The direct measurement theory studies linear functionals as applied to the problems of quantum mechanics in addition to considering quadratic functionals on the space of wave functions, well established since the beginning of the 20th century. The theory is based on the time invariance principle of an appropriate space for linear functionals. In this case, it turns out that the second order Schrödinger equation is factorized: factors "respect" the effect of one of two groups, i.e., the group of inertial gas motion or the nonlinear group. In the weakly dissipative Kolmogorov-Arnold-Moser (KAM) theory, the former group is of extraordinary interest in connection with the formation of caustic curves which, in turn, cause the appearance of advanced and delayed potentials, which makes it possible to estimate anew the ideas of Ito-Stratonovich in the theory of stochastic processes.
Within the framework of the simplest model of the weakly dissipative Kolmogorov-Arnold-Moser theory, known as the Bogdanov-map, we compute two families of phase portraits of vector fields. These phase curves describe the moving boundary in the variational problem governing a piecewise linear motion of a test particle in an inhomogeneous continuous medium. Bibliography: 6 titles. Illustrations: 1 figure.
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