Why do conjugate variables come in pairs? For example, in classical mechanics we have the generalized coordinates of position and momentum, and there is Jacobi's action-angle coordinates. Also, in the fundamental thermodynamic equations, all the relevant quantities appear in pairs, i.e. volume and pressure, entropy and temperature, particle number and chemical potential. Surely a dimensional argument can be made as to why we want these pairs (so that they give units of energy, etc...), but I am looking for a conceptual argument as to why conjugate variables come in pairs.

I apologize if this question is ill-posed.


This might be best answered by considering your former examples (position and momentum) and your latter examples (volume and pressure, etc.) separately. See this question for why I am considering these separately.

The first set of examples (ones from Hamiltonian mechanics) has its roots in Pontryagin dualities. The proof of this concept is a bit detailed but it essentially boils down to trying to find the conditions under which $\mathcal{F}(\mathcal{F}(f))$ is equivalent to $f$ in some sense. As it happens, this holds for all compact abelian groups. And also, as it happens, position is the Pontryagin dual of momentum, and vice versa.

The second set of examples (from thermodynamics) emerges when we look at equilibrium distributions in thermodynamics. In that case there is nothing 'special' about those variables coming in pairs. For example consider the internal energy equation for a canonical ensemble, relating the it to entropy (S), volume (V), temperature (T), and pressure (p):

$$ \mathrm {d} U=T\,\mathrm {d} S-p\,\mathrm {d} V $$ You can see that temperature and pressure are just the 'constants of proportionality' of changes in entropy and volume in this equation. As to why changes in temperature drive changes in entropy (heat transfer), it is because of the second law of thermodynamics seeking to increase the entropy of the system plus the bath. When the system is driven slightly away from equilibrium, for example by increasing $T$ of the bath by an very small amount, some amount of entropy is exchanged and the system equilibrates again.

There is a different kind of relationship between temperature and heat, say, and position/momentum. While generalized momentum and position are related by reciprocal differential equations, the same is not true for temperature and heat.

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