Nonconservative Stability Problems of Modern Physics by Oleg N. Kirillov 9783110655407 3110655403

Nonconservative Stability Problems of Modern Physics Oleg N. Kirillov – Ebook Instant Download/Delivery ISBN(s): 9783110655407,3110655403

Product details:

• ISBN 10: 3110655403
• ISBN 13: 9783110655407
• Author: Oleg N. Kirillov

Table contents:

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 Ponderomotive magnetism and the Cox–Levi theorem
1.2.6 Stability of Lagrange’s triangular libration points L4 and L5
1.2.7 Modulational instability
1.3 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
1.3.6 𝒫𝒯-symmetry, pseudo-Hermiticity, and G-Hamiltonian matrices
1.3.7 Heavy damping and high-order exceptional points
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 systems
2.2 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 oscillators
2.5 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, Bilharz, and Jury 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 stability
2.6.3 Abscissa minimization and the self-stability of bicycles
3 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
3.3.7 Inertia theorems and stability of gyroscopic systems
3.3.8 Positive and negative energy modes and Krein signature
3.3.9 Dispersive wave propagation in conservative systems
3.3.10 Absolute and convective instability
4 Reversible and circulatory systems
4.1 Reversible systems
4.2 Nonconservative positional forces
4.3 Circulatory systems
4.3.1 Divergence and flutter instabilities
4.3.2 Multiple parameter families of circulatory systems
4.3.3 Generic singularities on the stability boundary
4.4 Perturbation of eigenvalues
4.4.1 Simple eigenvalue
4.4.2 Double eigenvalue of geometric multiplicity 1
4.4.3 Double eigenvalue of geometric multiplicity 2
4.4.4 Triple eigenvalue of geometric multiplicity 1
4.5 Geometry of the stability boundary
4.5.1 Linear and quadratic approximations at smooth points
4.5.2 Singularities in two-parameter circulatory systems
4.5.3 Example. Stabilization of comfortable walking
4.5.4 Singularities in three-parameter circulatory systems
4.5.5 The cone αα and Merkin’s instability theorem
4.5.6 Example: a brake disk in distributed frictional contact
4.5.7 Example: stability of an airfoil in an inviscid flow
4.6 Eigencurves, their crossing and veering
4.6.1 Convex flutter domain: conical point αα
4.6.2 Convex/concave flutter domain: smooth points α2
4.7 Parametric optimization of circulatory systems
4.7.1 Example: optimization of Ziegler’s pendulum
4.7.2 A nonsmooth and nonconvex optimization problem
4.7.3 The gradient of the critical load
4.7.4 An infinite gradient at the crossing of the eigencurves
4.7.5 Improving variations and necessary conditions for optimality in the case where the eigencurves cross
5 Influence of structure of forces on stability
5.1 Undamped potential systems
5.1.1 Lagrange’s theorem and Poincaré instability degree
5.1.2 Rayleigh’s theorem on movement of eigenvalues
5.1.3 Steady-state bifurcation
5.2 Damped potential systems
5.2.1 Overdamped and heavily damped systems
5.2.2 Self-stable and heavily-damped TMS bicycle
5.2.3 Indefinitely damped systems
5.3 Undamped gyroscopic systems
5.3.1 Extension of Rayleigh’s theorem
5.3.2 Criteria of gyroscopic stabilization
5.4 Damped gyroscopic systems
5.4.1 Kelvin–Tait–Chetaev theorem
5.5 Circulatory systems with and without velocity-dependent forces
5.5.1 Merkin’s theorem and Bulatovic’s flutter condition
5.5.2 Bottema–Lakhadanov–Karapetyan theorem
5.5.3 Stabilizing and destabilizing damping configurations
6 Dissipation-induced instabilities
6.1 Crandall’s gyropendulum
6.1.1 Conservative gyroscopic stabilization and its destruction by stationary damping
6.1.2 Singular threshold of the nonconservative gyroscopic stabilization
6.1.3 Imperfect Krein collision and exchange of instability between negative and positive energy modes
6.2 Gyroscopic stabilization of nonconservative systems
6.2.1 The case of m=2 degrees of freedom
6.2.2 The case of arbitrary even m
6.3 Near-Hamiltonian systems
6.4 Gyroscopic and circulatory systems as limits of dissipative systems
7 Nonself-adjoint boundary eigenvalue problems for differential operators and operator matrices dependent on parameters
7.1 Adjoint boundary eigenvalue problems
7.2 Perturbation of eigenvalues
7.2.1 Semisimple eigenvalues
7.2.2 Multiple eigenvalues with the Keldysh chain
7.2.3 Higher order perturbation terms for double nonderogatory eigenvalues
7.2.4 Degenerate splitting of double nonderogatory eigenvalues
7.3 Example: a rotating circular string with an elastic restraint
7.4 Example: the Herrmann–Smith paradox
7.4.1 Formulation of the problem
7.4.2 Stationary flutter domain and mobile divergence region
7.4.3 Sensitivity of the critical flutter load to the redistribution of the elasticity modulus
7.5 Example: Beck’s column loaded by a partially follower force
7.5.1 The stability-divergence boundary (point A)
7.5.2 The flutter threshold of Beck’s column (point C)
7.5.3 The singularity 02 on the stability boundary (point B)
8 The destabilization paradox in continuous circulatory systems
8.1 Movement of eigenvalues under a velocity-dependent perturbation
8.1.1 Generalized boundary eigenvalue problem
8.1.2 Variation of parameters that is transversal to the stability boundary
8.1.3 Variation of parameters that is tangential to the stability boundary
8.1.4 Transfer of instability between modes
8.1.5 Drop in the critical frequency
8.2 Singular threshold of the flutter instability
8.2.1 Drop in the critical flutter load
8.2.2 The “no drop” condition and the tangent cone to the domain of asymptotic stability
8.3 Example: dissipation-induced instability of Beck’s column
8.3.1 Beck’s column without damping
8.3.2 Beck’s column with Kelvin–Voigt and viscous damping
8.3.3 Viscoelastic Beck’s column with a dash-pot
8.3.4 Ziegler’s pendulum with a dash-pot
8.4 Application to finite-dimensional systems
8.4.1 The destabilization paradox in Ziegler’s pendulum
8.5 Experimental detection of the Ziegler–Bottema destabilization paradox
8.5.1 The destabilizing effect of external damping
8.5.2 Ziegler–Bottema paradox due to vanishing external damping
8.5.3 Ziegler–Bottema paradox for the Pflüger column with external damping
8.5.4 The ‘flutter machine’: design, realization, and validation
8.5.5 Experimental results versus theoretical and computational predictions
9 The MHD kinematic mean field α2-dynamo
9.1 Eigenvalue problem for α2-dynamo
9.2 Uniform α-profiles generate only nonoscillatory dynamos
9.2.1 Conducting exterior: self-adjointness in a Krein space
9.2.2 Basis properties of eigenfunctions
9.2.3 Spectral mesh of eigencurves
9.2.4 Deformation of the spectral mesh via transition from conducting to insulating surrounding
9.3 Nonhomogeneous α-profiles and the conducting exterior
9.3.1 l≥0: definite Krein signature prohibits formation of complex eigenvalues
9.3.2 l=0: oscillating solutions from the repeated decaying modes with mixed Krein signature
9.3.3 l=0: Fourier components of α(x) determine the unfolding pattern of the spectral mesh
9.4 Insulating boundary conditions induce unstable oscillations
9.4.1 l=0: complex unfolding of double eigenvalues with definite Krein signature
10 Campbell diagrams of gyroscopic continua and subcritical friction-induced flutter
10.1 Friction-induced vibrations and sound generation
10.2 Example. Subcritical flutter of a rotating circular string
10.3 Axially symmetric rotor with anisotropic stator
10.3.1 Sensitivity analysis of the Campbell diagram
10.3.2 MacKay’s eigenvalue cones and instability bubbles
10.3.3 Double-coffee-filter singularity near the crossings with definite Krein signature
10.3.4 Unfolding MacKay’s cones with mixed Krein signature
10.3.5 Indefinite damping as a reason for subcritical flutter
10.3.6 Destabilizing role of circulatory forces
10.4 Example: eigenvalue surfaces of the rotating circular string
10.5 How to play a disk brake?
11 Non-Hermitian perturbation of Hermitian matrices with physical applications
11.1 Eigenvalue movement through a 1:1 resonance in complex matrices
11.1.1 Diabolical point (DP): passing of eigenvalues
11.1.2 Exceptional point (EP): splitting of eigenvalues
11.2 Eigensurfaces associated with DPs
11.2.1 Complex perturbation of a Hermitian matrix family
11.2.2 DP in the spectrum of real symmetric matrices
11.2.3 How a DP unfolds into the conical wedge of Wallis
11.2.4 Inflating the diabolical point into an exceptional ring
11.2.5 Example: flutter instability in granular flow
11.3 Unfolding conical singularities in crystal optics
11.3.1 DPs in Hamilton’s conical refraction
11.3.2 Approximation of the dispersion surface near a DP
11.3.3 Eigensurfaces of absorption- and chirality-dominated crystals
11.4 Eigensurfaces associated with EPs
11.5 Perturbation of eigenvectors and Berry phase
11.5.1 Hermitian case: geometric phase around a DP
11.5.2 Non-Hermitian case: geometric phase around an EP
11.5.3 Geometric phase around an EP in a microwave cavity
12 Double-diffusive instabilities in hydro- and magnetohydrodynamics
12.1 Standard, azimuthal, and helical magnetorotational instability
12.1.1 Cylindrical Couette–Taylor flow
12.1.2 Paradox of Velikhov and Chandrasekhar
12.1.3 Magnetorotational instability in astrophysics and its mechanical analogues
12.1.4 Laboratory experiments with CT-flow in axial, azimuthal, and helical magnetic fields
12.2 Mathematical setting
12.2.1 Nonlinear equations and a steady state
12.2.2 Linearization with respect to nonaxisymmetric perturbations
12.3 Geometrical optics approximation
12.3.1 Dispersion relation of AMRI
12.3.2 Dispersion relation of HMRI
12.3.3 The threshold of the standard MRI
12.3.4 Singularities and the Velikhov–Chandrasekhar paradox
12.3.5 The singular threshold of the HMRI and connection of HMRI and SMRI through a spectral exceptional point for magnetic Rossby number Rb=−1
12.4 Extending the range of HMRI/AMRI to Keplerian flows with arbitrary Rb
12.4.1 Inductionless approximation
12.4.2 Extremal properties of the critical hydrodynamic Rossby number
12.4.3 Continuation of the Liu limits to arbitrary Rb
12.4.4 Scaling law of the inductionless MRI
12.4.5 Growth rates of HMRI and AMRI and the critical Reynolds number
12.4.6 HMRI and AMRI as magnetically destabilized inertial waves
12.4.7 AMRI as a dissipation-induced instability of Chandrasekhar’s equipartition solution
12.4.8 Transition from AMRI to the Tayler instability
12.5 AMRI as a double-diffusive instability
12.5.1 G-Hamiltonian structure of the diffusionless AMRI
12.5.2 Krein signature and splitting of double eigenvalues with Jordan block
12.5.3 Neutral stability curves
12.5.4 The Krein collision at the linear Hamilton–Hopf bifurcation threshold
12.5.5 Dissipative perturbation of simple imaginary eigenvalues
12.5.6 Weak ohmic diffusion destabilizes positive energy waves at low Pm
12.5.7 Diffusionless and double-diffusive criteria are connected at Pm=1
12.5.8 Double-diffusive instability at Pm≠1 and arbitrary Re and Rm
12.6 Concluding remarks

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