Conjugate variables: Difference between revisions

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{{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>[http://www.springerlink.com/content/r40472577250313r/ Some remarks on time and energy as conjugate variables]</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).

==Examples==

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* The ''[[linear momentum]]'' of a particle is the derivative of its action with respect to its ''[[position (vector)|position]]''.

* The ''[[angular momentum]]'' of a particle is the derivative of its action with respect to its ''[[orientation (geometry)|orientation]]'' (angular position).

* The ''mass-moment'' ('''p'''''t''-''E'''''x''') of a particle is the derivative of its action with respect to its ''[[rapidity (physics)|rapidity]]''.

* The ''[[electric potential]]'' (φ, [[voltage]]) at an event is the negative of the derivative of the action of the electromagnetic field with respect to the density of (free) ''[[electric charge]]'' at that event. {{citation needed|date=April 2013}}

* The ''[[magnetic potential]]'' ('''A''') at an event is the derivative of the action of the electromagnetic field with respect to the density of (free) ''[[electric current]]'' at that event. {{citation needed|date=April 2013}}

* The ''[[electric field]]'' ('''E''') at an event is the derivative of the action of the electromagnetic field with respect to the ''electric [[polarization density]]'' at that event. {{citation needed|date=April 2013}}

* The ''[[magnetic field|magnetic induction]]'' ('''B''') at an event is the derivative of the action of the electromagnetic field with respect to the ''[[magnetization]]'' at that event. {{citation needed|date=April 2013}}

* The Newtonian ''[[gravitational potential]]'' at an event is the negative of the derivative of the action of the Newtonian gravitation field with respect to the ''[[mass density]]'' at that event. {{citation needed|date=April 2013}}