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
\((X, \mu, \phi)\) of a set, a probability measure, and a family of transformations
on \(X\), where (i) The measure \(\mu\) is invariant under \(\phi\), and (ii) For
all measurable \(A\), \(\mu(A) = \mu(\phi^{-1}(A))\).

* __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];
Goodson 16 [emphasis on chaos, fractals].

@ __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;
Ciaglia et al a1908-proc
[geometric structures emerging in Lagrangian, Hamiltonian and Quantum descriptions];
Curtright & Subedi a2101 [particle motion in a potential as geodesic motion].

> __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 \(m\, {\rm d}^2q^i/{\rm d}t^2 = F^i(q(t),{\rm d}q/{\rm d}t)\)
(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.

@ __Related topics__: Atkinson & Johnson FP(10)-a1908
[logical inconsistencies in systems with infinitely many elements].

**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];
Morgan AP(20)-a1901 [operator algebraic variant, unary classical mechanics].

@ __Mathematical__: Aldaya & Azcárraga FdP(87) [and group theory];
Giachetta et al a0911 [in terms of fibre bundles over the time-axis];
Salnikov et al a2007
[higher structures and differential graded manifolds].

@ __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__:
Ajanapon AJP(87)feb;
Gozzi PLB(88);
Thacker JMP(97) [reparametrization-invariant];
Rivero qp/98;
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];
Rivers in(11)-a1202;
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];
Cugliandolo et al JPA(19)-a1806 [towards a calculus with change of variables];
> s.a. field theory; 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 ch(17)-a1409-ln;
Leon & Fadragas a1412-book;
Bahamonde et al PRP(18)-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];
> s.a. condensed matter.

> __Other approaches__:
see classical mechanics [including generalizations]; hamiltonian
dynamics; lagrangian dynamics; poisson structures;
variational principles.

> __Other tools__: see chaos;
differential geometry; Feynman Diagrams;
lie algebras [and algebroids]; Peierls Bracket;
Reference Frame; time; Trajectory.

> __Other results__: see noether theorem.

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