ESTONIAN ACADEMY
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eesti teaduste
akadeemia kirjastus
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proceedings
of the estonian academy of sciences
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The concepts of Lie derivative for discrete-time systems; pp. 253–265

Full article in PDF format | doi: 10.3176/proc.2012.4.01

Authors
Tanel Mullari, Ülle Kotta, Zbigniew Bartosiewicz, Ewa Pawłuszewicz

Abstract

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.


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