Energy| Energy |
In General > s.a. technology.
* Idea: A conserved quantity
for a system, associated with invariance under time translation.
* History: Energy conservation
was introduced by Galileo [@ Galilei 1638],
but the concept was fully developed around 1850.
@ General references:
Pielou 01 [I];
issue SA(06)sep [future];
Lam PhSc(11) [need for a background structure];
> s.a. physics teaching.
@ History of the concept: Crease pw(02)jul;
Frontali PhysEd(14).
> Specific theories: see electromagnetism;
gravitational energy; newtonian gravity.
In Classical Physics
> s.a. conservation laws; energy-momentum
tensor [for a field]; Work-Energy Theorem.
* Energy theory: The computation
of a sufficient condition for stability of the laminar flow of a fluid.
* For a wave: Of the form E(t)
= ∫ [A f 2 + B
(df/dt)2] dx (with A = 0
for some granular systems, and B = 0 for electromagnetic waves).
@ General references: Schrödinger NC(58);
Arminjon AMP(16)-a1510-conf [conservation, for particles and fields, and the energy-momentum tensor].
@ Kinetic energy: Madhu Rao AJP(00)apr [and invariance];
Prentis AJP(05)aug [derivation];
Riggs TPT(16) [Newtonian vs relativistic dynamics].
> Related topics: see Equipartition
of Energy; physics teaching; tunneling [particles
with complex energy]; Virial Theorem.
In Relativistic Physics
> s.a. energy conditions; relativistic particle.
* Relativistic particle: If a particle's
4-momentum with respect to an observer ξa
is pa, its energy with respect to that
observer is E = −pa
ξa = m (1 −
v2)−1/2;
This represents the "inertial" energy of the particle (rest mass and kinetic energy);
The "total" energy is generally not a well-defined concept, but if there is a timelike
("stationary") Killing vector field Ka,
then the conserved quantity pa
Ka can be considered the particle's
energy in the gravitational field.
@ General references:
Sonego & Pin EJP(05),
Adkins AJP(08)nov [in special relativity];
Carini et al IJGMP(07) [covariant, non-inertial frames];
Serafin & Głazek AJP(17)apr-a1705 [extended physical systems in special relativity];
Grib & Pavlov Symm(20)-a2004 [particles with negative energies].
@ And gravity:
Bruschi a1701 [not all energy is a source of gravity];
Dewar & Weatherall FP(18)-a1707-conf [in Newtonian gravitation];
> s.a. matter near black holes [energy extraction].
@ Self-energy:
Arnowitt et al PR(60) [coupled to gravity];
Cheon IJTP(79) [in modified quantum electrodynamics];
de Souza ht/95,
ht/96,
JPA(97)ht/96 [electron self-field without renormalization];
van Holten NPB(98)ht/97;
Hirayama & Hara PTP(00)gq/99 [in curved spacetime];
Hod PRD(02) [black-hole background];
Barceló & Jaramillo a1112 [localization];
> s.a. non-linear electrodynamics; self-force.
In Quantum Physics
> s.a. measurement; quantum field theory effects [negative
energy density]; quantum information; Virial Theorem.
* For a particle in quantum mechanics:
For a photon, E = hν = \(\hbar\)ω.
@ General references:
Frank QIP(05)qp/04 [as rate of information processing];
Tejero & Vitolo IJGMP(14) [geometry of the energy operator].
@ Conservation: Prentis & Fedak AJP(04)may [and the work-energy theorem];
Sołtan a1907 [and the measurement context].
@ Reated topics: Boukas a0812 [minimal operating time for energy supply];
El Dahab & Tawfik CJP(14)-a1401 [maximal measurable energy].
For Curves or Loops
$ Def: For γ:
→M, relative to γ(u), the invariant
E(γ):= ∫ E(γ, γ(u)) |\(\dot\gamma\)(u)| du, where E(γ, γ(u)):= ∫ {|γ(v) − γ(u)|−2 − [D(γ(v), γ(u))]2} |\(\dot\gamma\)(v)| dv ,
and D(γ(v), γ(u))
is the distance along γ.
@ References:
Freedman et al AM(94);
Strzelecki & von der Mosel PRP(13) [Menger curvature as a knot energy].
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