Systems in Classical Mechanics

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 {x_n} → {x'_n}, with x'_n = x_n+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.

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