it.

It'll be a long time before the Belt is rich enough for us to have a free market, you see. And Huntley, Weatheral, Sasaki and the others are afraid that we'll be too beaten down by this daily grind, this constant fight just to breathe, in order to dream when we're done. I see it happening to my parents. Back on Earth Father wrote beautiful books and Mother wrote beautiful software. They went to parties and wore flamboyant clothes. They wove fantasies together. I've seen pictures: they glowed with elegance. Now, they go out and mine iron oxide so they can replenish the station air supplies. They collapse when they get home, too exhausted to think. It makes me cry, sometimes, to think of what else they could be doing with their lives.

We need a world. Not necessarily for all of us to live on, but something we can rely on and draw from, so that we don't have to grow all our own food or make all our own air. Just enough to make life a game again, a beautiful story like in the books I read, rather than deadly serious. The Founders know this. They don't know where they're going to find a planet, though, or how to get to one if we find one. I don't know either. I'm afraid we're going to end up just like those poor, dull buggers on Earth, rather than the dreamers we were meant to be.

"Come away, O human child, to the water and the wild
With a faerie hand in hand
For the world's more full of weeping
Than you can understand."

That's by Yeats. It's one of my favorite poems, and I hear it's one of Mark Huntley's favorites too. I just hope there's still going to be a wild for us somewhere.

-- Rigel.

The reply came only moments later:

From: "Doctor Fahd Johndpur" (drj@zero.monostation5.uas.com)
To: "Rigel A. McCaffrey" (rigel@kitsune.monostation5.uas.com)
Cc:
Subject: Re: Illusions

Well then for God's sake do something about it, Rigel. Doctor Burroughs tells me you've already got a good grasp of quantum mechanics, and, forgive my rudeness, but you're only twelve. Back on Earth I had colleagues who weren't as creative or as well-educated as you are now. You have gifts, girl, as do your peers. You can affect your world; this isn't Earth; the inertia isn't too great. Make life what you want it to be.

Is everlasting fame and a life of adventure among the stars enough extra credit to offer? Or do you want an A+ in the class, too? ;)

-- Doctor J.

She read this, and smiled. Way ahead of you, Doc. "Computer, open file slip.gcad and display. Media, play Pink Floyd, A Momentary Lapse of Reason, Track Two. Volume at maximum."

/* Pink Floyd -- Learning to Fly -- A Momentary Lapse of Reason */

Copyright 2001 Bjorn Townsend. All Rights Reserved.

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 * x2
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*m1*m2 / 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*m1 / r02
with m1 being the mass that doesn't show up in the local gravity formula, and r0 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 = ForceDistance
(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.


1Of 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.

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