>>>>
>>>> How do you determine amplitude?
>>>> What's the math (or physics) to derive
>>>> amplitude?

The fundamental formular Acos(B) + C is all you need to describe angular
modulation.
Changing the value of A over time determines the amplitude of an AM
modulated carrier.
Changing the value of B over time determines the amplitude of an FM
modulated carrier.
The rate of change of A or B changes the modulation frequency respectively.
C is DC, Y axis offset and has not been discussed here.

r, Bob F.