  We prove several interesting equalities for the integrals of higher order
derivatives on the homogeneous groups. As consequences, we obtain the sharp
Hardy--Rellich type inequalities for higher order derivatives including both
the subcritical and critical inequalities on the homogeneous groups. We also
prove several uncertainty principles on the homogeneous groups. Our results
seem to be new even in the case of Euclidean space $\mathbb R^n$ and give a
simple proof of several classical Hardy--Rellich type inequalities in $\mathbb
R^n$.
