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Two problems in the theory of differential equations

Dimitry Leites

Abstract

1) The differential equation considered in terms of exterior differential forms, as É.Cartan did, singles out a differential ideal in the supercommutative superalgebra of differential forms, hence an affine supervariety. In view of this observation, it is evident that every differential equation has a supersymmetry (perhaps trivial). Superymmetries of which (systems of) classical differential equations are missed yet? 2) Why criteria of formal integrability of differential equations are never used in practice?

Two problems in the theory of differential equations

Abstract

1) The differential equation considered in terms of exterior differential forms, as É.Cartan did, singles out a differential ideal in the supercommutative superalgebra of differential forms, hence an affine supervariety. In view of this observation, it is evident that every differential equation has a supersymmetry (perhaps trivial). Superymmetries of which (systems of) classical differential equations are missed yet? 2) Why criteria of formal integrability of differential equations are never used in practice?

Paper Structure

This paper contains 14 sections, 18 equations.

Table of Contents

  1. Introduction
  2. History: Faddeev--Popov ghosts as analogs of momenta in '' odd" mechanics
  3. Towards future
  4. Supersymmetries of differential equations
  5. Sophus Lie, Élie Cartan and supersymmetry
  6. Gauge fields from super point of view.
  7. Super specifics, unexplored to this day, cf. MaGDel
  8. Well-known example: SUSY of Maxwell and Dirac equations.
  9. Witten's discoveries: two more types of SUSY related to differential equations.
  10. Kirillov's observation: a SUSY-related interpretation of a known property of solutions of the Sturm--Liouville equation.
  11. Formal integrability of differential systems and non-holonomic structures
  12. Criteria for formal integrability of differential equations
  13. Hertz, É.Cartan, Carathéodori, V.Sergeev, and non-holonomic structures.
  14. Nonholonomic structures and the corresponding analogs of the curvature tensor