These forces and their associated displacements are called conjugate variables.

In the above description, the product of two conjugate variables yields an energy.

Volume is one of a pair of conjugate variables, the other being pressure.

In quantum mechanics, position and momentum are canonical conjugate variables.

Here, pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables.

In classical physics, the derivatives of action are conjugate variables to the quantity with respect to which one is differentiating.

The temperature/entropy pair of conjugate variables is the only heat term; the other terms are essentially all various forms of work.

Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, with accuracy.

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:

The corners represent common conjugate variables while the sides represent thermodynamic potentials.

The two thermodynamic parameters which form a generalized force-displacement pair are termed "conjugate variables".

(Equation for the conjugate variables momentum and position)

The common conjugate variables are:

This is due to the inverse relationship between the frequency bandwidth of a pulse and its time duration, due to their being conjugate variables.

If is conjugate to then we have the equations of state for that potential, one for each set of conjugate variables.

In mathematical terms, we say that is the Fourier transform of and that x and p are conjugate variables.

In mathematical terms, conjugate variables are part of a symplectic basis, and the uncertainty principle corresponds to the symplectic form.

One of the best known pairs of conjugate variables was found in 1925 by Werner Heisenberg and his co-workers.

Conjugate variables (thermodynamics)

The thermodynamic square can be used as a tool to recall and derive some of the thermodynamic potentials based on conjugate variables.

The chemical potential is a fundamental parameter in thermodynamics and it is conjugate variables (thermodynamics) to the particle number.

As conjugate variables to the composition , the chemical potentials are intensive properties, intrinsically characteristic of the system, and not dependent on its extent.

Liouville's theorem justifies the use of canonically conjugate variables, such as positions and their conjugate momenta.

In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature and entropy or pressure and volume.