periodic function?

Question

x(t) = Acos(w1 t)
y(t) = Bcos(w2 t - gamma)

here w1 is the angular frequency dealing with x and w2 is the angular frequency dealing with y.

I am trying to prove that if w1/w2 is rational then these functions are periodic
Then I want to prove that if w1/w2 is not rational then the motion never repeats itself.

Answer 1

Suppose that w1/w2=p/q where p and q are integers. Then q/w2=p/w1 and you will find both x and y have a period of 2pi*q/w2=2pi*p/w1.

Now suppose that the motion repeats itself after time T. Then w1*T is a multiple of 2pi, say w1*T=2pi*n. Also, w2*T is also a multiple of 2pi, so w2*T=2pi*m. Now divide these two equations to find that w1/w2=n/m is rational.

Answer 2

First I will make the two parametric equations as simple as possible keeping only w2 as variable apart from t.

x(t) = cos(t)
and
y(t) = cos(w2 t)

Now it comes to some kind of iterative formula:

The trace of the function will continue to move until it reaches an exact value so that it can restart.
However, for an irrational number such as pi, an exact value cannot be reached and the trace continues forever without reaching the point of restart.

If the motion is to repeat, the velocity of the trace at a specific point should have two values which will be exactly opposite.

Differentiating the parametric equations:

dx/dt = -w1(cos(w1 t)
dy/dt = -w2(cos(w2 t + gamma))

Using t = 0, we find for w1/w2 = rational that for different values of t that give the same set of coordinates as when t = 0, there are only two velocity vector that are opposite in directions. This can be tested for other points.

For irrational ratios, the restart is postponed to infinity because infinity is the number of cycles that will have to be done until the trace comes back onto an already passed point.

Answer 3

I am so not a math person. That side of my brain has all but fallen off...I'm more of the artsy type.

Answer 4

find the frequency of each parameterized equation and you have proven periodicity.