In this paper, we extend the Lie theory of integration by quadratures of systems of ordinary differential equations in two different ways. First, we consider a finite-dimensional Lie algebra of vector fields and discuss the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a prescribed way. It turns out that the conditions can be expressed in a purely algebraic way. In the second step, we generalize the construction to the case in which we substitute the Lie algebra of vector fields by a module (generalized distribution). We obtain a much larger class of explicitly integrable systems, replacing standard concepts of solvable (or nilpotent) Lie algebra with distributional solvability (nilpotency).
Journal of Physics A Mathematical General
- Pub Date:
- May 2015
- Mathematical Physics;
- Mathematics - Classical Analysis and ODEs;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- 18 pages