The Stability of Circulatory Systems
- PDF / 204,185 Bytes
- 3 Pages / 612 x 792 pts (letter) Page_size
- 111 Downloads / 211 Views
ANICS
The Stability of Circulatory Systems Academician V. V. Kozlov * Received June 17, 2020; revised June 17, 2020; accepted June 22, 2020
Abstract—The stability of linear mechanical systems in nonpotential force fields is studied. Under the presence of circulatory forces, the system is nonconservative, yet it can always be presented in a Hamiltonian form. A point where the potential energy has its maximum is an unstable equilibrium, regardless of the presence of circulatory forces. Keywords: circulatory force, Hamiltonian system, Frobenius theorem, degree of instability DOI: 10.1134/S1028335820090062
1. HAMILTONIAN REPRESENTATION OF CIRCULATION SYSTEMS We consider linear systems of second-order differential equations
Mx + Px = 0,
x ∈ R n.
(1)
Here M T = M > 0 is a symmetric positive definite matrix, and P is an arbitrary nth-order square matrix. It is unambiguously represented as the sum
P = K + N, where K is a symmetrical matrix and N is a skew-symmetric matrix. The quadratic form T = 1 (Mx, x ) 2 will be the kinetic energy of the system, and V = 1 (Kx, x) 2 is its potential energy. The term –Nx in (1) represents the actual circulation force. Due to the presence of circulating forces, the total energy T + V in the general case is not saved. The stability theory of such systems is discussed in detail in [1]. Theorem 1. The nondegenerate linear substitution equation (1) is reduced to a linear Hamiltonian system with n degrees of freedom. As a consequence, we find that the spectrum of the linear equation (1) is symmetric not only about the real axis of the complex plane, but also about the purely imaginary axis. This fact is well known (see, for example, [1]). It is less obvious that linear equation (1)
always admits a quadratic first integral. Moreover, according to Wintner [2] and Williamson [3], every linear Hamiltonian system is completely integrable: it has n independent quadratic integrals with zero pairwise Poisson brackets. In addition to a complete set of quadratic integrals, Eq. (1), like any Hamiltonian system, also admits integral invariants of various orders. It is interesting to note that the problems of bifurcations of eigenvalues of linear Hamiltonian systems and circulation systems are usually considered independently (and often in parallel) (see, for example, [4]). Proof of Theorem 1. As we know, any real (n × n)matrix can be represented as the product of two real symmetric matrices, the first of which is nondegenerate. This fact was already noted by Frobenius in 1910; a simple proof is contained in [5]. Reducing the kinetic energy to the sum of squares, we find that M = I. After that we put P = AB, where A and B are symmetric matrices, and |A| ≠ 0 . But then Eq. (1) is represented in the form of the Lagrange equations
A −1 x + Bx = 0 with Lagrangian
1 ( A −1 x, x ) − 1 (Bx, x ). 2 2 They can be written in the form of linear Hamilton equations
x = ∂H , ∂y
y = − ∂H ∂x
(2)
with the required quadratic Hamiltonian Steklov Institute of Mathematics, Russian Academy of Sciences, M
Data Loading...
