Non-Autonomous System: Definition & Explanation

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SUMMARY

A non-autonomous system is defined as a system where the acceleration explicitly depends on time, leading to phase trajectories that can intersect, indicating non-unique solutions to the equations of motion. For example, the equation of motion q'' = qt exemplifies a non-autonomous system due to its time-dependent nature. While both autonomous and non-autonomous systems generally possess unique solutions, the latter's phase space does not encapsulate the entire state of the system. Furthermore, non-autonomous systems can be transformed into an N+1 dimensional autonomous system.

PREREQUISITES
  • Understanding of classical mechanics and equations of motion
  • Familiarity with the concepts of autonomous and non-autonomous systems
  • Knowledge of phase space and its implications in dynamical systems
  • Basic mathematical skills for solving differential equations
NEXT STEPS
  • Study the characteristics of autonomous systems in detail
  • Explore the mathematical techniques for solving non-autonomous differential equations
  • Learn about phase space analysis and its applications in dynamical systems
  • Investigate the conversion methods from non-autonomous to autonomous systems
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Students of physics, engineers working with dynamical systems, and researchers interested in the behavior of time-dependent systems will benefit from this discussion.

aaaa202
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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?
 
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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.
 

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