On the Stability Degree

Let A be a nondegenerate operator: |A| 6 = 0 (This is equivalent to the condition |B| 6 = 0.) Then the eigenvalues of A can be of three types: real pairs ±a, pure imaginary pairs ±ib, and quadruples ±a± ib. It was shown in [1] that the linear system (2) with nondegenerate operator A that admits an integral in the form of a nondegenerate quadratic form (1) can always be reduced to a system of Hamiltonian equations. Therefore, the theory developed below applies to this (formally, more general) case. The instability degree u of system (2) is defined as the number of roots (counted according to their multiplicities) of the characteristic equation of A in the right half-plane, and the stability degree s is defined as the number of pure imaginary roots (counted according to their multiplicities) of the characteristic equation of A. By i and i− we denote the positive and negative indices of inertia of the quadratic form (1). By the Thomson theorem [2], u ≡ i−mod 2. (3) In particular, if i− (or i) is odd, then so is u. Consequently, the characteristic polynomial of A necessarily has a positive real root in this case; therefore, x = 0 is an unstable equilibrium. More precisely, the classical Thomson theorem holds for linear mechanical systems subjected to additional gyroscopic and dissipative forces. The congruence (3) was proved in [3] for systems of general form (with Ḣ ≤ 0). If Ḣ = 0, then the Thomson theorem is equivalent to the following assertion on the stability degree: s is even if and only if