Conjugate variables are pairs of variables mathematically defined in such a way that they become
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
duals
''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers.
Track listing
:* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, ...
, or more generally are related through
Pontryagin duality. The duality relations lead naturally to an uncertainty relation—in
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
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. Also, conjugate variables are related by
Noether's theorem, which states that if the laws of physics are invariant with respect to a change in one of the conjugate variables, then the other conjugate variable will not change with time (i.e. it will be conserved).
Conjugate variables in thermodynamics are widely used.
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
Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
: the longer a musical note is sustained, the more precisely we know its frequency, but it spans a longer duration and is thus a more-distributed event or 'instant' in time. Conversely, a very short musical note becomes just a click, and so is more temporally-localized, but one can't determine its frequency very accurately.
*
Doppler and
range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
: the more we know about how far away a
radar
Radar is a system that uses radio waves to determine the distance ('' ranging''), direction ( azimuth and elevation angles), and radial velocity of objects relative to the site. It is a radiodetermination method used to detect and track ...
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: ''γ'' d''A'' (''γ'' =
surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...
; ''A'' = surface area).
* Elastic stretching: ''F'' d''L'' (''F'' = elastic force; ''L'' length stretched).
* Energy and time: Units
being kg m
2 s
−1.
Derivatives of action
In
classical physics
Classical physics refers to physics theories that are non-quantum or both non-quantum and non-relativistic, depending on the context. In historical discussions, ''classical physics'' refers to pre-1900 physics, while '' modern physics'' refers to ...
, 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 uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
.
* The ''
energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
'' 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
Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
'' of the event.
* The ''
linear momentum'' of a particle is the derivative of its action with respect to its ''
position''.
* The ''
angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
'' of a particle is the derivative of its action with respect to its ''
orientation'' (angular position).
* The ''
mass-moment'' (
) of a particle is the negative of the derivative of its action with respect to its ''
rapidity
In special relativity, the classical concept of velocity is converted to rapidity to accommodate the limit determined by the speed of light. Velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velo ...
''.
* The ''
electric potential
Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
'' (φ,
voltage
Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...
) and ''
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
'' in a
quantum LC circuit.
* 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
An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge c ...
'' at that event.
* The ''
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
'' (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
In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Accordingly, physicists and engineers usually define magnetization as the quanti ...
'' at that event.
* The Newtonian ''
gravitational potential
In classical mechanics, the gravitational potential is a scalar potential associating with each point in space the work (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the ...
'' at an event is the negative of the derivative of the action of the Newtonian gravitation field with respect to the ''
mass density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter rho), although the Latin letter ''D'' (or ''d'') ...
'' at that event.
Quantum theory
In
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, conjugate variables are realized as pairs of observables whose operators do not commute. In conventional terminology, they are said to be ''incompatible observables''. Consider, as an example, the measurable quantities given by position
and momentum
. In the quantum-mechanical formalism, the two observables
and
correspond to operators
and
, which necessarily satisfy the
canonical commutation relation
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
hat x,\hat p ...
:
For every non-zero commutator of two operators, there exists an "uncertainty principle", which in our present example may be expressed in the form:
In this ill-defined notation,
and
denote "uncertainty" in the simultaneous specification of
and
. A more precise, and statistically complete, statement involving the standard deviation
reads:
More generally, for any two observables
and
corresponding to operators
and
, the
generalized uncertainty principle is given by:
Now suppose we were to explicitly define two particular operators, assigning each a ''specific'' mathematical form, such that the pair satisfies the aforementioned commutation relation. It's important to remember that our particular "choice" of operators would merely reflect one of many equivalent, or isomorphic, representations of the general algebraic structure that fundamentally characterizes quantum mechanics. The generalization is provided formally by the
Heisenberg Lie algebra , with a corresponding group called the Heisenberg group
.
Fluid mechanics
In
Hamiltonian fluid mechanics Hamiltonian fluid mechanics is the application of Hamiltonian mechanics, Hamiltonian methods to fluid mechanics. Note that this formalism only applies to non-dissipative fluids.
Irrotational barotropic flow
Take the simple example of a barotropic, ...
and
quantum hydrodynamics, the ''
action'' itself (or ''
velocity potential
A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.
It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a ca ...
'') is the conjugate variable of the ''
density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
'' (or ''
probability density).
See also
*
Canonical coordinates
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
Notes
{{DEFAULTSORT:Conjugate Variables
Classical mechanics
Quantum mechanics