Hamiltonian
Systems| Hamiltonian
Systems |
General Types > s.a. [classical mechanics; hamiltonian
dynamics]; phase space; schrödinger
equation;
statistical mechanics.
* Common forms: The classical ones are of the type H = T + V,
with T = (1/2) hij pi pj; > s.a. jacobi dynamics.
* Single particle: Configuration space = Physical space.
* Lagrangians linear in velocities: Use the Faddeev-Jackiw, or symplectic,
method.
* Non-canonical: The
equations of motion cannot be obtained from a variational principle,
and are of the form
u/t
= J(u) 
/u .
Specific Types of Systems > s.a. constrained
systems; integrable systems; oscillator; parametrized
theories;
particle physics.
* Coupled oscillators: The Hamiltonian is of the form
H = 
Gab
Pa Pb + Vab qaqb
.
@ Non-linear: Radak JMP(00)
[moments of distributions]; Choi & Nahm IJTP(07) [quadratic, t-dependent, and
SU(1,1) Lie algebra].
@ Non-local: Woodard PRA(00)
[non-locality of finite extent]; Bolonek & Kosinski qp/07 [and quantization].
@ Higher-order: Govaerts & Rashid ht/94;
Schmidt gq/95;
Hamamoto ht/95; > s.a. higher-order
lagrangians.
@ Discrete: Baez & Gilliams LMP(94);
Rosenau PLA(03)
[continuum approximations];
Talasila et al JPA(04);
Lall & West JPA(06);
Das a0811 [dicretized field theories].
@ Time-dependent: Sardanashvily JMP(98)
[in terms of fiber bundles]; Haas JPA(01)mp/02 [1D,
invariants].
@ Other types: Rosquist & Pucacco JPA(95)
[2D, geometric approach to invariants];
Vanneste & Shepherd PRS(99)
[non-canonical]; Casetti et al
RNC(99) [many degrees of freedom]; Capovilla et
al JPA(02)n.SI [curves];
Horwood et al CMP(05)mp/06 [orthogonally
separable, classification]; > s.a. Continuous
Media; lie
algebras
and
groups [t-dependent].
Field Theories > s.a. canonical
general relativity; dirac fields; higher-order
gravity; klein-gordon fields; membranes; yang-mills
gauge theory.
* With boundaries: For
each degree of freedom, each piece of boundary gives its conjugate momentum;
even a timelike one,
or a corner!
@ General references: Arnowitt et al in(62), DeWitt PR(67)
[general relativity]; Hájícek & Kijowski
PRD(98)gq/97 [with
discontinuities]; de León et al mp/02;
Gershgorin et al a0807 [waves in weakly inhomogeneous media].
@ Electrodynamics: Bogolubov & Prykarpatsky a0909-in
[and Lagrangian, quantization]
@ With boundary values: Soloviev JMP(93)ht,
NPPS(96)ht,
PRD(97)ht/96, ht/99,
JMP(02), JMP(02);
Bering
JMP(00)ht/98;
Zabzine JHEP(00)ht; > s.a. quasilocal
general relativity.
@ Variations: Hélein & Kouneiher mp/00 [pataplectic
form]; Echeverría-Enríquez
et al IJMMS(02)mp/01 [geometrical,
multivectors].
References
@ Perturbations: Abdullaev JPA(99) [Poincaré sections, method];
Fish mp/05 [3D,
dissipative].
@ Without Lagrangian: Rubio & Woodard CQG(94)gq/93,
CQG(94)
[from equations of motion and Poisson brackets]; Hojman ht/94,
JPA(96)
[including field theories];
Gomberoff & Hojman
JPA(97);
Herrera & Hojman mp/00.
@ Covariant: Zhao et al NCB(03); van Holten PRD(07)
[charged particles in external fields]; Struckmeier & Redelbach IJMPE(08)-a0811 [field
theory].
@ Complex Hamiltonian:
Bender et al JPA(06)mp [complex H,
trajectories].
@ Non-reversible systems: Figotin & Schenker JSP(07)
[dissipative, dispersive]; Eberard et al RPMP(07) [thermodynamics, on contact
manifolds]; > s.a. dissipative
systems
@ With fractional derivatives: Muslih & Baleanu CzJP(05)mp [Riewe's
formulation]; Tarasov JPA(05)m.CS/06; Baleanu
et al JMP(06)
[1+1 higher-derivative theories]; Rabei et al JMAA(07).
@ Other generalizations: Seke et al PLA(97)
[effective Hamiltonians]; Kozlov
JMP(01) [semidiscrete,
conservation
laws]; Morando & Tarallo mp/02 [quaternionic];
Hélein & Kouneiher JMP(02)
[pataplectic, for field theory]; Su & Qin mp/03 [Birkhoffian
generalization]; Hegseth qp/05 [for
quantum mechanics]; Bliokh mp/05 [for
Minkowski spacetime]; Tulczyjew mp/06 [with
discontinuities]; Tarasov & Zaslavsky mp/07 [systems
with long-range interaction
and memory]; Lázaro-Camí & Ortega RPMP(08)
[stochastic]; > s.a. Nambu
Brackets; non-commutative.
@ Related topics: Cabral & Gallas PRL(87)
[duality];
Kandrup PRD(94)
[Hred for subsystem].
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oct 2009