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 + \(1\over2\)mω2q2 = \(1\over2\)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:=
f0/Δf, 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 29 apr 2021