Hello everypony and thank you for all coming to listen to me today.

My name is Twilight Sparkle and I will be presenting on my research into the very nature of Equestria which was conducted under the guidance of my mentor, Princess Celestia.

First, we will begin by listing off some basic things that we know for fact.

1. Magic is a force that acts at a distance; it does not require contact.
2. A unicorn's horn is somehow involved in the unicorns ability to use magic.
3. The 'color' of magic can vary from pony to pony.
4. In at least some cases, using magic to lift things seems to apply a reactive response force (I submit to you when I met a unicorn named Trixie. I needed to dig my hooves into the ground to lift and ursa minor).
5. In some cases there does not seem to be a response force (The unicorn filly recently born to the Cake family was able to levitate itself. It's possible that this is some type of anti-gravity or mass-manipulation, but that seems unlikely). This seems, at first, to be a violation of one of the basic laws of motion.

I will make three fundamental assumptions.

1. We can, in fact, explain all aspects of magic within a logical and self consistent mathematical framework.
2. We can accomplish the above while remaining consistent with our current understanding of physics.
3. In order for assumption 1 and 2 to be true, we will need to formulate our theory of magic with physics as our inspiration.

To begin with, let us take one of the most fundamental spells that all unicorns learn while still just a foal: telekinesis.

Telekinesis is the ability to move objects without touching them through the power of one's mind by utilizing magic. Ordinarily the object being moved is enveloped in a glow of magic the same color as the unicorns horn. Telekinesis is able to move an object in all three spacial directions and affect rotation.

What makes telekinesis so odd, and difficult to understand, is that every other physical force that acts at a distance is either a attractive or repulsive force. Telekinesis seems to have much more freedom in its directionality.

It is at this point that I will be relying on assumption 3. All forces that are capable of acting at a distance are actually forces between fundamental 'charges.' In the case of electromagnetism, this is either an attractive or repulsive between positive and negative electric charges. In the case of gravity, it is an attractive force between positive 'charges' of mass. While some ponies have also theorized negative mass matter, because it is pure conjecture, we will be ignoring that for the sake of this discussion.

Obviously telekinesis is not just repulsive or attractive. It can also move objects perpendicular to the vector between the two objects.

For those of you who are unfamiliar with the term vector, imagine a vector is an arrow. When I refer to the vector between the two objects, I am talking about an arrow pointing from one to another. I recommend doing further reading on the subject. The human resource wikipedia can be of great help for beginners. In fact, if at any point I use a term that you do not understand, I recommend that you look it up as to avoid being lost. I will do my best to accommodate you, but I do want to avoid getting bogged down in the basics.

We also have the fact that most objects do not seem to be constantly affecting each other magically. In the case of telekinesis, it is only when a unicorn actively uses the magic that anything happens. A unicorn may be able to levitate two boxes, but the boxes will not levitate one another.

Therefore I posit that there must exist two types of magical charge.

1. Inactive magical charge. This will in no way interact with other inactive magical charges, but will respond to magical fields. However, it will not create a magical field of its own. The magnitude of inactive charge in an object will affect how easy the object is to affect with magic. Therefore we can also refer to inactive charge as magical conductivity.

2. Active magical charge. Active magical charge will exert a force on inactive magical charges. Active magical charge is the source of magical fields.

This right here explains fact 2 of magic, the importance of a unicorns horn. There are two likely explanations at this point of the role the horn plays in magic

1. The horn is like an active magical charge battery. The exterior surface of the horn acts as type of reverse faraday cage. A faraday cage is a construction which can protect whatever is inside from external electric fields. When the unicorn wishes to perform magic, some component of the pony's biology 'breaks' the cage allowing the internal magical field to escape from the confines of the horn. This would allow for the magical field to be directed in a specific direction. Further exploration of this possibility will need to wait for a later talk.

2. Accelerating inactive magical charge results in a magical field. Spirals in the horn conduct this charge, resulting in the generation in a field. This is the same basic way in which an electromagnet works. Once again, further investigation of this will wait for later.

While we will need to hold off on any definitive decision regarding the importance of a unicorn's horn, we nonetheless have established two reasonable possibilities.

We still have not explained the directionality of telekinesis, only its selectivity.

Because telekinesis is beyond a simple attractive and repulsive force, active magical charge, at least, must be more complicated than we have so far presumed.

Let us assume that active magical charge is composed of three components, an X, Y, and Z component. These components compose an 'active magical charge vector.' However, these components are dependent on the directionality of the main magic vector. The direction the telekinetic force applies will then be the cross product between the active magical charge vector and the unit vector along the direction between the unicorn and object the unicorn is levitating.

You may wish to read that one more time to make sure you have it.

To get the magnitude of the force we must also multiply it by the magical conductivity (keeping in mind that we are being inspired by the electromagnetic force, ordinarily you multiply two charges together, here we are multiplying two different types of charges together).

What then is the affect of distance? At worst, the magical force decreases as the inverse of the distance squared. However, that assumes the magical field is beaming off in all directions and that the field lines are decreasing in density. A skilled unicorn can 'tighten' his or her magical field and reduce the exponent on the distance to a theoretical zero at maximum tightness. Of course this ignores the magical interference caused by intervening air which we will discuss in a later talk.

In practice, however, because objects are not of infinitesimal size, the unicorn will need to allow some widening of the magical field in order to encompass an object. Ironically, because an objects angular size decreases with distance, a skilled unicorn will be able to us his or her magic more efficiently at greater distances.

So what we have is

F = S*(((X*Xhat+Y*Yhat+Z*Zhat)of object 1 x Rhat) * C of object 2)+((X*Xhat+Y*Yhat+Z*Zhat)of object 2 x Rhat) * C of object 1))/(R^A)

Where:
F is the force between the objects (a vector)
S stands for Sparkles constant, a number that I have yet to measure
X, Y, Z, represent the the X, Y, and Z components of an active magical charge
Xhat, Yhat, Zhat, are directions in the x, y, and z direction
Rhat is the direction between the two objects
C is the magical conductivity (inactive magical charge) of an object
R is the distance between the objects
A is a parameter between 2 and 0 based on the 'tightness' and focus of the magical field.

Note, this formulation obeys Newton's law that all forces must have an equal and opposite force.

I will freely admit that this does not explain unicorn self levitation nor the visual glow of magic. It also does not explain how a unicorn could levitate something heavy without the response force breaking the unicorns neck. There are, however, exploitations too these two issues based on our current understanding. However, I would like us to take a break to allow for discussion.

Please feel free to ask any questions, bring up any problems, or share any formulations of your own.