# canonical conjugate

## Thermodynamics

- Conjugate variables (thermodynamics) - the expression of all thermodynamic potentials are expressed in terms of conjugate pairs.

## Physics

A pair of variables mathematically defined in such a way that they become Fourier transform duals of one-another, or more generally are related through Pontryagin duality. 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, is given in the article canonical coordinates. 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 know its frequency very accurately.
- Time and energy - as energy and frequency in Quantum Mechanics are directly proportional to each other.
- Position and momentum: precise definition of position lead to ambiguity of momentum, and vice versa.
- Angle (angular position) and angular momentum;
- 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**.

**conjugate variables**such as pressure/volume or temperature/entropy. In fact all thermodynamic potentials are expressed in terms of conjugate pairs.

For a mechanical system, a small increment of energy is the product of a force times a small displacement.

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**thermodynamic potentials**are parameters associated with a thermodynamic system and have the dimensions of energy. They are called "potentials" because in a sense, they describe the amount of potential energy in a thermodynamic system when it is subjected to certain constraints.

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**Fourier transform**, named in honor of French mathematician Joseph Fourier, is a certain linear operator that maps functions to other functions. Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its

*frequency components*

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**Pontryagin duality**explains the general properties of the Fourier transform. It places in a unified context a number of observations about functions on the real line or on finite abelian groups:

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**Heisenberg uncertainty principle**, or HUP, gives a lower bound on the product of the standard deviations of position and momentum for a system, implying that it is impossible to have a particle that has an arbitrarily well-defined position and momentum simultaneously.

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**Mathematics**(colloquially,

**maths**or

**math**) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".

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**Hamiltonian mechanics**is a re-formulation of classical mechanics that was invented in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, another re-formulation of classical mechanics, introduced by Joseph Louis Lagrange in 1788.

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In mathematics and classical mechanics,

**canonical coordinates**are a particular set of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold. Canonical coordinates arise naturally in physics in the study of Hamiltonian mechanics.**.....**Click the link for more information.**.**

*time*One view is that time is part of the fundamental structure of the universe, a dimension in which events occur in sequence, and time itself is something that can be measured.

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**is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. A sequel, titled**

*FreQuency**Amplitude*was released in 2003.

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**.**

*time*One view is that time is part of the fundamental structure of the universe, a dimension in which events occur in sequence, and time itself is something that can be measured.

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**energy**(from the Greek ενεργός,

*energos*, "active, working")

^{[1]}is a scalar physical quantity that is a property of objects and systems of objects which is conserved by nature.

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**quantum mechanics**is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.

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**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

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**momentum**(pl. momenta; SI unit kg m/s, or, equivalently, N•s) is the product of the mass and velocity of an object. For more accurate measures of momentum, see the section "modern definitions of momentum" on this page.

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**angle**(in full,

**plane angle**) is the figure formed by two rays 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

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**angular momentum**of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external torque.

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**Doppler**can refer to:

- the Doppler effect, where the frequency of a wave changes with the relative velocity between the source and the observer
- Doppler broadening - Atoms in thermal motion emit doppler-shifted radiation. The net effect is to broaden a spectral line.

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**Radar**is a system that uses electromagnetic 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.

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In pulsed radar signal processing, an

**ambiguity function**is a two-dimensional function of delay and Doppler frequency showing the distortion of an uncompensated matched filter (sometimes called pulse compression) due to the Doppler shift of the return from a moving target.**.....**Click the link for more information.This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.