Conservation Laws, Conserved Quantities  

In General > s.a. asymptotic flatness at spatial and null infinity; lagrangian dynamics.
* Applications: They are useful for studying the evolution of a system without knowing details about the motion, to give a classification of all possible evolutions, etc, and to reduce the order of the differential equations describing the motion.

From Symmetries > s.a. differential equations; noether symmetries; killing fields [Killing vectors and tensors]; symmetries of dynamical theories.
* Idea: Generators of symmetries give rise to conserved currents, whose integrals are conserved quantities.
* Types: Conserved quantities can correspond to Noether, Mei and Lie symmetries in a Lagrangian dynamical system; Topological charges on the other hand do not give rise to symmetry transformations on the fields.
* Superpotential: An (n−2)-form U(ξ) in an n-dimensional spacetime, associated with an infinitesimal Lagrangian symmetry ξ, such that the boundary term in the conserved quantity Q(ξ) = Qbulk(ξ) + Qbdry(ξ) is

Qbdry(ξ) = ∂(Σ) U(ξ) ;

The term also denotes a potential θabc for a stress-energy pseudotensor, θmn = ∂a θm[na].
@ General references: Levi-Civita RAL(1899), translation Saccomandi & Vitolo RCD(12)-a1201 [for Hamilton's equations]; Wigner PT(64)mar; Katzin JMP(73) [quadratic constants of the motion]; Rosen & Freundlich AJP(78)oct [general framework]; Norbury EJP(88) [pedagogical, for momentum]; Schulte SHPMP(08) [and particle families]; Smith SHPMP(08) [in Lagrangian mechanics]; Ivanova et al PhyA(09)-a0806; Fang et al PLA(10) [types, and proposal of new types]; Ngome Abiaga PhD(11)-a1103 [Van Holten's covariant algorithm]; Nucci PLA(11) [Lie symmetries of Lagrangian systems]; Anco IJMPB(16)-a1512.
@ Non-Noether symmetries: Hojman JPA(92) + González-Gascón JPA(94) + Lutzky JPA(95); Anco & Bluman PRL(97) [for field theories]; Kara & Mahomed IJTP(00); Chavchanidze mp/02, JGP(03)mp/02; Birtea & Tudoran IJGMP(12); > s.a. lagrangian dynamics.
@ Superpotential: in Francaviglia & Raiteri CQG(02)gq/01; Katz & Livshits CQG(08)-a0807 [from variational derivatives, gravity theories]; Sardanashvily IJGMP(09)-a0906 [and gauge conservation laws]; Adamek & Novotny a1606 [new superpotential].
@ Related topics: Kara et al IJTP(99) [approximate symmetries]; Cicogna MMAS(13)-a1307 [and generalizations].

Related Concepts > s.a. Chevreton Tensor; energy; observables; stress-energy pseudotensor; superselection rules.
* Continuity equation: A local, differential form of conservation equation; If u is a velocity/flow 4-vector and f a source strength, then

a ua = f ,   or   ρ,t + ∇ · (ρv) = f   in 3+1 form .

@ General references: Horwood JMP(07) [higher-order first integrals]; Belevich JPA(09) [continuity equation].
@ Energy-momentum conservation: Giachetta et al G&C(99) [gauge approach];
Weatherall SHPMP-a1702 [as a consequence of the gravitational field equations]; > s.a. energy-momentum tensors; gravitational energy-momentum.
@ Time-dependent invariants of motion: Sarris & Proto PhyA(05) [complete sets of non-commuting observables].
@ Center of energy: Boyer AJP(05)oct [illustrations].

Specific Types of Theories > s.a. noether theorem; non-local theories.
* In Newtonian mechanics: Only 4 were known at the end of the XIX century, mass, linear momentum, angular momentum and energy.
* In relativistic mechanics: For geodesic motion in curved spacetime, each Killing vector field ξa gives rise to a constant of the motion, pa ξ a, and so does each Killing tensor.
* In relativistic field theory: They are expressed by T ab;b = 0; If the spacetime has a Killing vector field ξa, then T ab ξb is a conserved current.
@ Gauge theories: Chodos CMP(79) [Yang-Mills theory]; Przeszowski JPA(89) [non-abelian currents]; Deser & Henneaux MPLA(95)ht [abelian and non-abelian currents]; Barnich et al LMP(04)gq [n−2 forms in curved spacetime, classification]; Manno et al JGP(08) [second-order field equations and variational principles]; Berche et al AJP(16)aug [classical and quantum mechanics].
@ Spinor fields: Anco & Pohjanpelto PRS(03)mp/02 [any spin].
@ Gravity: Wald & Zoupas PRD(00)gq/99; Papadopoulos JMP(06)gq/05 [essential constants]; Obukhov & Rubilar PRD(06)gq, PRD(07)-a0712, PLB(08)-a0712; Fabris a1208-proc [and cosmology]; Kim et al PRL(13) [quasilocal charges]; Chen et al a1312 [quasilocal]; Hyun et al PRD(14)-a1406 [quasi-local, and holography]; García-Parrado a1606 [4D spacetime with a Killing vector]; Hopfmüller & Freidel a1802 [along null surfaces]; > s.a. gravitational action; higher-order theories; quasilocal energy.
@ Generalized gravity: Alves et al GRG(08)-a0710 [in massive graviton theory]; Lompay & Petrov JMP(13)-a1309 [metric-torsion theories]; Obukhov & Puetzfeld PRD(14)-a1405 [metric-affine gravity]; Setare & Adami NPB(16)-a1511 [Chern-Simons-like theories]; Obukhov et al PRD(15)-a1511 [generally covariant theories]; Ghodrati et al EPJC(16)-a1606 [in f(R) gravity].
@ Hamiltonian systems: Cariglia et al JMP(14)-a1404 [covariant algorithm using conformal Killing tensors]; Torres del Castillo a1408.
@ Quantum mechanics: Sharp & D'Amico PRB(14)-a1403 [metric-space formulation]; Ohsawa JMP(15)-a1410 [and symmetries, in semiclassical Gaussian wave packet dynamics]; Torres del Castillo & Herrera a1510 [and unitary symmetries of the Hamiltonian]; Aharonov et al a1609 [questions about the meaning]; > s.a. quantum spin models; relativistic quantum mechanics; schrödinger equation.
@ Particle dynamics: Katzin & Levine JMP(74); Cetto & de la Peña AJP(84)jun [relationship between energy and adiabatic invariant J]; Hojman et al JMP(86), Del Castillo & Hojman JMP(90) [geodesic motion]; Igata et al PRD(11)-a1005 [constrained, particle around a black hole]; Feroze MPLA(10)-a0905 [for curved spacetimes]; Cariglia et al CQG(14)-a1401 [Killing tensors and canonical geometry]; > s.a. diffusion.
@ Approaches, special situations: Popovych & Sergyeyev PLA(10) [two independent variables, normal form of evolution equations]; Peng a1607 [for differential-difference equations]; Ruggieri & Speciale a1612 [pdes, algorithmic computation]; Dray in(17)-a1701 [piecewise conserved quantities].
> Related topics: see angular momentum; electromagnetism; energy; energy-momentum tensors; interaction; momentum.

References > s.a. Superposition Principle.
@ General: Takens 77; Uhlenbeck in(83); Amigo & Reeh FdP(88) [additive constants of motion]; Serre 00; Jiang & Li IJTP(07) [classical vs quantum]; Popovych & Samoilenko JPA(08) [complete description for second-order 2-dimensional equations]; Haywood 10 [group theory]; Lange PhSc(11) [conceptual].
@ Other types: Gorni & Zampieri a1412 [non-local constants of motion in Lagrangian dynamics]; Post & Winternitz a1501 [N-th order integrals of motion].
@ Related topics: Olszewski AJP(83)oct [non-conservation of E, p, L in non-static spacetime]; Sidharth CSF(00)qp/98 [statistical view]; Fagotti a1408 [quasi-conserved operators]; Reintjes a1510 [constrained systems of conservation laws].

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