This is a project I started a while ago, and I want to document my progress on this blog. Being an imitator of the world, as all artists are, I decided to try to create a world with simple fundamental physics (simpler than ours at the current level of understanding). Using a simple formula for the theory of everything, I want to explore all the cool, complex things about the WorldMechanic universe.
Every fundamental particle, called a "fundy," has a location in Eucledian space (it can be any number of dimensions, but I'm sticking with two). They are differentiated only by identity, an intrinsic property that can have the value \(1\) or \(-1\). Each particle in the space has a Fundamental Force exerted upon it by every other particle, and the force acts in discrete units of time called moments (\(m\)). The force is deterministic: if you give specific values for the location of points in one moment, the next moment the points will be in a different location.
The notation for particles is to give every particle a number \(x\), which is quite arbitrary and can mean different things in different formulas. The identity of the particle is written as \(i_x\), while its location vector is written \(l_x\). The letters cleverly stand for identity and location.
Here's the formula for the force that a particle \(p\) has on particle \(n\):
\[F(p, n) = \frac{i_p v}{\|v\|^2}, v = l_p - l_n\]
The word version: the force that a particle \(p\) has on particle \(n\) is to go toward \(p\) in a direction proportional to \(i_p\) (so if it is \(-1\), to go away), and inversely proportional to the square of the distance. This inverse square law was not copied from the real world.
The direction that the particle actually moves in moment \(m+1\) is simply in the direction of the sum of all the forces at moment \(m\) (using a comma notation for the moment which the location describes), where \(N\) is the number of all particles in the universe:
\[l_{n, m+1} = l_{n, m} + \sum_{p = 1}^{N} F(p, n)\]
That's the formula that makes everything work. Now we can put particles in and see what happens!
