Involution - The Formal Theory of Differential Equations and its Applications in Computer Algebra

Standard books on the theory of differential equations deal with scalar equations and systems of equations in normal or Cauchy-Kovalevskaya form, i.e., systems that are solvable with respect to the highest-order derivative. Traditionally, one considers systems where a distinguished independent variable t exists such that the system can be written in evolution form, u t = φ(t, x, u, u x), where the remaining independent variables are collectively denoted by x. Despite the popularity of that assumption, non-normal systems are also ubiquitous in, e.g., classical and fluid mechanics, gas dynamics, electrodynamics, field theory, and general relativity. Classical examples include the incompressible Navier-Stokes equations, Maxwell's equations, the Yang-Mills equations, and Einstein's equations. In mathematics, non-normal systems of ordinary and partial differential equations (ODEs and PDEs) arise in differential geometry, the study of completely integrable systems, etc. For example, the geometric problems studied by Darboux and Goursat when formulated in local coordinates lead to overdetermined systems of PDEs in non-normal form. Seiler's book deals with the formal theory of mainly non-normal systems of both ODEs and PDEs. The treatment requires sophisticated tools from differential geometry and commutative abstract algebra, which many applied mathematicians might not be familiar with. Seiler's book has the catchy title: Involution. Naively, completing a system of differential equations to involutive form compares with bringing a system of linear algebraic equations into triangular form by Gauss elimination. That process eliminates redundant equations and reveals inconsistent equations, if present. Once brought into triangular form, the solvability and dimension of the solution space of the algebraic system become apparent. In the case of differential equations there is more freedom; one may not only perform algebraic operations but also differentiate equations. Yet, the goals of Gauss elimination and the completion process are similar: The completion of a system of PDEs will reveal the consistency of the system and allows one to determine the size of the formal solution space. For example in Lie-point symmetry analysis [3], one has to solve an overde-termined system of linear homogeneous PDEs for the coefficients of the vector field. To design a reliable and powerful integration algorithm for such a system, it needs to be brought into a canonical form (called the involutive or passive form). Roughly speaking, the original system is appended by all its differential consequences. Next, highest-derivatives (often arising from cross-differentiations) are eliminated, and, if they occur, integrability conditions are added to the system. …