# Non-autonomous system

I want to get some terminology straight.

A non-autonomous system is as far as I can understand a system, where the acceleration depends explicitly on time. This again means that the phase trajectory can cut itself since there is no unique solution to the equations of motion for non-autonomous systems. I think that is how my teacher said it.

I just want to get straight what a non-autonomous system is. Is it everything where q'' has an explicit time dependence? So what if we had the equation of motion:

q'' = qt

Does that equation describe a non-autonomous system? And is there in general no unique solution to the above equation given q(0),q'(0)? And why is that?

## Answers and Replies

An autonomous system is one whose right-hand side is independent of time. Both autonomous and non-autonomous systems typically have unique solutions. However, in non-autonomous system, its phase space does not describe the entire state of the system. A non-autonomous system can be trivially converted to an N+1 dimension autonomous system.