Nonconservative Stability Problems of Modern Physics.
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| Author / Creator: | Kirillov, Oleg N. |
|---|---|
| Imprint: | Berlin : De Gruyter, 2013. |
| Description: | 1 online resource (448 pages) |
| Language: | English |
| Series: | De Gruyter Studies in Mathematical Physics De Gruyter studies in mathematical physics. |
| Subject: | Stability -- Mathematical models. Eigenvalues. Oscillations. Mechanical impedance. Eigenvalues. Mechanical impedance. Oscillations. SCIENCE -- Energy. SCIENCE -- Mechanics -- General. Stability -- Mathematical models. Physik. SCIENCE -- Energy. SCIENCE -- Mechanics -- General. SCIENCE -- Physics -- General. Eigenvalues. Mechanical impedance. Oscillations. Stability -- Mathematical models. Stabilität Physikalisches System Nichtkonservative Kraft Electronic books. Electronic books. |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/11204593 |
Table of Contents:
- Preface; 1 Introduction; 1.1 Gyroscopic stabilization on a rotating surface; 1.1.1 Brouwer's mechanical model; 1.1.2 Eigenvalue problems and the characteristic equation; 1.1.3 Eigencurves and bifurcation of multiple eigenvalues; 1.1.4 Singular stability boundary of the rotating saddle trap; 1.2 Manifestations of Brouwer's model in physics; 1.2.1 Stability of deformable rotors; 1.2.2 Foucault's pendulum, Bryan's effect, Coriolis vibratory gyroscopes, and the Hannay-Berry phase; 1.2.3 Polarized light within a cholesteric liquid crystal; 1.2.4 Helical magnetic quadrupole focussing systems.
- 1.2.5 Modulational instability1.3 Brouwer's problem with damping and circulatory forces; 1.3.1 Circulatory forces; 1.3.2 Dissipation-induced instability of negative energy modes; 1.3.3 Circulatory systems and the destabilization paradox; 1.3.4 Merkin's theorem, Nicolai's paradox, and subcritical flutter; 1.3.5 Indefinite damping and parity-time (PT) symmetry; 1.4 Scope of the book; 2 Lyapunov stability and linear stability analysis; 2.1 Main facts and definitions; 2.1.1 Stability, instability, and uniform stability; 2.1.2 Attractivity and asymptotic stability.
- 2.1.3 Autonomous, nonautonomous, and periodic systems2.2 The direct (second) method of Lyapunov; 2.2.1 Lyapunov functions; 2.2.2 Lyapunov and Persidskii theorems on stability; 2.2.3 Chetaev and Lyapunov theorems on instability; 2.3 The indirect (first) method of Lyapunov; 2.3.1 Linearization; 2.3.2 The characteristic exponent of a solution; 2.3.3 Lyapunov regularity of linearization; 2.3.4 Stability and instability in the first approximation; 2.4 Linear stability analysis; 2.4.1 Autonomous systems; 2.4.2 Lyapunov transformation and reducibility; 2.4.3 Periodic systems.
- 2.4.4 Example. Coupled parametric oscillators2.5 Algebraic criteria for asymptotic stability; 2.5.1 Lyapunov's matrix equation and stability criterion; 2.5.2 The Leverrier-Faddeev algorithm and Lewin's formula; 2.5.3 Müller's solution to the matrix Lyapunov equation; 2.5.4 Inertia theorems and observability index; 2.5.5 Hermite's criterion via the matrix Lyapunov equation; 2.5.6 Routh-Hurwitz, Liénard-Chipart, and Bilharz criteria; 2.6 Robust Hurwitz stability vs. structural instability; 2.6.1 Multiple eigenvalues: singularities and structural instabilities.
- 2.6.2 Multiple eigenvalues: spectral abscissa minimization and robust stability3 Hamiltonian and gyroscopic systems; 3.1 Sobolev's top and an indefinite metric; 3.2 Elements of Pontryagin and Krein space theory; 3.3 Canonical and Hamiltonian equations; 3.3.1 Krein signature of eigenvalues; 3.3.2 Krein collision or linear Hamiltonian-Hopf bifurcation; 3.3.3 MacKay's cones, veering, and instability bubbles; 3.3.4 Instability degree and count of eigenvalues; 3.3.5 Graphical interpretation of the Krein signature; 3.3.6 Strong stability: robustness to Hamiltonian's variation.