Second Order Autonomous Systems

Almost all Newtonian mechanical systems are second order or higher. For a second order system, two variables are needed to model the system and locate a given point in phase space. In Cartesian coordinates the equations of motion take the form

and

These equations are coupled together and a solutiondetermines a curve in phase space.

The nature of the fixed points can be found by expressing the system as a linear system in the region of the fixed point and expressing the system in matrix form. For a second order linear system, provided the linearised matrix representing the system near the fixed point is non singular, there is a single fixed point. There are ten different types of simple fixed point for a second order system. These represent different types of solution corresponding to different sets of eigenvalues and eigenvalues of the linearised system.

Motion in one dimension represented by a second order differential equation- important since it represents an application of Newton's second law of motion – can be expressed as a second order system by writingto obtain the coupled equations

We can write the system in vector form as

As in the first order autonomous system, for a second order systemis independent ofso for a second order autonomous system,

Example: Express the systemas an autonomous second order system.

and

The system becomes

As for first order systems, ifis the solution satisfyingwhenthen ifsois also a solution.