Systems in Classical Mechanics  

In General, Space of Theories > s.a. formalism [dynamical systems]; physical 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]; Punzi & Wohlfarth PRE(09)-a0810 [geometric]; > s.a. chaos.
@ Related topics: Truesdell 84 [continuum mechanics]; Boccaletti & Pucacco 96, 99 [orbits]; Zhou et al mp/02 [volume-preserving].

Non-Linear Systems > s.a. chaos; ergodic and Mixing Systems; integrable systems; KAM Theorem.
* Types: They include unstable systems, which in turn include chaotic ones.
@ General references: Hilborn & Tufillaro AJP(97)sep [RL]; Rudolph & Hwa PhyA(04) [approach].
@ Texts: Kaplan & Glass 95 [II, bio-oriented]; Solari et al 96; Lam 98 [IIb]; Scheck 10; Stetz 16 [including numerical].
@ Non-integrable: Vernov mp/03-proc [solutions using Painlevé analysis].
@ Related topics: Kahn & Zarmi AJP(00)oct [method of normal forms]; Pelster et al PRE(03)mp/02 [periods of periodic orbits]; Awrejcewicz & Lamarque 03 [non-smooth systems]; Carpintero MNRAS(08)-a0805 [correlation dimension of an orbit]: Landa & McClintock PRP(13) [systems with fast and slow motions]; > s.a. Amplitude Death; Attractor; Topological Entropy; Topological Transitivity.

Few-Body Systems > s.a. classical particle models; motion of gravitating bodies.
* Two-body problem: It can be reduced to a one-body problem around the center of mass, with reduced mass

μ:= m1m2/(m1+m2) .

@ Two-body problem: Shchepetilov & Stepanova mp/05 [on constant-curvature spaces]; Droz-Vincent IJTP(11)-a1006 [two relativistic particles].
@ Three-body problem: Nauenberg PLA(01) [inverse-power-law forces, periodic orbits]; Jiménez-Lara & Piña JMP(03) [r–2 potential]; Calogero et al JMP(03) [solvable]; Valtonen & Karttunen 06; Yamada & Asada PRD(11)-a1011 [relativistic, collinear solutions]; Musielak & Quarles RPP(14)-a1508 [rev, historical and modern developments]; > s.a. newtonian orbits [including celestial mechanics]; dynamics of gravitating bodies; Three-Body Physics.

Many-Body Systems > s.a. composite quantum systems; correlations; Emergent Systems; stochastic processes; thermodynamics.
* Idea: It studies the emergence of new phenomena from interactions of many "elementary" objects, modeling the dynamics as stochastic and/or using statistical methods.
@ General references: Pines 61; Mattis ed-93 [exactly-solvable models]; Antoni et al cm/99-proc [infinite-range attractive interactions, and phase transitions]; Lipparini 08; Diacu et al JNS(12)+JNS(12)-a0808 [on constant-curvature spaces]; Kamenev 11 [non-equilibrium field-theoretical methods]; Vicsek & Zafeiris PRP(12) [collective motion]; Kuzemsky IJMPB(15)-a1507 [variational inequalities approach].
@ Relativistic: Louis-Martinez PLA(03)ht [solutions].
> Examples: see computational physics; gas; gravitating matter.

Examples > s.a. classical particle models; cosmological models; 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 the floor is flat and any obstacle or portion of wall is convex, or if the 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. causality violation; 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]; Ivashchuk & Melnikov G&C(09) [and cosmological models]; Chernov & Markarian 06 [chaotic].
@ Central potentials: Poole et al AJP(05)jan; > s.a. Bertrand's Theorem; Runge-Lenz Vector.
@ Other examples: Perelomov CMP(81)mp/01, TMP(02)mp/01 [Kovalevskaya top]; Abalmassov & Maljutin phy/04 [falling pen]; Tuleja et al AJP(07)mar [Feynman's wobbling plate]; Cherubini et al a0706/ARMA [bouncing ball with dissipation]; Lucarini JSP(12) [dynamical system with a stochastic perturbation]; Avanesov & Manko JRLR-a1304 [linear potential, Hilbert-space formalism and tomographic probability distribution]; MacKay & Salour AJP(15)jan-a1406 [simple exotic examples]; Simonella & Spohn BAMS-a1501 [review of book on Hard Spheres and Short-range Potentials].

Other Types of Systems > s.a. constrained, hamiltonian, lagrangian systems; physical systems; quantum systems.
@ Exactly solvable: 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]; Delphenich AdP(09)-a0812 [variational formulation]; Galley PRL(13)-a1210 [Lagrangian and Hamiltonian dynamics]; > s.a. dissipative systems; hamiltonian systems; quantum systems; variational principles.
@ With supersymmetry: Suen et al PLA(00); Heumann JPA(02)ht [Coulomb problem].
@ Classical time crystals: Shapere & Wilczek PRL(12)-a1202 + Zakrzewski Phy(12) ["time crystals", with motion in their lowest-energy state]; Yao et al a1801 [periodically driven Hamiltonian dynamics coupled to finite-temperature baths]; > s.a. crystals [quantum time crystals].
@ Other generalized types: Bender et al JPA(08) [with complex energy, quantum-like behavior]; > s.a. Discrete Models; Open Systems; p-Adic Systems; Pseudoclassical Systems.
> Related topics: see Ermakov and Hill System; Extended Objects; oscillator; Rigid Body; Rotor; turbulence.

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