Clifford algebraic perspective on second-order linear systems

A substantial proportion of all dynamic models arising naturally present themselves initially in the form of a system of second-order ordinary differential equations. Despite this, the established wisdom is that a system of first-order equations should be used as a standard form in which to cast the equations characterizing every dynamic system and that the set of complex numbers, and its algebra, should be used in dynamic calculations, particularly in the frequency domain. For any dynamic model occurring naturally in second-order form, it is proposed that it is both intuitively and computationally sensible not to transform the model into state-space form. Instead, it is proposed that Clifford algebra, Cl-2, be used in the representation and manipulation of this system. The attractions of this algebra are indicated in three contexts: 1) the concept of similarity transformations for second-order systems, 2) the solution for characteristic roots of self-adjoint systems, and 3) a model reduction for finite element models.