Lagrangian Systems  

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

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

= – |g|1/2 gac gbd Fab Fcd = –|g|1/2 gac gbd [a Ab][c Ad] .

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

= 2–1/2 G Ja(x) Ja(x) ,    = g Ja(x) Wa(x) + h.c. ,

where Ja = la + ha and 2–1/2 G = g2/mW2.
* Dirac field: With coupling to a gauge field (check conventions; could add a non-linear, ()2 term)

= (ic aDamc2) + YM .

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

= Ba(x) [ab( + m2) – ab] Bb(x) .

* Spin-2 field:

= (ahbc) (a hbc) – (a hab) (c hcb) .

@ References: Bracken IJTP(05) [electromagnetic].
> Other theories: see dilaton; gravity; klein-gordon field theory; monopoles; Nambu Mechanics; scalar-tensor; yang-mills gauge theory.

Other Theories > s.a. constraints; gauge theory; graph theory.
@ 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].
@ 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 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 [gab 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.
@ 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]; Baez & Gilliam LMP(94) [discrete]; 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 [nc space].


Main pageAbbreviationsJournalsCommentsOther sitesAcknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified 16 mar 2008