Generalization of the Difference Equation for a Simple Harmonic Oscillator
- Consider a finite set S which contains n distinct objects.
- Consider a map C on S such that C(a,b)=0 or 1 where a,b are any two objects in S.
We call such a map a Connection Map. If C(a,b)=1 we say that a and b are "connected" and a and b are "neighbors".
In general a member of S may have many neighbors, up to a maximum of n-1.
If C(a,b)=0 we say that a and b are not connected.
We require that..
An object cannot connect to itself. So C(a,a)=0 for all a in S.
Every object in S must have at least 1 connection. That is, for any object a in S it's possible to find at least one other object x in S such that C(a,x)=1.
Note that C is completely defined by a nxn matrix who's elements are 0 or 1 and who's diagonal elements are all 0.
- We say that the matrix C imposes a connection on S. A connection completely describes the geometry of S, so we can also say that C is a geometry on S.
- Clearly S can have many distinct geometries.
- Now define a field over S as a function f(a) for each member of S. We'll use a scalar field, so f(a) is simply a number.
- For the Simple Harmonic Oscillator (in 1 degree of freedom) we need to pick a specific geometry on S as follows..
Since members of S are distinct we can label them 1,2,3,..,n. These are just labels and do not imply any arithmetic operations. We then pick a specific Geometry on S defined by C(i,i+1)=1 for i=1,2,3,..,n-1 and all other C's are zero.
-The Simple Harmonic Oscillator is a field on S defined by..
where f(n) is the value of the field for object n and k is a constant. f(1), f(2) and k are given and these 3 numbers define the amplitude, wavelength and phase of the field.
We re-write this as..
We now interpret this equation in a more general way..
f(n) is the value of the field for object n and because of the geometry we picked we notice that n+1 and n-1 are the neighbors of n.
So this equation says that the value of f for object n is determined by the value of f at the neighbors of n. This is only true because of the particular C we used. This is the generalization of the difference equation for the Simple Harmonic Oscillator.
A few things to notice that differ from the usual treatment of the Simple Harmonic Oscillator..
No differential equation is used.
The difference equation has been generalized to a field over S.
There is no explicit coordinate system, and that includes the time coordinate.
Content written and posted by Ken Abbott email@example.com