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]; Bates & Śniatycki a1307 [extension of the Dirac theory]; García-Chung et al a1701 [Dirac theory for time-dependent Hamiltonians].
@ 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; Gotay et al phy/98 [GIMMsy 1, covariant approach, energy-momentum map]; Gotay et al mp/04 [GIMmsy II, covariant and canonical Lagrangian/Hamiltonian]; Gitman & Tyutin JPA(05); Zharinov TMP(10) [zero-divergence constraints].
@ Generally covariant: Schön & Hájíček CQG(90) [quadratic]; Montesinos & Vergara PRD(02)gq/01 [quadratic \(\mapsto\) 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) Γ' in (Γ, Ω) is first-class iff for all covectors n normal to Γ', Ωab nb is tangential to Γ'.
* Equivalently: The constraint (submanifold) Γ' 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 αijk 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 transformations.
* 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; Barbour & Foster a0808 [not always valid]; Wang et al IJTP(09); Henneaux et al a1004-proc [subtleties, and Hořava gravity]; Kiriushcheva et al a1112; Wang et al a1306 [proof of validity]; Pitts AP(14)-a1310 [not valid]; Hori a1812 [limits to validity].
@ Dirac conjecture, counterexample: Cawley PRL(79), PRD(80); Wu IJTP(94); Jin & Li JPA(01) [higher-order Lagrangian]; Hori a1902.
@ Related topics: Mena JMP(96)gq/95 [reality conditions]; Stoilov AdP(07)ht/05 [constraint-gauge duality]; Deriglazov JPA(07) [N-th stage constraints turned into secondary ones]; Dehghani & Shirzad a0811 [simplifying the algebra].

Second-Class Constraints > s.a. BRST transformations; Dirac Bracket; dissipation; quantization of second-class systems; Rotor.
$ Def: The constraint surface is a non-degenerate symplectic manifold; Equivalently, not all Poisson brackets between constraints are combinations of constraints.
* Procedure: One can either solve the second-class constraints explicitly, or implement them in the symplectic structure by working with Dirac brackets.
@ General references: Chaichian et al AP(94) [classification]; Vytheeswaran ht/99-in, IJMPA(02)ht/00 [and gauge invariance]; Bertin et al AP(08) [Hamilton-Jacobi formalism, generalized brackets and reduced phase space]; Bizdadea et al NPB(09)-a0904 [reducible].
@ 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).
@ Examples: Das & Ghosh FP(16)-a1508 [particle constrained to move on a torus knot].

Non-Holonomic Constraints > s.a. hamilton-jacobi theory.
@ General references: Bates & Śniatycki RPMP(93) [symmetry reduction]; 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); Cortés et al PRS(01); de León et al mp/02, mp/02-conf [integrators]; Chen JGP(04)mp/03 [stability]; Sarlet et al JGP(05) [adjoint symmetries and first integrals]; Benenti Sigma(07)m.DS [simple approach]; Ramírez & Sadovskaia RPMP(07); Grabowski et al JMP(09)-a0806 [new viewpoint]; Flannery AJP(11)sep [d'Alembert-Lagrange principle].
@ Geometry: Cushman et al RPMP(95); Morando & Vignolo JPA(98); Grifone & Mehdi JGP(99); Tavares JGP(03); Cushman et al 09.
@ Variational principle: Krupková JPA(09); Delphenich AdP(09)-a0908.
@ And Hamiltonian: Flannery AJP(05)mar; Bloch et al RPMP(09)-a0812 [equations of motion from unconstrained Hamiltonian]; Balseiro & García-Naranjo ARMA(12)-a1104 [geometric]; Mestdag et al a1105-proc [Hamiltonization and geometric integration]; Torres del Castillo & Sosa-Rodríguez IJGMP(14) [infinite number of Hamiltonians and symplectic structures].
@ Examples: Gersten et al AJP(92)jan [ball rolling on a surface]; Gatland AJP(04)jul [2-wheel cart driven by electrostatic forces].
@ 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 > s.a. variational principles [discrete systems].
@ Higher-order constraints: Krupková JMP(94), JMP(00); Cendra & Grillo JMP(07); Grillo JMP(09); Campos et al JPA(10) [higher-order classical field theories]; Martínez et al IJMPA(11) [perturbative Hamiltonian constraints].
@ With fractional derivatives: Muslih & Baleanu JMAA(05)mp [Hamiltonian formulation]; > s.a. hamiltonian systems.
@ Other types: Mišković & Zanelli ht/03-proc, JMP(03)ht [irregular]; Gomis et al NPB(04)ht/03 [non-local, reduced phase space]; Volkov & Zubelevich a1403 [non-smooth].

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