CLASSICAL MECHANICS AND ENTROPY

This note addresses a problem of nineteenth century applied mathematic s-is it possible in the context of Hamiltonian mechanics to define a f unction S of the generalized coordinates and momenta which is monotoni cally increasing along orbits? The question is of interest, because, f or a sytem not in thermodynamic equilibrium, entropy should increase s trictly monotonically along an orbit, and a negative answer implies th at mechanical principles different from those of Hamiltonian mechanics must be introduced to explain thermodynamics. This note answers the q uestion rigorously for Hamiltonian systems confined to an invariant re gion of finite volume in phase space; it is not possible to define a c ontinuous function which increases monotonically along orbits. An appe ndix gives a translation of an 1889 paper of Poincare addressing the s ame issue.