| Description |
1 online resource (448 pages). |
| Physical Medium |
polychrome |
| Description |
text file |
| Series |
De Gruyter Studies in Mathematical Physics
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De Gruyter studies in mathematical physics.
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| 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. |
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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. |
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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. |
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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. |
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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. |
| Note |
3.3.7 Inertia theorems and stability of gyroscopic systems. |
| Summary |
This work gives a complete overview on the subject of nonconservative stability from the modern point of view. Relevant mathematical concepts are presented, as well as rigorous stability results and numerous classical and contemporary examples from mechanics and physics. The book shall serve to present and prospective specialists providing the current state of knowledge in this actively developing field. The understanding of this theory is vital for many areas of technology, as dissipative effects in rotor dynamics orcelestial mechanics. |
| Bibliography |
Includes bibliographies (pages 387-422) and indexes. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Stability -- Mathematical models.
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Stability -- Mathematical models. |
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Stability. |
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Eigenvalues.
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Eigenvalues. |
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Oscillations.
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Oscillations. |
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Mechanical impedance.
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Mechanical impedance. |
| Genre/Form |
Electronic books.
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Electronic books.
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| Other Form: |
Print version: Kirillov, Oleg N. Nonconservative Stability Problems of Modern Physics. Berlin : De Gruyter, ©2013 9783110270341 |
| ISBN |
9783110270433 (electronic book) |
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3110270439 (electronic book) |
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9783110270341 |
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311027034X |
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