# External driving force on blackbox system: frequency response

External driving force on "blackbox" system: frequency response

Easy question:

I have a blackbox system (it's actually the Earth, but we can just treat it as a blackbox) driven externally by a periodic sinusoidal force (the Sun). If the driving force has a period of say 10 years, is there any way that the response of the Earth due to that force have anything but a period of 10 years once steady state has been reached? (which I think we can safely assume for the Earth-Sun system).

The answer seems intuitive enough, and obviously can be proved easily for pendulums and a lot of idealized systems, but I'm not 100% positive that it's always the case for ALL systems. Ideally there would be a math theorem or some physics proof perhaps, assuming a generic Lagrangian, that can prove that the response will also be sinusoidal with the same freq as the the driving frequency. Otherwise a counterexample would work just fine the other way.

Thanks for the help

mfb
Mentor

Let humans shoot a laser in space every 3.8557 years. The emission of light of this wavelength clearly has a frequency different from 10 years.
It is a blackbox - you cannot look at causal connections inside. And maybe the 10-year-cycle of earth inspired humans to build and use their lasers?

In terms of less intelligent setups: You can get different responses. The easiest things are higher harmonics from nonlinear reactions, but other frequencies can occur somehow, too.