# Conjugate variables

**Conjugate variables** are pairs of variables mathematically defined in such a way that they become Fourier transform duals of one another,^{[1]}^{[2]} 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.

## 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 more time). Conversely, a very short musical note becomes just a click, and so one can't determine its frequency very accurately.
^{[3]} - 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. In this case, the two dimensional function of doppler and range is known as a radar ambiguity function or
**radar ambiguity diagram**. - Surface energy: γdA (
*γ*= surface tension ;*A*= surface area). - Elastic stretching: FdL (
*F*= elastic force;*L*length stretched).

### Derivatives of action

In classical physics, the derivatives of action are conjugate variables to the quantity with respect to which one is differentiating. In quantum mechanics, these same pairs of variables are related by the Heisenberg uncertainty principle.

- The
*energy*of a particle at a certain event is the negative of the derivative of the action along a trajectory of that particle ending at that event with respect to the*time*of the event. - The
*linear momentum*of a particle is the derivative of its action with respect to its*position*. - The
*angular momentum*of a particle is the derivative of its action with respect to its*orientation*(angular position). - 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. - 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. - 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. - The
*magnetic induction*(B) at an event is the derivative of the action of the electromagnetic field with respect to the*magnetization*at that event. - 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.

### Fluid Mechanics

In Hamiltonian fluid mechanics and quantum hydrodynamics, the *action* itself (or *velocity potential*) is the conjugate variable of the *density* (or *probability density*).