Conjugate variables in thermodynamics vs. Hamiltonian mechanics

According to Wikipedia, the canonical coordinates $p, q$ of analytical mechanics form a conjugate variables' pair - not just a canonically conjugate one.

However, the "conjugate variables" I directly think of are the quantities of thermodynamics - e.g. Temperature and Entropy, etc.

So, why both these classes of variables are called "conjugate"? What is the relation among them?

Answer

1. 
   

   Conjugate variables $(q^i, p_i)$ are given in thermodynamics via contact geometry as the first law of thermodynamics

   $$\mathrm{d}U~=~ \sum_{i=1}^np_i\mathrm{d}q^i,\tag{1}$$ where $U$ is internal energy. See also Ref. 1 and this & this Phys.SE posts.
   
   


2. 
   

   Conjugate variables $(q^i, p_i)$ are given in Hamiltonian mechanics via symplectic geometry as Darboux coordinates, i.e. the symplectic 2-form takes the form

   $$\omega ~=~\sum_{i=1}^n\mathrm{d}p_i\wedge \mathrm{d}q^i.\tag{2}$$ Hamilton's principal function $S(q,t)$ satisfies $$\mathrm{d}S~=~ \sum_{i=1}^np_i\mathrm{d}q^i-H\mathrm{d}t,\tag{3}$$ cf. Ref. 2.
   
   

References:

1. 
   
   

   S. G. Rajeev, A Hamilton-Jacobi Formalism for Thermodynamics, Annals. Phys. 323 (2008) 2265, arXiv:0711.4319.

   
   


2. 
   

   J. C. Baez, Classical Mechanics versus Thermodynamics, part 1 & part 2, Azimuth blog posts, 2012.