Editing Perturbation theory (section)

==Examples==
Perturbation theory has been used in a large number of different settings in physics and applied mathematics.  Examples of the "collection of equations" <math>D</math> include [[algebraic equation]]s,<ref>{{Cite web |url=http://math.unm.edu/~lromero/2013/class_notes/poly.pdf |title=L. A. Romero, "Perturbation theory for polynomials", Lecture Notes, University of New Mexico (2013) |access-date=2017-04-30 |archive-date=2018-04-17 |archive-url=https://web.archive.org/web/20180417071534/http://math.unm.edu/~lromero/2013/class_notes/poly.pdf |url-status=dead }}</ref>
[[differential equation]]s<ref>Bhimsen K. Shivamoggi: ''Perturbation Methods for Differential Equations'', Springer, ISBN 978-1-4612-0047-5 (2003)</ref> (e.g., the [[equations of motion]]<ref>
[https://www.physik.uni-muenchen.de/lehre/vorlesungen/sose_11/T1_Theoretische_Mechanik/vorlesung/anharmonic-perturbation.pdf Sergei Winitzki, "Perturbation theory for anharmonic oscillations", Lecture notes, LMU (2006)]</ref>
and commonly [[wave equation]]s), [[thermodynamic free energy]] in [[statistical mechanics]], radiative transfer,<ref>
[https://www.researchgate.net/publication/222659054_Radiative_perturbation_theory_A_review Michael A. Box, "Radiative perturbation theory: a review", Environmental Modelling & Software 17 (2002) 95–106]
</ref>
and [[Hamiltonian (quantum mechanics)|Hamiltonian operators]] in [[quantum mechanics]].

Examples of the kinds of solutions that are found perturbatively include the solution of the equation of motion (''e.g.'', the [[trajectory]] of a particle), the [[average|statistical average]] of some physical quantity (''e.g.'', average magnetization), and the [[ground state]] energy of a quantum mechanical problem.

Examples of exactly solvable problems that can be used as starting points include [[linear equation]]s, including linear equations of motion ([[harmonic oscillator]], [[linear wave equation]]), statistical or quantum-mechanical systems of non-interacting particles (or in general, Hamiltonians or free energies containing only terms quadratic in all degrees of freedom).

Examples of systems that can be solved with perturbations include systems with nonlinear contributions to the equations of motion, [[Fundamental interaction|interaction]]s between particles, terms of higher powers in the Hamiltonian/free energy.

For physical problems involving interactions between particles, the terms of the perturbation series may be displayed (and manipulated) using [[Feynman diagram]]s.

===In chemistry===
Many of the [[ab initio quantum chemistry methods]] use perturbation theory directly or are closely related methods. Implicit perturbation theory<ref>{{cite journal | doi = 10.1021/ja00428a004 | title = Theory of the Chemical Bond | year = 1976 | last1 = King | first1 = Matcha | journal = Journal of the American Chemical Society | volume = 98 | issue = 12 | pages = 3415–3420 }}</ref> works with the complete Hamiltonian from the very beginning and never specifies a perturbation operator as such. [[Møller&ndash;Plesset perturbation theory]] uses the difference between the [[Hartree&ndash;Fock]] Hamiltonian and the exact non-relativistic Hamiltonian as the perturbation. The zero-order energy is the sum of orbital energies. The first-order energy is the Hartree&ndash;Fock energy and electron correlation is included at second-order or higher. Calculations to second, third or fourth order are very common and the code is included in most [[Computational chemistry#Software packages|''ab initio'' quantum chemistry programs]]. A related but more accurate method is the [[coupled cluster]] method.

===Shell-crossing===
A shell-crossing (sc) occurs in perturbation theory when matter trajectories intersect, forming a [[Singularity (mathematics)|singularity]].<ref>{{Cite journal |last1=Rampf |first1=Cornelius |last2=Hahn |first2=Oliver |date=2021-02-01 |title=Shell-crossing in a ΛCDM Universe |journal=Monthly Notices of the Royal Astronomical Society |volume=501 |issue=1 |pages=L71–L75 |doi=10.1093/mnrasl/slaa198 |doi-access=free |issn=0035-8711|arxiv=2010.12584 |bibcode=2021MNRAS.501L..71R }}</ref> This limits the predictive power of physical simulations at small scales.