The Concepts of Lie Derivative for Discrete-time Systems

Tanel Mullari ； Ülle Kotta ； Zbigniew Bartosiewicz ； Ewa Pawłuszewicz

《Proceedings of the Estonian Academy of Sciences》 61卷4期 (2012/12) Pp. 253-265

並列摘要

The paper extends the concept of the Lie derivative of the vector field, used in the study of the continuous-time dynamical systems, for the discrete-time case. In the continuous-time case the Lie derivative of a vector field (1-form or scalar function) with respect to the system dynamics is defined as its rate of change in time. In the discrete-time case we introduce the algebraic definition of the Lie derivative, using the concepts of forward and backward shifts. The definitions of discrete-time forward and backward shifts of the vector field are based on the concepts of already known forward and backward shifts of the 1-forms and on the scalar product of 1-form and vector field. Further we show that the interpretation of the discrete-time Lie derivative agrees with its interpretation as the rate of change in the continuous-time case. Finally, the geometric property of the discrete-time Lie derivative is also examined and shown to mimic the respective property in the continuous-time case.

並列關鍵字

differential geometry ； difference equations ； vector fields ； Lie derivative ； forward shift ； backward shift