Conjugate variables: Difference between revisions

Revision as of 13:43, 6 March 2011

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* [[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 (vector)|Position]] and [[linear momentum]]: a precise definition of position leads to ambiguity of momentum, and vice versa.

* [[Angle]] (angular position) and [[angular momentum]];

* [[Doppler effect|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'''.

* [[Electrical work]]: ℰde (''ℰ'' = [[electromotive force]]; ''e'' amount of charge)

* Magnetic work MdH (''M'' = magnetization; ''H'' = field)

* Surface energy: γdA (''γ'' = [[surface tension]] ; ''A'' = surface area)

* Elastic stretching: FdL (''F'' = elastic force; ''L'' length stretched)

* [[Gravitational potential energy]]: ψdm (''ψ'' = gravitational potential; ''m'' = mass)

===Derivatives of action===

In classical physics, the derivatives of [[action (physics)|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 (relativity)|event]] is 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 (vector)|position]]''.

* The ''[[angular momentum]]'' of a particle is the derivative of its action with respect to its ''[[angle]]'' (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 field|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.

=See also=