|Formulations of Classical Mechanics|
In General – Dynamical Systems
* Idea: The main ingredients are a space of states (phase space, ...), and an algebra of observables; A (predictable) dynamical system then specifies an evolution law on the former or, in the Heisenberg picture, an automorphism a \(\mapsto\) at on the algebra of observables.
* Formulations: The main distinction is between discrete, differential and integral ones (from a variational principle); In classical mechanics, a Hamiltonian or a Lagrangian is useful in practice, but conceptually by no means necessary; In quantum theory, the situation is different.
* Structure: One does not need a metric on phase space, only a symplectic structure, to calculate evolutions; But in order to extract physical meaning one does need a metric [@ Klauder qp/01].
$ Def: A dynamical system is a triple (X, μ, φ) of a set, a probability measure, and a family of transformations on X, where (i) The measure μ is invariant under φ, and (ii) For all measurable A, μ(A) = μ(φ–1(A)).
* Degrees of chaoticity: In order of increasing chaoticity, systems can be technically divided into integrable, ergodic, mixing, Kolmogorov, Bernoulli, exact; They are considered chaotic if they are mixing or more.
@ General texts: Abraham & Shaw 82-88; Arrowsmith & Place 90, 92; Marsden & Ratiu 94; Katok & Hasselblatt 95; Collet & Eckmann 97 [maps].
@ Geometrical: Akin 93; Aoki & Hiraide 94 [topological]; Klauder & Maraner AP(97)qp/96 [deformation and phase space geometry]; Kambe 09 [and fluids]; Giachetta et al 10.
> Types of dynamical systems: see types of classical systems [including non-linear]; Attractors; ergodic; integrable; Mixing.
Newtonian Mechanics > s.a. Newton's Laws.
* Idea: The equations of motion are of the form m d2qi/dt2 = Fi(x(t), dq/dt) (second law); To determine the evolution, solve the equations of motion, or use the symmetries present in the problem and the conservation laws to obtain first integrals.
* Limits: Newtonian dynamics is an approximation valid when relativistic effects are small, and there are no charged particles in motion – in that case, the energy-momentum of the radiated field must be taken into account.
@ References: Pflug pr(87) [limits]; Caticha & Cafaro AIP(07)-a0710 [from information geometry]; Tymms 16 [I/II]; > s.a. Newton's Laws.
Approaches and Techniques > s.a. hamilton-jacobi theory;
statistical mechanics; symplectic structure.
@ Koopman-von Neumann operatorial approach: Abrikosov et al MPLA(03)qp [and quantization]; Gozzi & Mauro IJMPA(04) [Hilbert space and observables].
@ Mathematical: Aldaya & Azcárraga FdP(87) [and group theory]; Giachetta et al a0911 [in terms of fibre bundles over the time-axis].
@ Probabilistic / stochastic aspects: Lasota & Mackey 94; Nikolić FPL(06)qp/05; Volovich FP(11)-a0910; > s.a. stochastic processes.
@ Path-integral / quantum-field-theory techniques: Thacker JMP(97) [reparametrization-invariant]; Gozzi & Regini PRD(00)ht/99; Gozzi NPPS(02)qp/01; Manoukian & Yongram IJTP(02)ht/04; Penco & Mauro EJP(06)ht; Gozzi FP(10) [and quantum path integrals]; Ovchinnikov Chaos(12)-a1203 [dynamical systems as topological field theories]; Cattaruzza & Gozzi PLA(12)-a1207 [local symmetries and degrees of freedom]; Cattaruzza et al PRD(13)-a1302 [and least-action principle]; > s.a. scalar field theories.
@ On the computer: Hubbard & West 92; Nusse & Yorke 97; Pingel et al PRP(04) [stability transformation].
@ Symbolic dynamics: Adler BAMS(98) [representations by Markov partitions]; Fedeli RPMP(06) [embeddigs].
@ Other approaches: Derrick JMP(87) [in terms of data on an observer's past light cone]; Drago AJP(04)mar [Lazare Carnot's 1783 formulation]; Ercolessi et al IJMPA(07)-a0706 [alternative linear structures on TQ]; Delphenich a0708 [from action of symmetry transformation groups]; Page FP(09) [in terms of diagonal projection matrices and density matrices]; Deriglazov & Rizzuti AJP(11)aug-a1105 [reparametrization-invariant formulation, and quantization]; Pérez-Teruel EJP(13)-a1309 [based on an analogy with thermodynamics]; Lecomte PRS(14) [frequency-averaging framework for complex dynamical systems].
@ Gravity and cosmology-related applications: Boehmer & Chan a1409-ln; Leon & Fadragas a1412-book; > s.a. MOND.
@ Related topics: Rosen AJP(64)aug [in terms of wave functions not in linear space]; Voglis & Contopoulos JPA(94) [invariant spectra]; Sarlet et al RPMP(13) [inverse problem].
> Other approaches: see classical mechanics [including generalizations]; hamiltonian dynamics; lagrangian dynamics; poisson structures; variational principles.
> Other concepts and tools: see chaos; differential geometry; Feynman Diagrams; lie algebras; Peierls Bracket; Reference Frame; time; Trajectory.
> Other results: see noether theorem.
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