Oscillators| Oscillators |
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-in
[harmonic oscillators in different theories].
Classical Harmonic Oscillator
* Lagrangian: L = 
m(x·)2 –
m
2x2,
with a parameter
(= (k/m)1/2 for a spring).
* Symplectic structure:
Phase space 
=
{(q, p)};
Symplectic 2-form 
=
dp 
dq = r d
dr.
* Hamiltonian: For a single oscillator, and for n coupled oscillators,
respectively,
H = p2/2m +
m2q2 =
r2 , H =
Gab pa pb
+ Vab qaqb
;
The Hamiltonian vector field is XH = –
/.
@ 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. Helmholtz
Resonator; non-commutative; Pendulum;
semiclassical quantum mechanics [coupled to quantum].
* Pais-Uhlenbeck fourth order
oscillator:
It has equation of motion
d4q/dt4 +
(12+22)
(d2q/dt2) +
1222 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-in
[superintegrable, position-dependent m]; Pereira et al PLA(07)
[chaotic,
phase and period]; Bervillier a0812 [conformal
mappings and other methods]; Fernández a0910.
@ 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 a0902 [2D].
@ Different configuration spaces: Cariñena et al a0709 [constant curvature, Cayley-Klein
approach].
@ Dirac
oscillator: de Lima PLA(08)-a0707;
Sadurní et al a0902/AP
[coupled to an external field].
@ 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 a0903 [dissipative,
coupled
to
a
bath]; Luo & Guo a0906 [infinite-dimensional Hamiltonian formalism].
Related Concepts > s.a. Detectors
[accelerated]; Separatrix; thermodynamics.
* Quality factor Q: For an oscillator, it is some measure of
its coupling to other systems, and gives the decay time for an oscillation
of
frequency as 
= Q/.
@ Coupled oscillators: Denardo et al AJP(99)mar
[parametric instability]; > s.a. Relaxation.
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send feedback and suggestions to bombelli at olemiss.edu – modified 6
oct
2009