Lagrangian Systems

Particle Dynamics > see classical particles; spinning particles; parametrized systems.

Field Theories [with metric signature (–,+,+,+)]
* Maxwell field:

= – |g|^1/2 g^ac g^bd F_ab F_cd = –|g|^1/2 g^ac g^bd _[a A_b][c A_d] .

(Or multiply by 1/4.) With matter, add a term c^–1 A_a J^a; > s.a. Proca Theory.
* Weak interactions: For the original theory and the one with vector bosons, respectively,

= 2^–1/2 G J^a(x) J_a(x) ,    = g J^a(x) W_a(x) + h.c. ,

where J^a = l^a + h^a and 2^–1/2 G = g²/m_W².
* Dirac field: With coupling to a gauge field (check conventions; could add a non-linear, ()² term)

= (ic ^aD_a – mc²) + _YM .

* Spin-1 field: In the massive case, the Proca Lagrangian [@ Proca CRAS(36)]

= B^a(x) [_ab( + m²) – _ab] B^b(x) .

* Spin-2 field:

= (_ah_bc) (^a h^bc) – (_a h^ab) (_c h^c_b) .

@ Electrodynamics: Bracken IJTP(05); Bogolubov & Prykarpatsky a0909-in [and Hamiltonian, quantization].
> Other theories: see dilaton; gravity; higher-order theories; klein-gordon; monopoles; Nambu Mechanics; scalar-tensor; yang-mills theory.

Other Theories > s.a. constraints; gauge theory; graph theory; perfect fluids.
@ Field theories: Szczyrba APM(76); Hájícek & Kijowski PRD(98)gq/97 [with discontinuities]; Boersma PRD(99)gq/00 [and boundary terms]; Echeverría-Enríquez et al IJMMS(02)mp/01; de León et al mp/02, mp/02; András gq/04 [coupled to general relativity]; Gravanis & Willison a0901 [distributional fields].
@ Other types: Barbero & Villaseñor PRD(02)ht [quadratic, Diff-invariant]; Scholle PRS(04) [continuum theories]; Castrillón López et al a0711 [interacting systems, "concatenating" variational principles]; Crampin & Mestdag JLT-a0801 [invariant Lagrangians on Lie groups]; Bernard & Contreras AM(08) [generic property of families of Lagrangian systems].

Generalized Theories > s.a. deformation quantization; higher-order lagrangians.
* Non-commutative fielod theory: For a scalar ³ theory, if * is the non-commutative star-product,

= – |g|^1/2 [g^ab _{a b} + (m²² + g * *)] .

@ Singular Lagrangians: Gràcia & Pons JPA(01)mp/00; Pugliese & Vinogradov JGP(00); Gitman & Tyutin NPB(02)ht [Hamiltonian]; Román-Roy mp/06-in; Duplij a0909/PLB [non-linear Hamiltonian formalism].
@ Discrete systems: Baez & Gilliam LMP(94); Caterina & Boghosian PhyA(08) [no-go theorem for least-action principle].
@ With fractional derivatives: Dreisigmeyer & Young JPA(03) [non-conservative]; Baleanu & Trujillo a0708 [including exact solutions].
@ Other systems: Hata PRD(94)ht/93 [theories/actions as variables]; Marolf PLB(95)gq/94 [partial system]; Soroka PAN(96)ht/95 [Grassmann-odd]; Fiziev & Kleinert gq/96 [action principle in spaces with torsion]; Vankerschaver et al RPMP(05) [non-holonomic, geometric].
@ Other backgrounds: Marolf CQG(94)gq/93 [degenerate metric]; Chamseddine & Connes CMP(97)ht/96, PRL(96)ht [non-commutative space].

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