Conjugate variables
variables that are Fourier transform duals
Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, 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 in thermodynamics are widely used.
Examples
There are many types of conjugate variables, depending on the type of work a certain system is doing (or is being subjected to). Examples of canonically conjugate variables include the following:
Time and frequency: the longer a musical note is sustained, the more precisely we know its frequency, but it spans a longer duration and is thus a more-distributed event or 'instant' in time. Conversely, a very short musical note becomes just a click, and so is more temporally-localized, but one can't determine its frequency very accurately.
Doppler and range: the more we know about how far away a radar target is, the less we can know about the exact velocity of approach or retreat, and vice versa.
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