Energy| 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,
t2
d3x
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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nov
2009