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
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