Singular relative integral invariants and adiabatic processes of thermodynamics

Abstract

UDC 517.987.5; 517.958 The vector field is rectified in the neighborhood of a nonsingular point. By means of local foliation into nonsingular hypersurfaces containing phase curves of the vector field, we obtain a two-dimensional vector quotient field. A quotient field in the phase plane defines the rectified phase pattern and the module (over the ring of first integrals) of invariant measures. The pair that is composed of the phase pattern and the invariant measure of the vector field in the plane has a stationary subaigebra in the algebra of all vector fields. The triple consisting of the initial vector field, its invariant measure, and an element of the stationary subalgebra of the pairs, admits a relative integral invariant, namely, the integral of the measure over sets that are bounded by the phase curves of two vector fields. The relative integral invariant is generated (over the ring of first integrals) by the action of an element of the stationary subalgebra (the product of the Hamiltonian or the potential by the time of the element of the subalgebra). This explicitly asymmetric construction relative to the pair of vector fields becomes symmetric if we consider singular invariant measures and smooth vector fields. Suppose that an element of the subalgebra is a nonsingular vector field at the point being considered. Then we can take it as the initial element for the next triple. Obviously, the iteration of this procedure generates a chain of triples and, respectively, a chain of measures. We are interested in the convergence of the arising measures (the sense of convergence is defined in the article). We say that a chain of triples is subjected to the weight k > 0 (say, k is a positive integer) if, in each triple, the invariant measure generated by the action of the element of the subaigebra is generated by the first integral of the first element of the triple of degree k (The invariant measure of a triple makes it possible to determine two numbers, namely, the successive weight, i.e., the order of singularity of the measure generated by the action of the element of the subalgebra over the ring of the first integrals of the subalgebra and the preceding weight, which is the same as above, only with the replacement of the element of the subalgebra by the initial vector field.). The order of singularity of an invariant measure is the weight of the measure relative to the action for the element of the subalgebra. Theorem. The orders of singularity of the invariant measures of a chain of triples subjected to the weight k correspond to the orbits of the Feigenbaum mapping.