Non stationary rough procesess (motion with friction)
Institute of Physics and Technology(State University) 9 Institutskiy per., Dolgoprudny,Moscow Region, 141700, Russian Federation1 pp. (accepted)
Mathematical models of real processes contain along with an independent variable (time) and state variables certain constants, parameters, coefficients. A part of them are definite numbers, such as (relation of length of a circle to its diameter): e (the basis of the natural logarithm), another part of them are known with some accuracy (roughness). At last, the values of some parameters can change during the process, for example, atmosphere parameters. These parameters can be substituted rather by arbitrary functions of time (a non-stationary roughness ) In this report systems with friction are discussed . A parameter, changing in time, is the coefficient of friction [1, 2]. Really, on a long skiing distance such variation is caused by change of temperature and humidity of the atmosphere: the friction coefficient between skis and a ski track can continuously change. Another example: curling team, manipulating sweeps, operates friction coefficient.
In the report one-dimensional process with friction is modeled by an ordinary differential equation. It is supposed that coefficient of friction entering into the right part is more or less arbitrary function of time. Group-theoretic properties of the model are discussed: groups of shifts along trajectories and group of symmetry are calculated.
1. Yakovenko G.N. Non-stationary rough systems as a generalization of the class of controllable systems. //Automatica i Telemechanica, 2011 . — № 7. — Pp. 75–82.
2. Ivanov A.P. Foundation of theory of systems with friction. Moscow-Izhevsk, NIC “Regular and Chaotic Dynamics”, 2011