LIOUVILLE AND JACOBI THEOREMS FOR VECTOR DISTRIBUTIONS

Olga Krupkov\'{a}

Abstract. The paper contains a generalization of the Liouville and Jacobi
integration methods, known from the classical calculus  of variations, to the
case of characteristic distributions of closed $2$-forms of constant rank on
smooth manifolds. It is shown that the results can be used to find integral
manifolds of completely integrable distributions on smooth manifolds. In this
way, the theorems of Liouville and Jacobi are generalized to a wide class of
generally higher order ordinary and partial differential equations, possibly
degenerate, which are not supposed to come from a Lagrangian.

Keywords: completely integrable distribution, closed $2$-form, characteristic
distribution of a closed $2$-form, Liouville theorem, Jacobi theorem,
symmetries of differential equations, integration of differential equations,
degenerate equations, non-variational equations, connections on fibered
manifolds.

MS classification: 58A30, 34A05, 34A08, 70H10, 70H20.