Additive and multiplicative doubling variables in autonomous equations

The paper analyzes the Cauchy problem for a dynamic equation or for a dynamic system (1) of equations. In doing so, we take a flow and a tangential space Tx, the flow being the dynamic system or mapping X basis of segregation of Em subset on the Tx layer or a trivial tangential segregation. The segregation is given by diffeomorphism from X on E1, of Crr(T) class, r ≥ 1. The aim of this work is to elucidate the possibility of embedding this segregation into the field of extremals. A necessary and a sufficient condition for a trivial embedding is formulated as the intensification of Liouville’s condition in order to keep the phase volume or a condition of existence of invariant measure. The application of this condition allows to construct the dynamic system energy. The distribution which is a dual of Hamiltonian gives the Lagrangian density. In this manner a variational problem is obtained, in which the initial system plays the role of an intermediate integral and of coordination conditions for LaGrange’s system of equations. Procedures of different types of doubling variables (additive and multiplicative, external and internal) have been used in non-trivial embedding of the initial Cauchy problem characteristic into the field of extremals. Multiplicative doubling is analogous to the application of the integrating factor, and the additive one is identical to the addition of equations in variations to the initial system.