Singular relative integral invariants and adiabatic processes of thermodynamics (Q1125635)

The paper deals first, more mathematically, with singular relative integral invariants and Poincaré's first return map in the form of Dulac and second, more physically, with thermodynamic systems having one degree of freedom. The main result of the first part is that the orders of singularity of the invariant measures of a chain of triples subjected to a weight correspond to the orbits of the Feigenbaum mapping. The triples consist of an initial vector field, its invariant measure and an element of the stationary subalgebra of the pairs represented by the phase pattern and the invariant measure of the vector field in the plane. The rectified phase pattern as well as the module of invariant measures are defined by the quotient field in the phase plane which is obtained by means of local foliations into nonsingular hypersurfaces containing phase curves of the vector field. Invariant measures and its singularities are considered both in the neighborhood of a resonance and nonresonance saddle. The arguments given in the paper simplify considerably the answers to Dulac's problem and reveal the close relation between the study of the triples and the classical theory of dynamical systems. A new interpretation of the classical direct and inverse dissipation problems, presenting an alternative to Kepler's classical problem, the three body problem, etc. can be given. In the second part a thermodynamic system with one degree of freedom is defined in a 3D-space having the intrinsic energy, the pressure and the volume as components, with the aid of an absolutely integrable differential form, the so-called contact structure, that specifies the entropy of the system. On a Lie algebra of isoentropic vector fields the adiabatic action of the Lie group of internal symmetries of a thermodynamic system are deeply investigated as well as the adiabatic invariants and the orders of its singularities at a nonsingular point.