Types of Constrained Systems| 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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