Classical
Mechanics |

**In General** > s.a. history
of physics; state.

* __Idea__: The class of
physical theories in which the system has a well-defined history, with dynamics
described by (differential or functional) equations of motion on a
configuration space *C* (infinite-dimensional in field theory);
The logical structure is that of a Boolean lattice.

* __History__: Since the
1970s, when it was realized that chaos arises even with few degrees of freedom
in non-linear systems, the perspective in the field has changed.

* __Subjects of interest__:
Various general aspects of non-linear dynamical systems, like chaos and turbulence.

* __Important recent applications__: Galaxy formation; Saturn ring structure.

**Related Topics**

> __Approaches__: see formulations [including dynamical systems]; hamiltonian dynamics; lagrangian dynamics.

> __Concepts__: see Degrees of Freedom; energy; entropy; force;
inertia; information;
statistical mechanics; time; Trajectory.

> __Systems__: see classical systems; field theory.

> __Phenomena__: see
chaos;
Friction.

> __Results__: see Bertrand's
Theorem; noether theorem; Work-Energy Theorem.

**Variations and Generalizations** > s.a. hilbert space;
higher-order lagrangians; MOND.

* __Standard ones__: Special
and general relativistic dynamics; Quantum dynamics.

* __And quantum theory__:
Quantum corrections, if taken into account, introduce modifications to classical dynamics.

* __Barbour-Bertotti__: Classical,
without the ideal elements of inertial frames and external time (> see parametrized
theories).

* __Super classical quantum mechanics__:
A proposed theory which is equivalent to the Heisenberg, Schrödinger,
and Dirac non-relativistic quantum mechanics, with the addition of Born's probabilistic
interpretation of the wave function built in from the start.

@ __Relationship with special and general relativity__:
Havas RMP(64);
NCB 102(88)495 [Newton's third law].

@ __Relationship with quantum mechanics__:
Dittrich & Reuter 01;
Savickas AJP(02)aug [and general relativity];
Valentini PLA(04)qp/03 [non-quantum
systems]; Bojowald et al PRD(12)-a1208 [higher time derivatives in effective dynamics];
Kurihara et al JTAP(14)-a1312 [classical mechanics as an equilibrium state of statistical mechanics].

@ __Quantum correction__: Bouda & Djama PLA(01)
[second law]; Ward MPLA(02).

@ __Post-Newtonian__: Chicone gq/01-conf
[equations of motion are functional differential equations].

@ __Nambu mechanics__: Lassig & Joshi LMP(97)
[constrained systems]; > s.a. poisson structure.

@ __Supermechanics, anticommuting degrees of freedom__: Cariñena & Figueroa JPA(97)
[Hamiltonian and Lagrangian]; Bruce et al a1606 [geometric].

@ __Other examples__: Salesi IJMPA(02)qp/01 [spinning particles].

@ __Stochastic__: Guerra PRP(81);
Streater RPMP(93)
[and Markov chains]; Zambrini a1212 [path-integral inspired stochastic deformation of Lagrangian and Hamiltonian approaches]; > s.a. stochastic processes.

@ __Other generalizations__: Lamb AJP(01)apr
[super-classical quantum mechanics]; Kisil JPA(04)qp/02,
Brodlie & Kisil in(03)qp,
Brodlie JMP(04)
[*p*-mechanics]; Khrennikov & Nilsson 04 [*p*-adic; r BAMS(06)];
Kisil RPMP(05)
[*p*-mechanics and field theory]; Lämmerzahl & Rademaker PRD(12)-a0904 [higher-order equations of motion];
García-Morales CNSNS(16)-a1507 [semipredictable]; > s.a. conformal invariance.

**References** > s.a. BRST transformations;
parametrized systems [including
relationalism]; spacetime; topological
field theories.

@ __Resources__: issue AJP(00)apr [reviews].

@ __Texts__: Hertz re-56 [classic]; Mercier 59;
Bergmann 62 [I]; Pars 65;
Aharoni 72; Desloge 82;
Raychaudhuri 83; Griffiths 85;
Fowles 86; Kibble 86;
Reichert 90;
Matzner & Shepley 91; Marsden 92;
Barger & Olsson 95; Marion & Thornton
95; Hestenes 99; Teodorescu 07, 08, 09 [comprehensive].

@ __Texts, II__: Chow 95; Kibble & Berkshire
04;
Taylor 05; Verma 09; Johnson 10;
Kleppner & Kolenkow 10 [II advanced]; Chaichian et al 12;
Chow 13; Rajeev 13; Englert 15; Iro 15; Nolte 15 [geometry, non-linear dynamics, complex systems, networks, relativity; r PT(15)]; Bettini 16; Nolting 16.

@ __Texts, III__: Synge & Griffith 59;
Saletan & Cromer 71; Sudarshan & Mukunda
75; Abraham & Marsden
78; Goldstein 80;
Gallavotti 83; Woodhouse 87;
Arnold 89; Calkin 96 [Lagrangian and Hamiltonian];
Thirring 97; Hand & Finch 98;
Corinaldesi 99;
Greiner 02; Fasano
& Marmi 06; DiBenedetto 10; Shapiro & de Berredo-Peixoto 13.

@ __Geometrical emphasis__:
Marmo
et al 85; Giachetta et al 10;
Holm 11; Lessig a1206 [primer].

@ __Problems and solutions__:
Tonti 77 [method]; Lim 94;
de Lange & Pierrus 10.

@ __Other emphasis__: Lanczos 49 [variational methods];
Rasband 83, Abraham & Ratiu
94 [symmetries]; Katok & Hasselblatt
95, Scheck 10 [non-linear/chaos];
José & Saletan 98;
Johns 05 [relativity and quantum mechanics];
Müller-Kirsten 08 [relativity];
Thorne & Blandford 15 [applications];
Sussman & Wisdom 15 [conceptual-computational]; Hentschke 17 [numerical, theory of elasticity, engineering applications]; > s.a. computational physics.

@ __Foundations__: Hesse AJP(64)dec
[philosophical]; Desloge AJP(89)aug;
Gallavotti in(06)mp/05;
Darrigol SHPMP(07)
[necessary nature]; Prestron SHPSA(08)
[Mach and Hertz]; Sklar 13;
Hartmann a1307-PhD;
Alonso-Blanco & Muñoz-Díaz a1404,
a1411; Lubashevsky a1603 [from "microlevel reducibility"].

> __Online resources__:
Internet
Encyclopedia of Science pages.

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send feedback and suggestions to bombelli at olemiss.edu – modified 6
dec 2017