# The Lie Brackets on Time Scales

Authors: H. Kusak; A. Caliskan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1–4] some geometric notions are trying to research on the time scale. In the paper [5] the curve and surface description were made for the first time. The reference [6] direction nabla derivative and its properties were investigated. The properties of vector field, derivative mapping, and delta connection were investigated in [7, 8]. In [6] nabla covariant derivative definition was given. Also some fundamental properties about time scale can be obtained in the references [9–12].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.


## 2. Preliminaries

The following definitions and theorems will serve as a short primer on time scale calculus; they can be found in [6, 10, 11]. A time scale 𝕋 is any nonempty closed subset of ℝ. Within that set, define the jump operators ρ:𝕋→𝕋 by ρ(t)=sup{s∈𝕋:s<t}, where infϕ:=sup𝕋, and supϕ:=inf𝕋, where ϕ denotes the empty set. Also the graininess function is defined by ν:=ρ(t)-t. If f:𝕋→ℝ is a function, then we define the function fρ:𝕋→ℝ by fρ(t)=f(ρ(t)) for t∈𝕋, that is, fρ=f∘ρ.Theorem 2.1.
Iff,g:𝕋→ℝ are ∇-differentiable at t∈𝕋k, then(i)
f+g is ∇-differentiable at t and
(2.1)(f+g)∇(t)=f∇(t)+g∇(t),(ii)
for any constantc, c·f is ∇-differentiable at t and
(2.2)(cf)∇(t)=cf∇(t),(iii)
f·g is ∇-differentiable at t and
(2.3)(fg)∇(t)=f∇(t)g(t)+f(ρ(t))g∇(t)=g∇(t)f(t)+g(ρ(t))f∇(t),(iv)
ifg(t)·g(ρ(t))≠0, then f/g is ∇-differentiable at t and
(2.4)(fg)∇(t)=f∇(t)g(t)-f(t)g∇(t)g(t)⋅g(ρ(t)).Definition 2.2.
Let two vector fieldsZ and W be given. The covariant nabla differentiation with respect to W at the point P(t10,t20,…,tn0) is defined as the vector
(2.5)DWZ=(∂Z∇W)(P)=Y∇(0),
provided it exists, where Y(ξ)=Z(t10+ξw1,…,tn0+ξwn) for ξ∈𝕋.Theorem 2.3.
Let two vector fieldsZ, W be given. The covariant nabla differentiation with respect to W at the point P(t10,t20,…,tn0) exits and is expressed by the formula
(2.6)∂Z(P)∇wP=∑i=12‍∂gi(P)∇wP∂∇ixi.Theorem 2.4.
Leta,b∈ℝ and two vector fields X an Y be given. For any two tangent vectors Vp and Wp, the following properties are proven:
(2.7)(i)∂X∇(aV+bW)=a∂X∇V+b∂X∇W,(ii)∂(aX+bY)∇V=a∂X∇V+b∂Y∇V.


## 3. Nabla Lie Bracket on Time Scales

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 [7, 8, 11] that is why we are defining our study as Nabla-Lie.Definition 3.1.
Let one has the two-dimensional space which is calledΛ2=𝕋×𝕋 on T time scales. As f:M⊂Λ2→ℝ function, that is, on P point, the nabla vector field Vp(f) is given with these coordinates
(3.1)Vp(f)=∑i=12‍Vpi∂f∇xi=Vp1∂f∇x1+Vp2∂f∇x2,
which was introduced in the reference [7].Definition 3.2.
Let one shows all the set of vector fields withΨ(Λ2) on Λ2 space. By varying the point P along the curve on M, one can obtain another smooth function [V(f)]∇ which is the nabla derivative of f along the vector field. The function
(3.2)[V(f)]∇(p)=Vp(f)
is called the nabla Lie derivative on time scales. It is common to denote the nabla Lie derivative of a f function which has vector field along V that is shown as [V(f)]∇. As shown, two-dimensional nabla Lie multiplication is a function from Ψ(Λ2)×Ψ(Λ2) set to Ψ(Λ2) set.Lemma 3.3.
f function will be the completely nabla differentiable as regarded to be equal the inner multiplication vector field of V at point p with the derivative of f function at the same time interval of the nabla Lie derivation. In coordinates,
(3.3)[V(f)]∇(p)=∑i=12‍∂f∇xiVi=(Vp1∂f∇x1+Vp2∂f∇x2)=〈f∇,Vp〉.Theorem 3.4.
Let one has thef function which is completely nabla differentiable; ρ(t) will be the backward jump operator of taken time scale and area of V vector is expressed by Vρ notation at ρ(t) point. All smooth vector fields of the time scale space Ψ(Λ2) are vector space on a M manifold. V1 and V2 are the two vector fields, given a function with Lie nabla derivative which is defined on M⊂Ψ(Λ2). Then both V1[V2f]∇ and V2[V1f]∇ will help the calculation of nabla Lie bracket. The following equations can be expressed by nabla derivations on time scales:
(3.4)V1[V2f]∇(p)=V11(∂V21∇x1∂f∇x1+∂V22∇x1∂f∇x2)+V11(V21ρ∂2f∇x12+V22ρ∂2f∇x1∇x2)+V12(∂V21∇x2∂f∇x1+∂V22∇x2∂f∇x2)+V12(V21ρ∂2f∇x2∇x1+V22ρ∂2f∇x22),(3.5)V2[V1f]∇(p)=V21(∂V11∇x1∂f∇x1+∂V12∇x1∂f∇x2)+V21(V11ρ∂2f∇x12+V12ρ∂2f∇x1∇x2)+V22(∂V11∇x2∂f∇x1+∂V12∇x2∂f∇x2)+V22(V11ρ∂2f∇x2∇x1+V12ρ∂2f∇x22).Proof.
V
1 and V2 vector fields are written as the following:
(3.6)V1=∑i=12‍V1i∂∇xi=V11∂∇x1+V12∂∇x2,(3.7)V2=∑i=12‍V2i∂∇xi=V21∂∇x1+V22∂∇x2,
because of Definitions 3.1 and 3.2. V1[(V2f)]∇ vector field can be calculated as the following by the help of nabla derivation and the definition of Lie derivation which is used on the time scales:
(3.8)V1[V2]∇(f)=[∑i=12‍V1i∂∇xi(∑i=12‍V2j∂f∇xj)]∇=[(V11∂∇x1+V12∂∇x2)(V21∂f∇x1+V22∂f∇x2)]∇=V11∂∇x1(V21∂f∇x1)+V11∂∇x1⋅(V22∂f∇x2)+V12∂∇x2(V21∂f∇x1)+V12∂∇x2(V22∂f∇x2)=V11(∂V21∇x1⋅∂f∇x1+V21ρ∂2f∇x12)+V11(∂V22∇x1⋅∂f∇x2+V22ρ∂2f∇x1∇x2)+V12(∂V21∇x2⋅∂f∇x1+V21ρ∂2f∇x2∇x1)+V12(∂V22∇x2⋅∂f∇x2+V22ρ∂2f∇x22)=V11(∂V21∇x1⋅∂f∇x1+∂V22∇x1∂f∇x2)+V11(V21ρ∂2f∇x12+V22ρ∂2f∇x1⋅∇x2)+V12(∂V21∇x2⋅∂f∇x1+∂V22∇x2⋅∂f∇x2)+V12(V21ρ∂2f∇x2∇x1+V22ρ∂2f∇x22).
With the similar idea, the following equation:
(3.9)V2[V1]∇(f)=[∑i=12‍V2i∂∇xi(∑i=12‍V1j∂f∇xj)]=[(V21∂∇x1+V22∂∇x2)(V11∂f∇x1+V12∂f∇x2)]=V21∂∇x1(V11∂f∇x1)+V21∂∇x1(V12∂f∇x2)+V22∂∇x2(V11∂f∇x1)+V22∂∇x2(V12∂f∇x2)=V21(∂V11∇x1+V11ρ∂2f∇x12)+V21(∂V12∇x1∂f∇x2+V12ρ∂2f∇x1∇x2)+V22(∂V11∇x2∂f∇x1+V11ρ∂2f∇x2∇x1)+V22(∂V12∇x2∂f∇x2+V12ρ∂2f∇x22)=V21(∂V11∇x1∂f∇x1+∂V12∇x1∂f∇x2)+V21(V11ρ∂2f∇x12+V12ρ∂2f∇x1∇x2)+V22(∂V11∇x2∂f∇x1+∂V12∇x2)+V22(V11ρ∂2f∇x2∇x1+V12ρ∂2f∇x22)
can be found. Thus, obtained equations are desired at the theorem.Definition 3.5.
V
1
,
V
2
∈
Ψ
(
Λ
2
) are vector fields. The equation
(3.10)[V1,V2]∇:=V1[V2]∇-V2[V1]∇
is called nabla Lie bracket on time scales. Here V1[V2]∇ and V2[V1]∇ are nabla Lie derivations.Lemma 3.6.
Let one has the two completely nabla-differentiable vector fields of f functions andV1,V2∈Ψ(Λ2) vector fields. Due to the definitions of V1[V2f]∇ and V2[V1]∇, nabla Lie is equal to the substraction of the following equation:
(3.11)[V1,V2]∇(f):=V1[(V2f)]∇-V2[(V1f)]∇=V11(∂V21∇x2∂f∇x1+∂V22∇x1∂f∇x2)-V21(∂V11∇x1∂f∇x1+∂V12∇x1∂f∇x2)+V12(∂V21∇x2∂f∇x1+∂V22∇x2∂f∇x2)-V22(∂V11∇x2∂f∇x1+∂V12∇x2∂f∇x2)+(V21ρV11-V11ρV21)∂2f∇x12+(V21ρV12-V12ρV21)∂2f∇x1∇x2+(V22ρV12-V22V12ρ)∂2f∇x22+(V22ρV11-V11ρV22)∂2f∇x1∇x2=∑i=12‍[∑i=12‍(∂V2j∇xiV1i-∂V1j∇xiV2i)]∂∇xj+(V21ρV11-V11ρV21)∂2f∇x12+(V21ρV12-V12ρV21)∂2f∇x1∇x2+(V22ρV12-V22V12ρ)∂2f∇x22+(V22ρV11-V11ρV22)∂2f∇x1∇x2.Theorem 3.7.
Let one has the two completely nabla differentiable vector fields asV1 and V2. Nabla Lie multiplication can be defined as nabla covariant derivation because of the definition of nabla covariant derivative at [8] regarding N graininess function on time scales. The above representation gives a different geometrical dimension to Lie bracket. Lie parenthesis operator has an expression
(3.12)[V1,V2]∇(f)=DV1V2-DV2V1+[∂V2∇x1V1[f]-∂V1∇x1V2[f]]ν,
which is like that in the nabla covariant defined with Definition 2.2.Proof.
Let us briefly write the substraction equations of nabla lie derivations by the help of equations that take place at Lemma3.6. Then, we will, respectively, add and remove the expressions in brackets. In the next step, we will try to reach nabla covariant derivations from the nabla derivation definition by multiplying and dividing with (ρ(x1)-x1):
(3.13)[V1,V2]∇(f)=V1[(V2f)]∇-V2[(V1f)]∇=V11(∂V21∇x1∂f∇x1+∂V22∇x1∂f∇x2)-V21(∂V11∇x1∂f∇x1+∂V12∇x1∂f∇x2)+V12(∂V21∇x2∂f∇x1+∂V22∇x2∂f∇x2)-V22(∂V11∇x2∂f∇x1+∂V12∇x2∂f∇x2)+(V21ρV11-V11ρV21)∂2f∇x12+(V21ρV12-V12ρV21)∂2f∇x1∇x2+(V22ρV12-V22V12ρ)∂2f∇x22+(V22ρV11-V11ρV22)∂2f∇x1∇x2=∂f∇x1(V11∂V21∇x1+V12∂V21∇x2)+∂f∇x2(V11∂V22∇x1+V12∂V22∇x1)-∂f∇x1(V21∂V11∇x1+V22∂V11∇x2)-∂∇x2(V21∂V12∇x1+V22∂V12∇x2)+(V21ρV11-V1ρV21+V21V11-V21V11)∂2f∇x12+(V21ρV12-V12ρV21+V21V12-V21V12)∂2f∇x1∇x2+(V22ρV12-V22V12ρ+V22V11-V22V12)∂2f∇x22+(V22ρV11-V11ρV22+V22V11-V22V11)∂2f∇x1∇x2=∂f∇x1V1[V21]+∂f∇x2V1[V22]-∂f∇x1V2[V11]-∂∇x2V2[V12]+V11(V21ρ-V21)-V21(V11ρ-V11)ρ(x1)-x1⋅(ρ(x1)-x1)∂2f∇x12+V12(V21ρ-V21)-V21(V12ρ-V12)ρ(x1)-x1⋅(ρ(x1)-x1)⋅∂2f∇x1∇x2+V12(V22ρ-V22)-V22(V12ρ-V12)ρ(x1)-x1⋅(ρ(x1)-x1)⋅∂2f∇x22+V11(V22ρ-V22)-V22(V11ρ-V11)ρ(x1)-x1⋅(ρ(x1)-x1)⋅∂2f∇x1∇x2=∂f∇x1V1[V21]+∂f∇x2V1[V22]-∂f∇x1V2[V11]-∂∇x2V2[V12]+(V11∂V21∇x1-V21∂V11∇x1)∂2f∇x12ν+(V12∂V21∇x1-V21∂V12∇x1)∂2f∇x1∇x2ν+(V12∂V22∇x1-V22∂V12∇x1)∂2f∇x22ν+(V11∂V22∇x1-V22∂V11∇x1)∂2f∇x1∇x2+∂f∇x1V1[V21]+∂f∇x2V1[V22]-∂f∇x1V2[V11]-∂∇x2V2[V12]+ν∂V21∇x1∂f∇x1(V11∂f∇x1+V12∂f∇x2)+ν∂V22∇x1∂f∇x2(V12∂f∇x2+V11∂f∇x1)-ν∂V11∇x1∂f∇x1(V21∂f∇x1+V22∂f∇x2)-ν∂V12∇x1∂f∇x2(V21∂f∇x1+V22∂f∇x2)=∂f∇x1V1[V21]+∂f∇x2V1[V22]-∂f∇x1V2[V11]-∂∇x2V2[V12]+ν∂V21∇x1∂f∇x1V1[f]+ν∂V22∇x1∂f∇x2V1[f]-ν∂V11∇x1∂f∇x1V2[f]-ν∂V12∇x1∂f∇x2V2[f]=∂V2∇V1-∂V1∇V2+(∂V21∇x1∂f∇x1+∂V22∇x1∂f∇x2)V1[f]ν-(∂V11∇x1∂f∇x1+∂V12∇x1∂f∇x2)V2[f]ν=∂V2∇V1-∂V1∇V2+[∂V2∇x1V1[f]-∂V1∇x1V2[f]]ν=DV1V2-DV2V1+[∂V2∇x1V1[f]-∂V1∇x1V2[f]]ν.
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 isℝ, the grannies function ν is zero, so it is seen that nabla Lie bracket is transformed to Lie bracket at real. Thus, as a result of the equation it was proved to achieve with nabla lie bracket for both different time scales and its equivalents at real.Theorem 3.8.
The vector fieldsV1,V2,V3∈Ξ(Λ2) are completely nabla differentiable. From the following equations, it is seen that the nabla Lie bracket is a Lie bracket operator:
(3.14)(i)[V1,V2]∇=-[V2,V1]∇,(ii)[αV1+βV2,V3]∇=α[V1,V3]+β[V2,V3]∇,(iii)[V1,[V2,V3]∇]∇+[V2,[V3,V2]∇]∇+[V3,[V1,V2]∇]∇=0.Proof.
The following proofs are obtained by the definition of Lie bracket and the results of nabla covariant derivation.(i)
Here we will proof that the nabla Lie bracket is not commutative:(3.15)[V1,V2]∇=V1[V2]∇-V2[V1]∇=-[V2[V1]∇-V1[V2]∇]=-[V2,V1]∇.(ii)
Let us denote the sumαV1+βV2 with W in the following equations:
(3.16)[αV1+βV2,V3]∇=DV3W-DWV3+[∂V3∇x1W-∂W∇x1V3]ν=α∂V3∇V1+β∂V3∇V2-α∂V1∇V3-β∂V2∇V3+[α∂V3∇x1V1-β∂V3∇x1V2-α∂V1∇x1V3-β∂V2∇x1V3]ν=α[∂V3∇V1-∂V1∇V3+[∂V3∇x1V1-∂V1∇x1V3]ν]+β[∂V3∇V2-∂V2∇V3+[∂V3∇x1V2-∂V2∇x1V3]ν]=α[V1,V3]∇+β[V1,V3]∇.(iii)
From the definition of nabla Lie derivative we can obtain the following equations:(3.17)[V1,[V2,V3]∇]∇=V1([V2,V3]∇)-[V2,V3]∇(V1)=V1[V2[V3]∇]∇-V1[V3[V2]∇]∇-V2[V3[V1]∇]∇+V3[V2[V1]∇]∇,[V2,[V3,V1]∇]∇=V2([V3,V1]∇)-[V3,V1]∇(V2)=V2[V3[V1]∇]∇-V2[V1[V3]∇]∇-V3[V1[V2]∇]∇+V1[V3[V2]∇]∇,[V3,[V1,V2]∇]∇=V3([V1,V2]∇)-[V1,V2]∇(V3)=V3[V1[V2]∇]∇-V3[V2[V1]∇]∇-V1[V2[V3]∇]∇+V2[V1[V3]∇]∇.It is easy to see that by the addition of three equations above, we can obtain the result as zero:
(3.18)[V1,[V2,V3]∇]∇+[V2,[V3,V2]∇]∇+[V3,[V1,V2]∇]∇=0.


## 4. A Numeric Example

Let us give a functionf(x1,x2)=(x1x2+3x2)(∂/∇x1)+(2x1-x2)(∂/∇x2) and the vector fields on Λ2 as the following:
(4.1)V1=(x1-x22)∂∇x1+(x12)∂∇x2,V2=(x2+5)∂∇x1+(x1-4x2)∂∇x2.Here we will calculate the nabla Lie multiplication, then we will try to obtain[V1,V2]∇ with using different time scales:
(4.2)[V1,V2]∇(f):=V1[(V2f)]-V2[(V1f)]=(x1-x22)(0∂f∇x1+1∂f∇x2)-(x2+5)(1∂f∇x1+(ρ(x1)+x1)∂f∇x2)+x12(1∂f∇x1-4∂f∇x2)-(x1-4x2)(-(ρ(x2)+x2)∂f∇x1+0∂f∇x2)+((x2+5)ρ(x1-x22)-(x1-x22)ρ(x2+5))∂2f∇x12+((x2+5)ρx12-(x12)ρ(x2+5))∂2f∇x1∇x2+((x1-4x2)ρ(x12)-(x1-4x2)(x12)ρ)∂2f∇x22+((x2+5)ρ(x1-x22)-(x1-x2)ρ(x2+5))∂2f∇x1∇x2=({∂∇x1}x1+3-x22-5x2+(x2+5)(ρ(x1)+x1)(x1+3)-x12x2-4x12(x1+3)+(x1-4x2)(ρ(x2)+x2)x2+ρ(x2+5)x12-ρ(x12)(x2+5))∂∇x1+(-x1+x22-(x2+5)(ρ(x1)+x1)+4x12)∂∇x2.Let us firstly identify the𝕋=ℝ situation that obtained for nabla Lie bracket:
(4.3)[V1,V2]∇(f)=(-4x13+3x12x2-12x12+31x1+2x1x22+6x1x2-8x23-x22-5x2)∂∇x1+(4x12-11x1-2x1x2+x22)∂∇x2.From these equations, when the time scale is𝕋=ℝ, we have seen that we could obtain Lie brackets which are known from the geometry, see in Figure 1.Figure 1
The graph of the vector fields[V1,V2]∇ on 𝕋=ℝ.When the time scale is𝕋=ℤ as an example, we have obtained a different equation from to have known Lie bracket at real:
(4.4)[V1,V2]∇(f)=(36x1-9x2-3x12-x22+3x2x12+6x1x2-4x13+2x1x22-17)∂∇x1+(-11x1-2x1x2+4x12+x22+x2+5)∂∇x2.This is seen as a difference of natural consequence of the continuous and discrete structures, see in Figure2.Figure 2
The graph of the vector fields[V1,V2]∇ on 𝕋=ℤ.


## 5. Conclusion

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.



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