Classical Relativistic Particles  

In General > s.a. particles; particle models / geodesics; self-force; test-body motion.
$ Action, reparametrization-invariant: Two versions, which give the trajectory only as a locus of points, are

S1[x] = m0c ds |gab x·a x·b|1/2 = dt (pa x·a – \(1\over2\)g p2)   or   S2[x] = \(1\over2\) dt (u–1 gab x·a x·bum2) ;

Here, S1 (Jacobi-type) is proportional to the world-line length, g(t) is a 1D metric/Lagrange multiplier, as in 1D gravity with scalars, and u(t) is an additional variable.
$ Action, proper-time-gauge:

S[x] = \(1\over2\)m0 dt gab x·a x·b .

* Equation of motion: If a particle is coupled to a field, its equation of motion is just the expression of momentum exchange between particle and field, so it can be obtained from ∇a Ttotab = 0; For a scalar particle in electromagnetic field and linearized gravity, respectively,

dpm/dτ = q Fmn un = (q/m) Fmn pn ,   dpm/dτ = (κ/2) m hab,m uaub .

* Issue: The descriptions using t and the proper time τ are not equivalent [@ Kalman PR(61); Sonego PRA(91)].
* Issue: If a point particle is included among the field sources, the treatment cannot be made fully consistent.
@ General references: Walstad a1512 [momentum and kinetic energy, thought experiment].
@ Lagrangian formulation: Potgieter AJP(83)jan, comment Berger AJP(84)may; Desloge & Eriksen AJP(85)jan.
@ In curved spacetime: Muñoz IJTP(77) [weak-field approximation, Lorentz-force form]; Modanese JMP(92) [fluctuating gravitational field]; Piechocki CQG(03)gq/02 [de Sitter, different topologies]; Bini et al CQG(03)gq/02 [in gravitational wave collision]; Barrabès & Hogan CQG(04)gq/03 [deflection]; Chicone & Mashhoon CQG(05)gq/04 [in Fermi coordinates]; Fukumoto et al PTP(06)gq [finite-size, fast-moving]; in Franklin 10; Sardanashviky IJGMP(10) [in terms of jets of one-dimensional submanifolds]; Arraut et al CEJP(11)-a1005 [static spherically-symmetric metrics]; Corichi IJMPD(15)-a1207 [stationary black-hole background, energy]; > s.a. kerr spacetimes; kerr-newman solutions; scattering.
@ Interacting: Bergmann & Komar GRG(82); Tretyak & Nazarenko CondMP(00)ht; Damour et al PLB(01)gq [3PN]; Lompay ht/05; Tarasov AP(10) [non-Hamiltonian, subject to a general force]; Alesci & Arzano PLB(11)-a1108 [coupled to 3D Einstein gravity]; Novello & Bittencourt GRG(13)-a1201 [accelerated motions as geodesics in dragged metrics].
@ Geometric: Balachandran et al JMP(84); Freidel et al PRD(07) [Dirac observables and effective non-commutative geometry]; Chanda & Guha a1706.
@ Related topics: Gill et al IJTP(93), IJTP(98) [proper-time formulation]; Parrott gq/02 [Rohrlich equation]; Uggerhøj RMP(05) [in strong crystalline fields]; Russo & Townsend JPA(09) [jerk, snap, and higher-order derivatives]; Prosekin et al PRD(15)-a1506 [propagation, transition from ballistic to diffusion regime].
> Other topics: see gravitomagnetism; mass; non-commutative physics; radiation; scalar field theories; specific heat; statistical-mechanical systems.

Variations and Generalizations > s.a. dissipative system; quantum particles [including superparticle]; spinning particles.
Charged particles: The length-of-worldline action becomes

S[x] = ds [m0c |gab x·a x·b|1/2 + q Aa x·a] .

* Modified Lagrangian: It can depend on the curvature of the worldline (rigidity) and its torsion; > s.a. higher-order lagrangians.
@ Charged particles: Hyman AJP(97)mar, Muñoz AJP(97)may [with constant fields]; Horwitz ht/98 [Lorentz force equation from Stückelberg mechanics]; Aldaya et al JPA(02) [group cohomology]; Carvalho et al IJMPA(04)gq [with defect distribution]; Timoumi RPMP(04) [closed trajectories]; Marmo & Tulczyjew RPMP(06) [Poincaré-covariant, and T reflection]; Rohrlich 07; Chin JMP(09)-a0809 [constant field]; Boozer JPA(08) [advanced effects]; Arrayás & Trueba JPA(10) [in a knotted electromagnetic field]; Kar & Rajeev AP(11)-a1010 [with magnetic moment, and radiation reaction]; Gallo & Moreschi PRD(12)-a1112 [without explicitly divergent quantities]; Torromé IJGMP-a1207 [pointlike]; Franklin & LaMont BJP(14)-a1310 [two oppositely charged particles in 1D, with retarded potentials and no radiation reaction]; Azzurli & Lechner AJP(14)aug [massless]; > s.a. electromagnetism; modified electromagnetism [non-linear]; self-force [including preacceleration].
@ Charged particles, radiation: Lidsky TMP(05) [radiating]; Franklin & Griffiths AJP(14)aug [particle in hyperbolic motion].
@ Charged particles, curved spacetime: Frolov & Shoom PRD(10) [near a weakly magnetized Schwarzschild black hole]; Shiose et al PRD(14)-a1409 [around a weakly-magnetized rotating black hole]; Noble & Jentschura PRA(16)-a1603 [in Reissner-Nordström spacetime].
@ With torsion: Barros et al CQG(05) [and rigidity]; Gaitan et al a0912-conf; > s.a. torsion in physics.
@ In gravity's rainbow spacetime: Ling et al MPLA(07)gq/06; Garattini & Mandanici PRD(12)-a1109.
@ In generalized backgrounds: Johnson & Hu PRD(02) [modified Abraham-Lorentz-Dirac equation in a quantum field]; Cabo-Bizet & Cabo Montes de Oca PLA(06) [stochastic medium]; Punzi et al PRD(09)-a0901 [in area-metric manifolds, Finsler norm]; Stern PLA(11) [on quantum spacetime]; Calcagni PRD(13)-a1306 [multiscale spacetimes]; > s.a. kaluza-klein phenomenology; non-commutative geometry [Snyder spacetime]; types of field theories [double field theory].
@ Frenet-Serret formalism: Arreaga et al CQG(01)ht [Frenet-Serret curvature action]; Bini et al CQG(06)-a1408 [for null world lines].
@ Other generalizations: Pavón JMP(01)qp [stochastic dynamics]; Sonego & Pin JMP(09)-a0812 [relativistic, anisotropic]; Markov et al JPG(10)-a1008 [with color charge]; Lahiri & Lee a1011 [with non-Abelian charge]; Tarakanov a1010-conf [internal degrees of freedom, potential depending on v and a]; Arzano & Kowalski-Glikman CQG(11) [with de Sitter momentum space]; Amelino-Camelia et al PRD(12)-a1206 [in de Sitter spacetime, with deformed Lorentz symmetry and relative locality]; Romero & Vergara MPLA(15)-a1501 [Lifshitz field theories, Snyder non-commutative spacetime and momentum-dependent metric]; > s.a. classical systems [two particles].

Coupled to Gravity > s.a. 3D gravity; dynamics of gravitating particles; particle models [point particles]; spinning particles; test-body motion.
@ Non-spinning: Kaniel & Itin gq/01; Blanchet & Faye JMP(01)gq/00 [regularization].
@ Charged: Rosen AP(62) [field of particle in motion]; Khriplovich & Pomeransky SHEP(99)gq/98-conf; Khriplovich ht/00-proc [spinning]; Das et al mp/05 [review]; Gorbatenko TMP(05) [Einstein-Infeld-Hoffmann, order (v/c)3].
@ With torsion: Fiziev & Kleinert gq/96 [action]; Kleinert & Pelster GRG(99)gq/96 [autoparallels]; Barros e Sá gq/97; Geyer et al IJMPA(00)ht/99; Pezzaglia gq/99/IJTP; Arroyo et al GRG(04)ht/03; Barros et al IJMPA(04).
> Related topics: see multipoles [extended objects]; orbits of gravitating bodies; quantum particles; radiation.

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