Formulations
of Classical Mechanics| Formulations
of Classical Mechanics |
In General – Dynamical Systems
* Idea: The main ingredients are a space of states (phase space,
...), and an algebra of observables; The dynamical system then specifies
an evolution
law on the former or, in the Heisenberg picture, an automorphism a 
at on
the algebra of observables.
* Formulations: The main distinction is between differential and
integral ones (from a variational principle).
* 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, 
, 
)
of a set, a probability measure, and a family of transformations on X,
where (i) The measure is
invariant under ,
and (ii) For all measurable A, (A)
= (–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.
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]; > s.a. Newton's
Laws.
Approaches, Types, and Techniques > s.a. hamilton-jacobi
theory;
statistical mechanics; symplectic
structure; types of systems.
@ 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; Nikolic
FPL(06)qp/05;
Volovich a0910; > s.a.
stochastic processes.
@ Path-integral / quantum-field-theory techniques: Thacker JMP(97)
[reparametrization-invariant];
Gozzi & Regini PRD(00)ht/99;
Gozzi NPPS-qp/01;
Manoukian & Yongram IJTP(02)ht/04;
Penco & Mauro EJP(06)ht.
@ On the computer: Hubbard & West 92; Nusse & Yorke 97; Pingel
et al PRP(04) [stability transformation].
@ Related topics: Rosen AJP(64)aug
[in terms of wave functions not in linear space]; Voglis & Contopoulos
JPA(94)
[invariant spectra].
@ 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]].
> Other approaches:
see classical mechanics; hamiltonian
dynamics; lagrangian
dynamics; MOND; variational
principles.
> Other concepts and tools:
see Feynman Diagrams; lie
algebras; Peierls
Bracket; Reference Frame; time; Trajectory.
> Other results:
see noether theorem.
References
@ General texts: Abraham & Shaw 82-88; Arrowsmith & Place 90;
Marsden & Ratiu
94; Katok & Hasselblatt 95; Collett & Eckmann 97 [maps].
@ Geometrical: Akin 93; Aoki & Hiraide 94 [topological]; Klauder & Maraner
AP(97)qp/96 [deformation
and phase space geometry].
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send feedback and suggestions to bombelli at olemiss.edu – modified 4
nov 2009