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].

@ __History of the concept__: Crease pw(02)jul;
Frontali PhysEd(14).

> __Specific theories__: see electromagnetism;
gravitational energy; newtonian gravity.

**In Classical Physics** > s.a. 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* (d*f*/d*t*)^{2}] d*x* (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 *ξ*^{a}
is *p*^{a}, its energy with respect to that observer is
*E* = –*p*^{a}*ξ*_{a}
= *m* (1–*v*^{2})^{–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 *K*^{a}, then the conserved quantity *p*^{a}*K*_{a} 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-a1705 [extended physical systems
in special relativity].

@ __And gravity__: Bruschi a1701 [not all energy is a source of gravity];
Dewar & Weatherall 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* = *h**ν* = \(\hbar\)*ω*.

@ __General references__:
Prentis & Fedak AJP(04)may [conservation, and work-energy theorem];
Frank QIP(05)qp/04 [as rate of information processing];
Tejero & Vitolo IJGMP(14) [geometry of the energy operator].

@ __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*:
I → *M*, relative to *γ*(*u*), the invariant

*E*(*γ*):=
∫ *E*(*γ*, *γ*(*u*))
|\(\dot\gamma\)(*u*)|
d*u*, where *E*(*γ*, *γ*(*u*)):=
∫ {|*γ*(*v*)–*γ*(*u*)|^{–2} – [*D*(*γ*(*v*), *γ*(*u*))]^{2}}
|\(\dot\gamma\)(*v*)| d*v* ,

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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