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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