Editing Conjugate variables (section)

{{Short description|Variables that are Fourier transform duals}}

'''Conjugate variables''' are pairs of variables mathematically defined in such a way that they become [[Fourier transform]] [[dual (mathematics)|duals]],<ref>{{Cite web |url=http://www.aip.org/history/heisenberg/p08a.htm |title=Heisenberg – Quantum Mechanics, 1925–1927: The Uncertainty Relations |access-date=2010-08-07 |archive-date=2015-12-22 |archive-url=https://web.archive.org/web/20151222204440/https://www.aip.org/history/heisenberg/p08a.htm |url-status=dead }}</ref><ref>{{cite journal | url=https://doi.org/10.1007%2FBF02731451 | doi=10.1007/BF02731451 | title=Some remarks on time and energy as conjugate variables | year=1962 | last1=Hjalmars | first1=S. | journal=Il Nuovo Cimento | volume=25 | issue=2 | pages=355–364 | bibcode=1962NCim...25..355H | s2cid=120008951 | url-access=subscription }}</ref> or more generally are related through [[Pontryagin duality]]. The duality relations lead naturally to an uncertainty relation—in [[physics]] called the [[Heisenberg uncertainty principle]]—between them. In mathematical terms, conjugate variables are part of a [[symplectic basis]], and the uncertainty relation corresponds to the [[symplectic form]]. Also, conjugate variables are related by [[Noether's theorem]], which states that if the laws of physics are invariant with respect to a change in one of the conjugate variables, then the other conjugate variable will not change with time (i.e. it will be conserved).
[[Conjugate variables (thermodynamics)|Conjugate variables in thermodynamics]] are widely used.