Cartan-Geometric Approaches to Submaximally Symmetric Ordinary Differential Equations

Abstract

This thesis is concerned with a symmetry classification problem for ordinary differential equations (ODEs) that dates back to Sophus Lie. We focus on higher order ODEs, i.e. scalar ODEs of order greater than or equal to 4 or vector ODEs of order greater than or equal to 3, up to contact transformations. The maximal contact symmetry algebra dimensions for these ODEs are known. We determine for all higher order ODEs the submaximal (i.e. next largest realizable) contact symmetry dimensions. Using the known contact fundamental (generalized Wilczynski or C-class) invariants for higher order ODEs, we also determine submaximal symmetry dimensions for several classes of the ODEs that are contact invariant. Moreover, we give a complete local classification of all submaximally symmetric vector ODEs of C-class, i.e. ODEs with symmetry dimensions realizing submaximal symmetry dimensions that are characterized by the vanishing of all generalized Wilczynski invariants. Our results refine the classical results for scalar ODEs, and also provide generalizations of those results to vector ODEs.