Canonical conjugate

Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation—in physics 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.

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:

In classical physics, 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.

In Hamiltonian fluid mechanics and quantum hydrodynamics, the action itself (or velocity potential) is the conjugate variable of the density (or probability density).

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