Oscillators and Vibrations  

In General > s.a. hamiltonian systems; Perturbation Methods; quantum oscillators; resonance.
* Excitation: Can be direct (small drive gives small response), or parametric.
* Modification – Mathieu equation: A harmonic oscillator with a small oscillating correction to m; It has a parametric resonance which may lead to chaotic instability (like a child on a swing).
@ References: Pippard 89; Dattoli & Torre NCB(95) [phase space, coherent states]; Roelofs AJP(01)aug [book reviews]; Kim & Noz qp/04-conf [harmonic oscillators in different theories]; Jenkins PRP(13)-a1109 [self-oscillation]; Fitzpatrick 13; Balachandran & Magrab 18; Franklin 20; Bistafa a2104 [history, Krafft].

Classical Harmonic Oscillator
* Lagrangian: L = \(1\over2\)m (x·)2 − \(1\over2\)m ω2 x2, with ω a parameter (= (k/m)1/2 for a spring).
* Symplectic structure: Phase space Γ = {(q, p)}; Symplectic 2-form Ω = dp ∧ dq = r dφ ∧ dr.
* Hamiltonian: For a single oscillator, and for n coupled oscillators, respectively,

H = p2/2m + \(1\over2\)2q2 = \(1\over2\)r2 ,   H = \(1\over2\)Gab pa pb + \(1\over2\)Vab qaqb ;

The Hamiltonian vector field is XH = −∂/∂φ.
@ Symmetries: Lutzky JPA(78) [and conservation laws]; Cariñena et al JPA(02)ht [rational, non-symplectic].
@ Other topics: Hojman JMP(93) [small oscillations]; Degasperis & Ruijsenaars AP(01) [equivalent Hamiltonians].

Other Types of Oscillator > s.a. Dirac Oscillator; Helmholtz Resonator; non-commutative; Pendulum; semiclassical quantum mechanics [coupled to quantum].
* Pais-Uhlenbeck fourth-order oscillator: It has equation of motion

d4q/dt4 + (ω12 + ω22) (d2q/dt2) + ω12 ω22 q = 0 .

@ Anharmonic / non-linear / perturbed: Gottlieb & Sprott PLA(01) [driven, chaotic]; Amore & Aranda PLA(03) [method]; Amore & Fernández EJP(05)mp/04 [period]; Cariñena et al mp/05-proc [superintegrable, position-dependent mass]; Pereira et al PLA(07) [chaotic, phase and period]; Bervillier JPA(09)a0812 [conformal mappings and other methods]; Fernández a0910; He PLA(10) [Hamiltonian approach]; Quesne EPJP(17)-a1607 [quartic and sextic]; Turbiner & del Valle a2011 [quartic, solution].
@ Relativistic: Beckers & Ndimubandi PS(96) [quantum]; Li et al JMP(05)hp; Kim & Noz JOB(05)qp [coupled]; Solon & Esguerra PLA(08)-a0806 [even polynomial potentials, periods]; Nagiyev et al NCB(09)-a0902 [2D]; Kowalski & Rembieliński PRA(10)-a1002 [massless]; Babusci et al a1209; Ivanov & Pavlovsky a1411 [Path Integral Monte-Carlo approach].
@ Different configuration spaces: Cariñena et al JMP(08)-a0709 [constant curvature, Cayley-Klein approach]; Quesne PLA(15)-a1411 [on the sphere and the hyperbolic plane].
@ Other generalized: Finkelstein & Villasante PRD(86) [anticommuting/Grassmann]; Meißner & Steinborn PRA(97) [anharmonic, iterative Ens]; Finkelstein IJMPA(98), Ellinas PS(99)*, add PS(00) [deformed]; Frydryszak RPMP(08)-a0708 [nilpotent].
@ Time-dependent: Colegrave et al PLA(88) [complex invariants]; Kim & Page PRA(01) [action-phase variables].
@ Damped: Maamache & Choutri JPA(00); Chee et al JPA(04)mp/02, JPA(04)mp/02 [N oscillators, phase space structure]; Chandrasekar et al JMP(07) [Lagrangian and Hamiltonian description]; Kumar et al PRE(09)-a0903 [dissipative, coupled to a bath]; Luo & Guo a0906 [infinite-dimensional Hamiltonian formalism].

Related Concepts > s.a. Detectors [accelerated]; Separatrix.
* Quality factor Q: A measure of an oscillator's coupling to other systems, defined by Q:= f0f, where Δf is the frequency width at half magnitude; It gives the decay time for an oscillation of frequency ω as τ = Q/ω.
@ Coupled oscillators: Denardo et al AJP(99)mar [parametric instability]; > s.a. Relaxation.
> Thermodynamics: see non-equilibrium thermodynamics [perturbed]; non-extensive statistics; statistical mechanical and thermodynamical systems.

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