Clifford algebraic perspective on second-order linear systems

A substantial proportion of all dynamic models arising naturally present th emselves initially in the form of a system of second-order ordinary differe ntial equations. Despite this, the established wisdom is that a system of f irst-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, particula rly in the frequency domain. For any dynamic model occurring naturally in s econd-order form, it is proposed that it is both intuitively and computatio nally sensible not to transform the model into state-space form. Instead, i t is proposed that Clifford algebra, Cl-2, be used in the representation an d manipulation of this system. The attractions of this algebra are indicate d in three contexts: 1) the concept of similarity transformations for secon d-order systems, 2) the solution for characteristic roots of self-adjoint s ystems, and 3) a model reduction for finite element models.