Types of Constrained Systems  

In General
* Reducible constraints: Constraints that are not independent; For example, of the form pab, a = 0, with pab antisymmetric.
@ General references: Chmielowski JMP(93) [classification].
@ Local field theories, gauge theories: Anderson & Bergmann PR(51) [covariant field theories]; Cabo et al JMP(93); Wald in(92); Grundling & Lledo RVMP(00)mp/98; GIMMsy phy/98 [covariant approach, energy-momentum map]; GIMmsy mp/04 [covariant and canonical Lagrangian/Hamiltonian]; Gitman & Tyutin JPA(05).
@ Generally covariant: Schön & Hájícek CQG(90) [quadratic]; Montesinos & Vergara PRD(02)gq/01 [quadratic linear].
@ Time-dependent: de León et al JPA(96), FdP(02)mp/01; Mangiarotti & Sardanashvily JMP(00) [Hamiltonian].

First-Class Constraints [> s.a. constraints in general relativity; gauge theories]; quantization of first-class systems.
$ Def: The constraint (submanifold) ' in (, ) is first-class iff for all covectors n normal to ', ab nb is tangential to '.
$ Equivalently: The constraint (submanifold) ' is first-class if the Poisson bracket between any two constraints is again a combination of constraints; The constraint functions have to form a closed Poisson algebra (not necessarily a Lie algebra),

{Ci, Cj} = k ijk Ck .

* Dirac conjecture: All first-class constraints generate gauge transformations; The Dirac Hamiltonian can then be enlarged to an extended Hamiltonian including all first-class constraints, without changes in the dynamics; > s.a. gauge.
* Examples: Generally covariant theories, for which solving the field equations is equivalent to finding the dynamical variables.
@ Reduction: Gogilidze et al PRD(96) [admissible gauges]; Pons et al JPA(99)mp/98 [reduced phase space for gauge theories]; Cantrijn et al JMP(99).
@ And gauge, Dirac conjecture: Castellani AP(82), Costa et al PRD(85), Cabo & Louis-Martinez PRD(90) [proof]; Gogilidze et al JPA(94), TMP(95); Maraner AP(96); Wang & Ruan PRA(96); Rothe PLB(02)ht, Rothe & Rothe JPA(03), AP(04) [Lagrangian and Hamiltonian]; Pons SHPMP(05)phy/04 [incompleteness of Dirac's analysis]; Gitman & Tyutin IJMPA(06)ht/05.
@ Dirac conjecture, counterexample: Wu IJTP(94); Jin & Li JPA(01) [higher-order Lagrangian]; & Cawley.
@ Related topics: Mena JMP(96)gq/95 [reality conditions]; Stoilov ht/05 [constraint-gauge duality]; Deriglazov JPA(07) [N-th stage constraints turned into secondary ones].

Second-Class Constraints > s.a. BRST; dissipation; quantization of second-class systems.
$ Def: The constraint surface is a nondegenerate symplectic manifold.
@ General references: Chaichian et al AP(94) [classification]; Vytheeswaran ht/99-in, IJMPA(02)ht/00 [and gauge invariance].
@ Observables: Lyakhovich & Marnelius IJMPA(01)ht [extended]; Bratchikov JGP(06)ht/03.
@ Made first-class: Batalin & Fradkin NPB(87); Harada & Mukaida ZPC(90); Batalin & Marnelius MPLA(01)ht [as gauge theory]; Krivoruchenko et al IJMPA(07)ht/05 [underlying gauge symmetry]; Deriglazov & Kuznetsova PLB(07)ht/06 [deformation of local symmetries].
@ Faddeev-Jackiw approach: Barcelos-Neto & Wotzasek IJMPA(92), MPLA(92); Seiler JPA(95).

Non-Holonomic
@ General references: Bates & Sniatycki RPMP(93) [symmetry reduction]; Cushman et al RPMP(95) [geometry]; de León & de Diego JMP(96); Ibort et al RSMT(96)mp/01 [as implicit differential equations]; Chen et al PRS(97) [classification]; Krupková JMP(97); Marle RPMP(98); Morando & Vignolo JPA(98) [geometric]; Grifone & Mehdi JGP(99) [geometry]; Cortés et al PRS(01); Tavares JGP(03) [Cartan geometrization]; de León et al mp/02, mp/02-in [integrators]; Chen JGP(04)mp/03 [stability]; Sarlet et al JGP(05) [adjoint symmetries and first integrals]; Flannery AJP(05) [and Hamiltonian]; Benenti Sigma(07)m.DS [simple approach]; Ramírez & Sadovskaia RPMP(07); Grabowski et al a0806 [new viewpoint].
@ Examples: Gersten et al AJP(92) [ball rolling on a surface]; Gatland AJP(04).
@ Field theories: Binz et al RPMP(02)mp, Vankerschaver et al RPMP(05)mp [multisymplectic]; Vankerschaver & Martín de Diego JPA(08) [symmetries]; > s.a. symplectic structures.

Generalized Types
@ Higher-order constraints: Krupková JMP(94), JMP(00); Cendra & Grillo JMP(07).
@ With fractional derivatives: Muslih & Baleanu JMAA(05)mp [Ham formulation]; > s.a. hamiltonian systems.
@ Other types: Miskovic & Zanelli ht/03-in, JMP(03)ht [irregular]; Gomis et al NPB(04)ht/03 [non-local, reduced phase space].

Main page – Abbreviations – Journals – Comments – Other sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified 11 jun 2008