{{For|conjugate variables in context of thermodynamics|Conjugate variables (thermodynamics)}}
'''Conjugate variables''' are pairs of variables mathematically defined in such a way that they become [[Fourier transform]] [[dual (mathematics)|duals]],<ref>[http://www.aip.org/history/heisenberg/p08a.htm Heisenberg – Quantum Mechanics, 1925–1927: The Uncertainty Relations]</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 }}</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).
