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The Lie derivative, which has a wide range of application in physics and geometry, is trying to be examined on time scales. Firstly, nabla Lie bracket is defined on two-dimensional time scales. Secondly, the nabla Lie multiplication and some properties are given on the time scales. Lastly, for analyzing the differences between the real Lie multiplication and the nabla Lie multiplication, a numerical example is given.

By unifying continuous and discrete calculus a different kind of calculus was exposed which recently takes the attentions as time scale theory has been introduced by Hilger in 1988 with his doctoral dissertation. The time scale theory by means of calculus every passing day lots of new theories and different implementation parts are quickly composed. Time scale is really very important and has an useful role at a great deal of sciences which are studying with dynamical systems. Differential geometry is one of these sciences. In [

In physics, the use of Lie derivative is based on very old time. Especially the use of the Lie brackets is quite important in nonlinear control system and field of neural networks domains. Lie derivative studies, maintained until today, always show their effects in continuous space; however, in discrete space Lie multiplication is not studied. In this study in order to eliminate this problem, we will try to survey the Lie brackets which combines discrete space and continuous space on time scale. Thus, defined nabla Lie operator simultaneously in real terms and in different time scales, their reciprocities will be able to find easily. Additionally, for analyzing the differences between the reel Lie multiplication and the nabla Lie multiplication a numerical example is given.

The following definitions and theorems will serve as a short primer on time scale calculus; they can be found in [

If

for any constant

if

Let two vector fields

Let two vector fields

Let

Lie multiplication and derivative are indispensable notions for algebra and geometry. Up to now we used them for discrete structures. In this study we have a chance for analyzing both discrete structures and indiscrete structures. Under both circumstances, we will research the structure of time scale. Because of covariant derivative was proved as nabla covariant derivative in the references [

Let one has the two-dimensional space which is called

Let one shows all the set of vector fields with

Let one has the

Let one has the two completely nabla-differentiable vector fields of f functions and

Let one has the two completely nabla differentiable vector fields as

Let us briefly write the substraction equations of nabla lie derivations by the help of equations that take place at Lemma

Thus, it has been proven that may be thinking of the substraction of nabla covariant vectors which are belonging to two vector fields of nabla Lie derivatives. When the time scale is

The vector fields

The following proofs are obtained by the definition of Lie bracket and the results of nabla covariant derivation.

Here we will proof that the nabla Lie bracket is not commutative:

Let us denote the sum

From the definition of nabla Lie derivative we can obtain the following equations:

Let us give a function

Here we will calculate the nabla Lie multiplication, then we will try to obtain

Let us firstly identify the

From these equations, when the time scale is

The graph of the vector fields

When the time scale is

This is seen as a difference of natural consequence of the continuous and discrete structures, see in Figure

The graph of the vector fields

In this study, it is seen that the Lie derivative which is frequently used in physics handles instead of continuous derivative in the way that the nabla Lie multiplication which is created by using time scale both continuous and discrete spaces that are obtained at the same time. This is possible to observe. This is the way to perform the easier and smoother transmission from the continuous space to discrete space. If we consider that the Lie multiplication cannot be analyzed yet in differential geometry discrete space and we can understand how important practice transmission to discrete space with nabla Lie multiplication. It is possible to use nabla Lie multiplication theoretically and practically in many fields of physics with this study.

For instance, with the leading of our work it will be possible to use nabla Lie derivative in nonlinear control systems, field of neural networks, and periodic orbits of a dynamical system, which are important fields.