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];
Krishnaswami & Senapati Res-a1901 [intro];
Gevorkian a2008 [on a conformally flat manifold, irreversibility];
> 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];
Rrapaj & Roggero a2005 [RBM neural networks];
Miller et al JGP-a2004 [generally covariant framework].
@ 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].
@ Conservative systems: Finn et al a1806 [as geodesic motion in curved manifolds].
@ 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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