Lagrangian Systems| 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; morse theory [on spaces of braids];
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\(\hbar\) γa Da − mc2) ψ + \(\cal L\)YM .
* Spin-1 field: In the massive case, the Proca Lagrangian [@ Proca CRAS(36)]
\(\cal L\) = Ba†(x) [ηab(\(\square\) + m2) − ∇a∇b] 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];
Vollick a2101
[in terms of electric and magnetic fields, without potentials].
@ 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;
newtonian gravity; 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 [including 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 [gab ∂aφ ∂bφ + (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].
@ Non-holonomic systems:
Vankerschaver et al RPMP(05) [geometric];
Anahory et al a2003 [discrete, exact mechanics].
@ 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];
Gubbiotti a1910 [fourth-order difference equations].
@ 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];
Neiman a1212 [with null boundary data];
Bosso a1804 [theories with minimal length];
> s.a. differential equations.
@ 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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