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Conjugate variables

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Conjugate variables

In physics

Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, conjugate variables are pair of variables mathematically defined in such a way that they become Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions....
duals of one-another, or more generally are related through Pontryagin duality

Pontryagin duality

In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform....
. The duality relations lead naturally to an uncertainty (Heisenberg uncertainty principle) relation between them.

A more precise mathematical definition, in the context of Hamiltonian mechanics

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without recourse to Lagrangian mechanics using sym...
, is given in the article canonical coordinates

Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are particular sets of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold....
.

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Encyclopedia

Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, conjugate variables are pair of variables mathematically defined in such a way that they become Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions....
duals of one-another, or more generally are related through Pontryagin duality

Pontryagin duality

In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform....
. The duality relations lead naturally to an uncertainty (Heisenberg uncertainty principle) relation between them.

A more precise mathematical definition, in the context of Hamiltonian mechanics

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without recourse to Lagrangian mechanics using sym...
, is given in the article canonical coordinates

Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are particular sets of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold....
.

Examples

Examples of canonically conjugate variables include the following:

Time
Time

Time is a component of the measurement used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects....
and frequency
Frequency

Frequency 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.
Time
Time

Time is a component of the measurement used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects....
and energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
- as energy and frequency in Quantum Mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
are directly proportional to each other.
Position
Position

Position may refer to:* A location in a coordinate system, usually in two or more dimensions; the science of position and its generalizations is topology...
and momentum
Momentum

In classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section Momentum#Modern definitions of momentum on this page....
: precise definition of position lead to ambiguity of momentum, and vice versa.
Angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
(angular position) and angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
;
Doppler
Doppler

Doppler can refer to:...
and range: the more we know about how far away a radar
Radar

Radar is a system that uses electromagnetic radiation waves to identify the range, altitude, direction, or speed of both moving and fixed objects such as aircraft, ships, motor vehicles, weather formations, and terrain....
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
Radar ambiguity function

In pulsed radar and sonar signal processing, an ambiguity function isa two-dimensional function of time delay and Doppler frequency showing the distortion of an uncompensated...
or radar ambiguity diagram.