Local relative integral invariants determined by the phase portrait of a vector field on the plane

Abstract

by the phase portrait of a vector field on the UDC 517.92 We/Tx some invariant measure for a given vector hem on the plane. The pair (the phase portrait of the vector field, the invariant measure of the vector held) determines a Lie suba/gebra in the algebra of all smooth vector 5elds on the plane, namely the stationary suba/gebra of the pair. An element of the suba/gebra has a relative integral invariant, namely: the integral of the above measure along sets bounded by phase curves of the initial vector/Te/d and the element of suba/gebra. The main result of the paper is the following:Theorem. Relative integral invariants in the genera/situation (more exactly, for the finite-modal case) are expressed in terms of elementary functions of suitable phase coordinates.Degenerate relative integral invariants, for which the above theorem is not valid, appear in twoparametric families of the above objects in the general situation.* Unfortunately, this connection is not a straightforward one, because it depends not only on the parity of j, but also on the parity of the first integral's homogeneous part.