**Conjugate variables** are pairs of variables mathematically defined in such a way that they become

Fourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

duals of one-another, or more generally are related through

Pontryagin dualityIn mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...

. The duality relations lead naturally to an uncertainty in

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

called the Heisenberg uncertainty principle relation between them. In mathematical terms, conjugate variables are part of a symplectic basis, and the uncertainty principle 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
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

and frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the 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 know its frequency very accurately.
- Doppler
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...

and range: the more we know about how far away a radarRadar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...

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 functionIn pulsed radar and sonar signal processing, an ambiguity function is a two-dimensional function of time delay and Doppler frequency\chi showing the distortion of a returned pulse due to the receiver matched filter due to the Doppler shift of the return from a moving target...

or **radar ambiguity diagram**.
- Surface energy: γdA (
*γ* = surface tensionSurface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...

; *A* = surface area)
- Elastic stretching: FdL (
*F* = elastic force; *L* length stretched)

### Derivatives of action

In classical physics, the derivatives of

actionIn physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...

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 principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

.

- The
*energy*In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

of a particle at a certain eventIn physics, and in particular relativity, an event indicates a physical situation or occurrence, located at a specific point in space and time. For example, a glass breaking on the floor is an event; it occurs at a unique place and a unique time, in a given frame of reference.Strictly speaking, the...

is the negative of the derivative of the action along a trajectory of that particle ending at that event with respect to the *time*Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

of the event.
- The
*linear momentum* of a particle is the derivative of its action with respect to its *position*.
- The
*angular momentum*In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

of a particle is the derivative of its action with respect to its *angle*In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

(angular position).
- The
*electric potential*In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

(φ, voltageVoltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

) 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*Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

at that event.
- The
*magnetic potential*The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...

(A) at an event is the derivative of the action of the electromagnetic field with respect to the density of (free) *electric current*Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

at that event.
- The
*electric field*In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

(E) at an event is the derivative of the action of the electromagnetic field with respect to the *electric polarization density*In classical electromagnetism, polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is...

at that event.
- The
*magnetic induction*A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

(B) at an event is the derivative of the action of the electromagnetic field with respect to the *magnetization*In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...

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.