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 = \(1\over2\)hij
pi pj;
> s.a. jacobi dynamics.
* Degenerate: The rank
of the symplectic structure is not constant throughout phase space;
Phase space is divided into causally disconnected, non-overlapping
regions in each of which the rank of the symplectic matrix is constant,
and there are no classical orbits connecting two different regions.
* Time-dependent:
Cosymplectic structures play a central role in the theory.
* Non-canonical: The
equations of motion cannot be obtained from a variational principle,
and are of the form
\(\partial u/\partial t = J(u)\) δ\(\cal H\)/δu .
@ References: Rosquist & Pucacco JPA(95) [2D, geometric approach to invariants];
Casetti et al RNC(99) [many degrees of freedom];
Horwood et al CMP(05)mp/06 [orthogonally separable, classification];
de Micheli & Zanelli JMP(12) [degenerate, quantum].
@ Non-canonical:
Vanneste & Shepherd PRS(99);
Junginger et al a1409 [construction of Darboux coordinates];
Yoshida & Morrison PS(16)-a1409 [hierarchy].
@ 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 & Kosiński qp/07 [and quantization];
Gibbons et al JGP(10);
De Sole & Kac a1210;
> s.a. higher-order lagrangians.
@ Higher-order:
Govaerts & Rashid ht/94;
Schmidt gq/95;
Hamamoto ht/95;
Miron 02-a1003;
> s.a. higher-order lagrangians.
@ Time-dependent: Sardanashvily JMP(98) [in terms of fiber bundles];
Haas JPA(01)mp/02 [1D, invariants];
de León & Sardón a1607 [geometric Hamilton-Jacobi theory];
> s.a. lie algebras and groups.
@ Discrete:
Baez & Gilliams LMP(94);
Rosenau PLA(03) [continuum approximations];
Talasila et al JPA(04);
Lall & West JPA(06);
Das CJP(10)-a0811 [discretized field theories];
Elze PRA(14)-a1312 [cellular automata];
Káninský a2008 [linear dynamical systems];
> s.a. lagrangian systems.
Specific Types of Systems > s.a. constrained
systems; integrable systems; oscillator;
parametrized theories; particle physics.
* Coupled oscillators:
The Hamiltonian is of the form
H = \(1\over2\)Gab Pa Pb + \(1\over2\)Vab qa qb .
* Single particle: Configuration space = Physical space.
* Lagrangians linear in
velocities: Use the Faddeev-Jackiw, or symplectic, method.
@ Quadratic Hamiltonians: Suslov PS(10) [integrals of motion].
@ Other types: Capovilla et al JPA(02)n.SI [curves];
Cariñena et al IJGMP(13) [Lie-Hamilton systems];
Ghosh a2104 [with balanced loss and gain];
> s.a. Continuous Media; projective geometry.
Field Theories > s.a. canonical general relativity;
dirac fields; higher-order gravity;
klein-gordon fields; membranes;
yang-mills theory.
* With boundaries:
For each degree of freedom, each piece of boundary gives its conjugate
momentum, even a timelike one or a corner!
* Lorentz invariance:
There is no simple way to check whether a given Hamiltonian field theory
is relativistic or not, and one normally has to either solve for the
equations of motion or calculate the Poisson brackets of the Noether charges.
@ General references: Giachetta et al 97;
Hájíček & Kijowski PRD(98)gq/97 [with discontinuities];
de León et al mp/02;
Gershgorin et al JMP(09)-a0807 [waves in weakly inhomogeneous media];
Danilenko TMP(13)-a1302 [modified formalism];
Kajuri MPLA(16)-a1606 [and Lorentz invariance];
Vines et al PRD(16)-a1601 [extended, spinning test body in curved spacetime];
Campoleoni et al JHEP(16)-a1608 [massless higher-spin fields].
@ Gravity: Arnowitt et al in(62),
DeWitt PR(67) [general relativity];
Gomes & Shyam JMP(16)-a1608 [uniqueness result for general relativity].
@ Quantum field theories:
Rinehart a1505 [foundations];
Teufel & Tumulka a1505 [without ultraviolet divergences].
@ Electrodynamics: Bogolubov & Prykarpatsky UJP-a0909 [and Lagrangian, quantization];
Heninger & Morrison a1808 [with magnetic monopoles];
Vollick a2101
[in terms of electric and magnetic fields, without potentials].
@ 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;
Barbero et al CQG(14)-a1306 [geometric approach];
Troessaert a1506 [gauge theories];
> s.a. quasilocal general relativity.
@ Variations:
Hélein & Kouneiher mp/00,
JMP(02) [pataplectic form];
Echeverría-Enríquez et al IJMMS(02)mp/01 [geometrical, multivectors].
References > s.a. Contact Manifolds.
@ 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].
@ With complex parameters: Bender et al JPA(06)mp [complex H, trajectories];
Nanayakkara & Mathanaranjan Pra(14)-a1406 [complex H and time].
@ Non-conservative systems: Bravetti & Tapias JPA(15)-a1412;
Galley PRL(13)-a1210;
> s.a. classical systems
and variational principles.
@ Non-reversible systems: Figotin & Schenker JSP(07) [dissipative, dispersive];
Eberard et al RPMP(07) [thermodynamics, on contact manifolds];
Buliga a1902 [dissipative version, and information];
> 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);
Malinowska & Torres FCAA(12)-a1206 [and quantization].
@ Quantum-gravity motivated theories: Colladay PLB(17)-a1706 [lorentz-violating theories, extended Hamiltonian formalism];
Bosso PRD(18)-a1804 [theories with minimal length];
Singh & Carroll a1806
[finite-dimensional generalization, based on generalized Clifford algebra];
> s.a. non-commutative field theories.
@ Other generalizations: Seke et al PLA(97) [effective Hamiltonians];
Kozlov JMP(01) [semidiscrete, conservation laws];
Morando & Tarallo mp/02 [quaternionic];
Su & Qin CTP(04)mp/03 [Birkhoffian generalization];
Hegseth qp/05 [for quantum mechanics];
Bliokh mp/05 [for Minkowski spacetime];
Tulczyjew mp/06 [with discontinuities];
Tarasov & Zaslavsky CNSNS(08)mp/07 [systems with long-range interaction and memory];
Lázaro-Camí & Ortega RPMP(08) [stochastic];
Ushakov IJTP(11)-a1004 [non-symplectic generalization];
Bender et al a1509 [for any linear constant-coefficient evolution equation];
> s.a. Nambu Brackets.
@ Related topics:
Cabral & Gallas PRL(87) [duality];
Kandrup PRD(94)
[Hred for subsystem].
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