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* \(\mapsto\)
*a*_{t} 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]; Brin & Stuck 16 [III].

@ __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* d^{2}*q*^{i}/d*t*^{2}
= *F*^{i}(*x*(*t*),
d*q*/d*t*)
(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 T*Q*];
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;
Bahamonde et al a1712 [intro];
> 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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