Conservation
Laws, Conserved Quantities| Conservation
Laws, Conserved Quantities |
In General > s.a. asymptotic
flatness at spatial and null
infinity; differential equations; 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. Noether
Symmetries; killing fields; symmetries.
* Idea: Generators of symmetries give rise to conserved currents,
whose integrals are conserved quantities.
* Superpotential: An
(n–2)-form 
(
)
in an n-dimensional spacetime, associated with an infinitesimal Lagrangian
symmetry
, such
that the boundary term in the conserved quantity Q()
= Qbulk()
+ Qbdry()
is
Qbdry()
= 
bdry(Sigma) ()
;
The term also denotes a potential for a stress-energy
pseudotensor, 
mn
= 
a m[na].
@ General references: Katzin JMP(73)
[quadratic constants of the motion]; Rosen & Freundlich AJP(78)
[general framework]; Norbury EJP(88)
[pedagogical, for momentum]; Schulte SHPMP(08)
[and particle families]; Smith SHPMP(08)
[in Lagrangian mechanics]; Ivanova et al a0806.
@ 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;
> s.a. lagrangian dynamics.
@ Related topics: Kara et al IJTP(99)
[approximate symmetries]; in Francaviglia & Raiteri CQG(02)gq/01 [superpotential].
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 
,t
+ · (v)
= f in
3+1 form .
@ General references: Horwood JMP(07) [higher-order first integrals].
@ Energy-momentum conservation: Giachetta et al G&C(99) [gauge approach]; > s.a. energy-momentum.
@ Time-dependent invariants of motion: Sarris & Proto PhyA(05)
[complete sets of non-commuting observables].
@ Center of energy:
Boyer AJP(05)
[illustrations].
Specific Theories > s.a. electromagnetism;
energy; energy-momentum;
hamiltonian systems.
* In Newtonian theory: Only
4 were known at the end of the XIX cy, mass,
linear
momentum, angular momentum (these are multipole moments), and energy.
* In relativistic theory:
Expressed
by T ab;b =
0; If the spacetime
has
a
Killing vector field a,
then T ab b
is a conserved
current.
@ Gauge theories: Chodos CMP(79)
[Yang-Mills theory]; Przeszowski JPA(89)
[non-abelian currents]; Gauge & Henneaux MPLA(95)ht [abelian
and non-abelian
currents]; Barnich
et
al
LMP(04)gq [n–2
forms in curved spacetime,
classification].
@ 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; Alves et
al a0710-GRG [in
massive graviton theory]; > s.a.
gravitational action, quasilocal
energy.
@ Particle dynamics: Katzin & Levine JMP(74);
Cetto & de la Peña AJP(84)
[relationship between energy and adiabatic invariant
J]; Hojman
et al JMP(86),
Del Castillo & Hojman JMP(90)
[geodesic motion]; > s.a. diffusion.
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].
@ Related topics: Olszewski AJP(83)
[non-conservation of E, p, L in
non-static spacetime]; Sidharth
CSF(00)qp/98 [statistical
view].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jun 2008