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 = = $$\hbar$$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].