Conjugate variables

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. 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).
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:
Derivatives of action.
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.
Quantum theory.
In quantum mechanics, 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 formula_2 and momentum formula_3. In the quantum-mechanical formalism, the two observables formula_4 and formula_5 correspond to operators formula_6 and formula_7, which necessarily satisfy the canonical commutation relation: 
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, formula_10 and formula_11 denote "uncertainty" in the simultaneous specification of formula_4 and formula_5. A more precise, and statistically complete, statement involving the standard deviation formula_14 reads:
More generally, for any two observables formula_16 and formula_17 corresponding to operators formula_18 and formula_19, 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 formula_21, with a corresponding group called the Heisenberg group formula_22.
Fluid mechanics.
In Hamiltonian fluid mechanics and quantum hydrodynamics, the "action" itself (or "velocity potential") is the conjugate variable of the "density" (or "probability density).