On Linear Equations of Dynamics

We consider linear autonomous systems of second-order differential equations that do not contain first derivatives of independent variables. Such systems are often encountered in classical mechanics. Of particular interest are cases where external forces are not potential. An important special case is given by the equations of nonholonomic mechanics linearized in the vicinity of equilibria of the second kind. We show that linear systems of this type can always be represented as Lagrange and Hamilton equations, and these equations are completely integrable: they admit complete sets of independent involutive integrals that are quadratic or linear in velocity. The linear integrals are Noetherian: they appear due to nontrivial symmetry groups.