Energy  

In General > s.a. quantum field theory effects [negative energy density].
* Idea: A conserved quantity for a system, associated with invariance under time translation.
* History: Energy conservation was introduced by @ Galilei 1638.
@ References: Pielou 01 [I]; Crease pw(02)jul [history of concept]; issue SA(06)sep [future].
> Specific theories: see electromagnetism; gravitational energy; newtonian gravity.

In Classical Physics > s.a. energy-momentum tensor; Work-Energy Theorem.
* Equipartition principle: In a classical theory, every canonical variable which appears in the action or Hamiltonian only quadratically, in a term of the form bpi2 (or similar for qi), contributes an amount kT/2 to the mean energy at temperature T; (But if energy levels are quantized, the way energy is distributed in a system will depend on T ); Applications: Dulong-Petit law on specific heat of solids.
* Virial theorem: The relationship between the time average of the kinetic energy of a system of mass points and the virial of Clausius; When averages are taken over a period (or, for non-periodic motion, over a long time),

avg{K} = – avg{i Fi · ri} ;

Very useful in the kinetic theory of gases, e.g., to derive Boyle's law.
* 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 (A = 0 for some granular systems, B = 0 for electromagnetic waves).
@ General references: Schrödinger NC(58); Madhu Rao AJP(00)apr [invariance and kinetic energy]; Prentis AJP(05)aug [derivation of kinetic energy].
@ Virial theorem: in Goldstein 80; Milgrom PLA(94)ap [from action principle]; Böhmer et al JCAP(08)-a0710, Sefiedgar et al PRD(09)-a0908 [in f(R) gravity]; Amore & Fernández EJP(09)-a0904 [in non-linear problems].
@ Equipartition: Patrasciou pr(81); Komar GRG(96) [relativistic]; Berchialla et al PLA(04) [time, Fermi-Pasta-Ulam model]; > s.a. specific heat.

In Relativistic Physics
* Relativistic particle: If the 4-momentum wrt an observer a is pa, E = –a pa = m (1–v2)–1/2.
* Virial theorem: The tensor virial theorem states that, for a system with vanishing stresss-energy outside a bounded V,

t2d3x T 00 x i x j = 2 d3x T ij .

@ General references: Sonego & Pin EJP(05), Adkins AJP(08)nov [in special relativity]; Carini et al IJGMP(07) [covariant, non-inertial frames].
@ Virial theorem: in Schutz 85, ex.4.23; Lucha & Schöberl PRL(90); Gourgoulhon & Bonazzola CQG(94); Bonazzola & Gourgoulhon CQG(94); Nowakowski et al PRD(02) [with a cosmological constant]; Georgiou CQG(03) [rotating charged pfluids in general relativity]; Tan AP(08) [generalized, for two-component Fermi gas]; Sefidgar et al PRD(09)-a0908 [for f(R) gravity].
@ Self-energy: Arnowitt et al PR(60) [coupled to gravity]; 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]; > s.a. non-linear electrodynamics; self-force.

In Quantum Physics > s.a. measurement.
* For a particle in quantum mechanics: For a photon, E = h = .
@ References: Georgescu & Gérard CMP(99) [virial theorem]; Prentis & Fedak AJP(04)may [conservation, and work-energy theorem]; Frank QIP(05)qp/04-in [as rate of information processing]; Boukas a0812 [minimal operating time for energy supply].

For a Curve or Loop
$ Def: For : I → M, relative to (u), the invariant

E():= E(,(u)) | ·(u)| du,      where      E(,(u)):= {|(v)–(u)|–2 – [D((v),(u))]2} | ·(v)| dv ,

and D((v),(u)) is the distance along .
@ References: Freedman et al AM(94).


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