Editing Virial theorem (section)

== Generalizations ==

Lord Rayleigh published a generalization of the virial theorem in 1900,<ref>{{cite journal|doi=10.1080/14786440009463903 |title=XV. On a theorem analogous to the virial theorem |date=August 1900 |author=Lord Rayleigh |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |series=5 |volume=50 |issue=303 |pages=210–213 |url=https://zenodo.org/record/1871519 }}</ref> which was partially reprinted in 1903.<ref>{{Cite book |url=https://books.google.com/books?id=S-sPAAAAYAAJ&pg=PA491 |title=Scientific Papers: 1892–1901 |date=1903 |author=Lord Rayleigh |publisher=Cambridge: Cambridge University Press |pages=491–493 }}</ref> [[Henri Poincaré]] proved and applied a form of the virial theorem in 1911 to the problem of formation of the Solar System from a proto-stellar cloud (then known as [[cosmogony]]).<ref>{{cite book
| last = Poincaré | first = Henri | author-link = Henri Poincaré | title = Leçons sur les hypothèses cosmogoniques |lang=fr |trans-title=Lectures on Theories of Cosmogony | publisher = Hermann | year = 1911 | pages = 90–91 et seq | location = Paris }}</ref> A variational form of the virial theorem was developed in 1945 by Ledoux.<ref>{{cite journal
 | last = Ledoux
 | first = P.
 | year = 1945
 | title = On the Radial Pulsation of Gaseous Stars
 | journal = The Astrophysical Journal
 | volume = 102
 | pages = 143–153
 | doi = 10.1086/144747
 | bibcode = 1945ApJ...102..143L
| doi-access = free
 }}</ref> A [[tensor]] form of the virial theorem was developed by Parker,<ref>{{cite journal
 | last = Parker
 | first = E. N.
 | year = 1954
 | title = Tensor Virial Equations
 | journal = Physical Review
 | volume = 96
 | issue = 6
 | pages = 1686–1689
 | doi = 10.1103/PhysRev.96.1686
 | bibcode = 1954PhRv...96.1686P
}}</ref> Chandrasekhar<ref>{{cite journal
 | last1 = Chandrasekhar
 | first1 = S.
 | author1-link = Subrahmanyan Chandrasekhar
 | last2 = Lebovitz
 | first2 = N. R.
 | year = 1962
 | title = The Potentials and the Superpotentials of Homogeneous Ellipsoids
 | journal = Astrophys. J.
 | volume = 136
 | pages = 1037–1047
 | doi = 10.1086/147456
 | bibcode = 1962ApJ...136.1037C
| doi-access = free
 }}</ref> and Fermi.<ref>{{cite journal
 | last1 = Chandrasekhar
 | first1 = S.
 | author1-link = Subrahmanyan Chandrasekhar
 | last2 = Fermi
 | first2 = E.
 | year = 1953
 | title = Problems of Gravitational Stability in the Presence of a Magnetic Field
 | journal = Astrophys. J.
 | volume = 118
 | pages = 116
 | doi = 10.1086/145732
 | bibcode = 1953ApJ...118..116C
}}</ref> The following generalization of the virial theorem has been established by Pollard in 1964 for the case of the inverse square law:<ref>{{cite journal
 | last = Pollard
 | first= H.
 | year = 1964
 | title = A sharp form of the virial theorem
 | journal = Bull. Amer. Math. Soc.
 | volume = LXX
 | pages = 703–705
 | doi = 10.1090/S0002-9904-1964-11175-7
 | issue = 5
| doi-access = free
 }}</ref><ref>{{cite book
 | last = Pollard
 | first = Harry
 | title = Mathematical Introduction to Celestial Mechanics
 | publisher = Prentice–Hall, Inc.
 | location = Englewood Cliffs, NJ
 | year = 1966
 | isbn=978-0-13-561068-8
}}</ref>{{Failed verification|date=December 2023}}
<math display="block">
 2\lim_{\tau\to+\infty} \langle T\rangle_\tau =
 \lim_{\tau\to+\infty} \langle U\rangle_\tau \quad \text{if and only if} \quad \lim_{\tau\to+\infty}{\tau}^{-2}I(\tau) = 0.
</math>
A ''boundary'' term otherwise must be added.<ref>{{cite journal
 | last1 = Kolár
 | first1 = M.
 | last2 = O'Shea
 | first2 = S. F.
 | date = July 1996
 | title = A high-temperature approximation for the path-integral quantum Monte Carlo method
 | journal = Journal of Physics A: Mathematical and General
 | volume = 29
 | issue = 13
 | pages = 3471–3494
 | bibcode = 1996JPhA...29.3471K
 | doi = 10.1088/0305-4470/29/13/018
}}</ref>