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
Conjugate variables (qi,pi) are given in thermodynamics via contact geometry as the first law of thermodynamics dU = n∑i=1pidqi, where U is internal energy. See also Ref. 1 and this & this Phys.SE posts.
Conjugate variables (qi,pi) are given in Hamiltonian mechanics via symplectic geometry as Darboux coordinates, i.e. the symplectic 2-form takes the form ω = n∑i=1dpi∧dqi. Hamilton's principal function S(q,t) satisfies dS = n∑i=1pidqi−Hdt, cf. Ref. 2.
References:
S. G. Rajeev, A Hamilton-Jacobi Formalism for Thermodynamics, Annals. Phys. 323 (2008) 2265, arXiv:0711.4319.
J. C. Baez, Classical Mechanics versus Thermodynamics, part 1 & part 2, Azimuth blog posts, 2012.
No comments:
Post a Comment