Lagrangian Systems  

Particle Dynamics and Other Systems > s.a. classical particles; relativistic particles; spinning particles; parametrized systems.
@ General references: Barbero & Villaseñor PRD(02)ht [quadratic, Diff-invariant]; Scholle PRS(04) [continuum theories]; Castrillón López et al a0711 [interacting systems, "concatenating" variational principles]; Bernard & Contreras AM(08) [generic property of families of Lagrangian systems].
@ On Lie groups: Crampin & Mestdag JLT(08)-a0801 [invariant Lagrangians]; Lucas a1111 [finite-dimensional, pedagogical].
> Other systems: see constrained systems; graph theory; higher-order lagrangians [including non-local systems]; monopoles; Nambu Mechanics.

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

\(\cal L\) = –\(1\over4\)|g|1/2 gac gbd Fab Fcd = –|g|1/2 gac gbd[a Ab][c Ad] .

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

\(\cal L\) = \(1\over\sqrt2\)G J a(x) Ja(x) ,   \(\cal L\) = g J a(x) Wa(x) + h.c. ,

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

\(\cal L\) = ψ* (ic\(\hbar\) γaDamc2) ψ + \(\cal L\)YM .

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

\(\cal L\) = Ba(x) [ηab(\(\square\) + m2) – ∇ab] Bb(x) .

* Spin-2 field:

\(\cal L\) = \(1\over2\)(∇ahbc) (∇a hbc) – (∇a hab) (∇c hcb) .

@ Electrodynamics: Bracken IJTP(05); Bogolubov & Prykarpatsky UJP(09)-a0909, FP(10) [and Hamiltonian, quantization]; Saldanha BJP(15)-a1509 [alternative].
@ Other field theories: Szczyrba APM(76); Giachetta et al 97; Hájíček & 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 JMP(09)-a0901 [distributional fields]; Cattaneo et al a1207 [with boundaries]; Neiman PRL(13)-a1310 [imaginary part, and entanglement]; > s.a. dilaton; fluids [dissipative] and perfect fluids; gauge theory; klein-gordon fields; yang-mills theory.
> Gravitational theories: see action for general relativity; higher-order theories; scalar-tensor gravity.

Non-Lagrangian Systems > s.a. noether theorem [generalization]; path integrals [Lagrange structures, Peierls brackets].
* Example: A charged particle in the field of magnetic monopole.
@ References: Nucci & Leach PS(11) [systems without a Lagrangian].

Other Generalized Theories > s.a. deformation quantization; thermodynamics; dissipative systems and variational principles [non-conservative].
* Non-commutative field theory: For a scalar φ3 theory, if * is the non-commutative star-product,

\(\cal L\) = –\(1\over2\)|g|1/2 [gabaφbφ + (m2φ2 + \(1\over3\)g φ *φ *φ)] .

@ Singular Lagrangians: Gràcia & Pons JPA(01)mp/00; Pugliese & Vinogradov JGP(00); Gitman & Tyutin NPB(02)ht [Hamiltonian]; Román-Roy mp/06-conf; Duplij JKNU-a0909 [non-linear Hamiltonian formalism]; Langerock & Castrillón IJGMP(10)-a1007 [Routh reduction procedure]; Sardanashvily a1206 [Grassmann-graded Lagrangian theory]; Duplij IJGMP(15)-a1308 [partial Hamiltonian formalism]; > s.a. hamilton-jacobi theory; statistical-mechanical systems [inequivalent Lagrangians].
@ Discrete systems: Baez & Gilliam LMP(94); Caterina & Boghosian PhyA(08) [no-go theorem for least-action principle]; Elze PRA(14)-a1312 [cellular automata]; Höhn JMP(14)-a1407 [constraints and degrees of freedom].
@ With fractional derivatives: Dreisigmeyer & Young JPA(03) [non-conservative]; Baleanu & Trujillo ND-a0708 [including exact solutions]; Golmankhaneh et al IJTP(10), IJTP(12) [Hamiltonian structure].
@ 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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