In General, Space of Theories > s.a. formalism [dynamical
systems]; states; symmetries;
symplectic structures.
* Construction of theories: Use symmetries and invariances.
@ New theories from old: Khorrami & Aghamohammadi NCB(98)ht/96.
@ Stability of an equilibrium point: Birtea & Puta JMP(07)
[equivalence of methods].
@ Stability of a solution:
González & Hernández a0705 [inequivalence
with geodesic deviation for Jacobi metric]; > s.a. chaos.
@ Related topics: Truesdell 84 [continuum mechanics]; Boccaletti & Pucacco
96 [orbits];
Zhou et al mp/02 [volume-preserving].
Non-Linear Systems > s.a. Attractor; chaos;
integrable; KAM.
* Types: They include
unstable systems, which in turn include chaotic ones.
@ General references: Hilborn & Tufillaro AJP(97)RL;
Rudolph & Hwa PhyA(04)
[approach].
@ Texts: Kaplan & Glass 95 [II, bio-oriented]; Lam 96, 98 [IIb];
Solari et al 96; Scheck 97.
@ Non-integrable: Vernov mp/03-in
[solutions using Painlevé analysis].
@ Related topics: Kahn & Zarmi AJP(00)
[method of normal forms]; Pelster et al mp/02 [periods
of periodic orbits]; Carpintero a0805 [correlation dimension of an orbit].
Few-Body Systems > s.a. orbits
of gravitating bodies.
* Two-body problem: It can be reduced to a one-body problem around the
CM with reduced mass
:= m1m2/(m1+m2)
.
@ Two-body problem: Shchepetilov & Stepanova mp/05 [on
constant curvature spaces].
@ Three-body systems: Nauenberg PLA(01)
[periodic orbits]; Jiménez-Lara & Piña JMP(03)
[r–2 potential]; Calogero et
al JMP(03)
[solvable]; Valtonen & Karttunen 06 [r PT(07)apr]; > s.a. newtonian
orbits [including celestial mechanics].
Many-Body Systems > s.a. correlations;
gas; gravitating
matter; quantum
systems; thermodynamics.
* Idea: It studies the emergence
of new phenomena from interactions of
many "elementary" objects.
@ General references: Pines 61; Mattis ed-93; Antoni et al cm/99-in
[infinite-range attractive interactions, and phase transitions].
@ Relativistic: Louis-Martinez PLA(03)ht [solutions].
Discrete Systems > s.a. hamiltonian
systems; integrable systems; Sequential
Dynamical Systems.
* Bernoulli shift: The map {xn}
→ {x'n}, with x'n = xn+1,
between doubly infinite sequences (n
Z)
of binary numbers;
As
a dynamical system, it is Kolmogorov, with Kolmogorov entropy S = ln
2.
@ General references: Easton 98 [geometric methods].
@ Continuum limit:
Bergman
& Inan ed-04 [continuum
models]; Tarasov JPA(06)
[with long-range interactions].
Central Potentials > s.a. potential.
* Bertrand's theorem:
The only central potentials leading to closed orbits (for a range of initial
conditions) are the harmonic oscillator and
the Newtonian
potential.
@ General references: Poole et al AJP(05).
@ Bertrand's theorem: Bertrand CR(1873)
[translation Santos et al a0704];
Brown AJP(78);
in Goldstein 80; Tikochinsky AJP(88);
Gurappa et al MPLA(00)qp/99 [quantum
analog]; Zarmi AJP(02)
[using Poincaré-Lindstedt perturbation method].
>
Online resources: see ScienceWorld
page,
Wikipedia page.
Examples > s.a. Gyroscope;
Pendulum; Projectile
Motion.
* Billiard: If the walls
move in time, the bounce law states that the angle of incidence is not equal
to the angle of reflection; A billiard is known
to be chaotic if flat and any obstacle or portion of wall is convex,
or
if space has negative curvature (Hadamard 1898); In general, so is motion
in
a closed environment (compact configuration space), like a damped pendulum
with
an external periodic force, for some parameter values; > s.a. spectral
geometry.
@ Billiard: Liboff
PLA(01)
[wedge billiard]; Chernov JSP(06)
[Sinai billiard, statistical]; Rapoport et al CMP(07)
[as approximation to Hamiltonian
flow]; Wojtkowski CMP(07) [hyperbolic].
@ Other examples: Perelomov CMP(81)mp/01,
TMP(02)mp/01 [Kovalevskaya
top]; Abalmassov & Maljutin phy/04 [falling
pen]; Tuleja et al AJP(07)
[Feynman's wobbling plate]; Cherubini et al a0706-ARMA
[bouncing ball with dissipation].
Other Types of Systems > s.a. constrained, hamiltonian, lagrangian,
Open Systems; physical
systems; quantum
systems.
@ Exactly solvable: Alhaidari mp/03 [larger
class]; Calogero JMP(04); Bruschi
& Calogero JMP(06); > s.a. chaotic
systems.
@ Quasi-exactly solvable: Brihaye et al JMP(95);
Avinash & Bhabani
PLA(98)
[and orthogonal polynomials]; Bender & Boettcher JPA(98)
[quartic family].
@ Metrizable system: Sharipov TMP(95); > s.a. jacobi
metric.
@ Non-conservative: Dreisigmeyer & Young JPA(03), JPA(04)
[and fractional derivatives]; Bucataru & Miron RPMP(07) [and non-linear connections].
@ With supersymmetry: Suen et al PLA(00);
Heumann JPA(02)ht [Coulomb
problem].
> Other: see dissipation; Ermakov and Hill
System; Extended
Objects; oscillator;
Rigid Body; Rotor; turbulence.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
3 jul 2008