Autonomous Systems

In one dimension the motion of a system may be described by a function of position and time satisfying the equationIn general the velocity of the particle will be a function of time. If howeveris actually independent ofthen the position of the particle will depend only on the differencesince if

separation of variables gives

Integration now gives

Example: The equation of radioactive decay,is an autonomous system. Separation of variables and integration gives

Rearrangement gives

The original equation implies that the probability of decay does not depend when the decay takes place. This is a vital property, used in radioactive dating.

In general all the solutions of an autonomous system are relation. If two solution areand withthenmust satisfy

The above discussion can be extended to higher order systems in the natural way.