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 solution determines 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 writing to obtain the coupled equations  We can write the system in vector form as As in the first order autonomous system, for a second order system is independent of so for a second order autonomous system, Example: Express the system as an autonomous second order system. and The system becomes As for first order systems, if is the solution satisfying when then if so is also a solution. 