On an invariant measure for nonholonomic \(RL\)-systems on Lie groups (Q1358184)

A class of nonholonomic systems on Lie groups \((RL\)-system) has been introduced by \textit{A. P. Veselov} and \textit{L. E. Veselova} [Funct. Anal. Appl. 20, 308-309 (1986); translation from Funkts. Anal. Prilozh. 20, No. 4, 65-66 (1986; Zbl 0621.58024) and Math. Notes 44, No. 5, 810-819 (1988); translation from Mat. Zametki 44, No. 5, 604-619 (1988; Zbl 0694.58021)]. For these systems, kinetic energy is defined by a left-invariant metric and bonds are determined by a set of right-invariant 1-forms. It has also been demonstrated in these papers that the Euler-Arnold equations of such systems have an invariant measure on the Lie group (such groups are called unimodular). As a corollary they prove that the measure is preserved on the phase space of the systems on unimodular Lie groups. The authors quoted above conjectured that the condition of unimodularity can be discarded. In this short communication we prove the validity of this conjecture.