Noether’s theorems of variable mass systems on time scales

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Abstract

This paper deals with the Noether’s theory for variable mass system on time scales. The calculus on time scales unifies and extends variable mass system continuous model and discrete model into a single theory. Firstly, Hamilton’s principle of the variable mass system on time scales is given. Secondly, based on the quasi-invariance of the Hamilton’s action under a group of infinitesimal transformations, Noether’s theorem and its inverse theorem of the variable mass system on time scales are presented. Finally, two examples are given to illustrate the applications of the results.

Abstract

This paper deals with the Noether’s theory for variable mass system on time scales. The calculus on time scales unifies and extends variable mass system continuous model and discrete model into a single theory. Firstly, Hamilton’s principle of the variable mass system on time scales is given. Secondly, based on the quasi-invariance of the Hamilton’s action under a group of infinitesimal transformations, Noether’s theorem and its inverse theorem of the variable mass system on time scales are presented. Finally, two examples are given to illustrate the applications of the results.

1 Introduction

The theory of time scales was born in 1988 with the work of Stefen Hilger in order to unify and generalize continuous and discrete analysis [1, 2]. The calculus of variations on time scales has been developing rapidly in the past thirteen years, after the pioneering work of Bohner in 2004 [3]. Cai and Fu established the Noether symmetries of the non-conservative and non-holonomic systems on time scales, and obtained the symmetry theorem for constrained mechanical systems on time scales [4, 5]. More recently, Noether theory for Bikhoffian systems on time scales was established by Song and Zhang [6]. Zhai and Zhang obtain the Noether theorem for non-conservative systems with time delay on time scales [7].The time scales has a tremendous potential for applications and has recently received much attention in other areas such as engineering, biology, economics, and physics [9, 10, 11, 12].

In 1918, Noether proposed famous Noether symmetry theorem which deal with the invariance of the Hamilton action under the infinitesimal transformations: when a system exhibits a symmetry, then a conservation law can be obtained [16]. The symmetries and conservation laws can also be studied by using differential variational principles [17]. The calculus of variations and control theory are disciplines in which there appears to be many opportunities for application of time scales [13, 14]. The Noether method is making good progress, such as Herglotz variational problems [15]. And in recent years, a series of important results have been obtained on the study of the Noether symmetry and conservation law of classical mechanical systems, such as Torres made use of the Euler-Lagrange equations on time scales to generalize one of the most beautiful results of the calculus of variations-the celebrated Noether’s theorem [18, 19].

The problem of variable mass has attracted people’s attention as early as the middle of the nineteenth century. With the development of space technology and other industrial technologies, the study of variable mass system dynamics becomes more and more important. There are many studies on variable mass systematics have been done by Mei [20, 21]. A series of new theories and methods have been put forward, and a series of innovative research results have been obtained [22, 23, 24, 25, 26].

In this article, we will study the Noether theorems and its inverse problem of variable mass on time scales. In Section 2, we review some basic definitions and properties about the calculus on time scales. In Section 3, we obtain the Lagrange equations of systems by deriving Hamilton’s principle for variation mass systems with delta derivative. In Section 4, based on the quasi-invariance of Hamiltonian action of the variation mass systems under the infinitesimal transformations with respect to the time scales and generalized coordinates, the Noether’s theorem and the conservation laws for variation mass systems on time scales are obtained. In Section 5, the Noether’s inverse theorem of variable mass systems on time scales is given. In the end, two examples are given to illustrate the applications of the results.

2 Basics on the time scales calculus

In this section we give basic definitions and facts concerning the calculus on time scales. More can be found elsewhere [27].

A time scales is a nonempty closed subset of real numbers, and we usually denote it by symbol T. The two most popular examples are (T = ) and (T = ). We define the forward and backward jump operators σ,ρ.

Definition 2.1

Let T be a time scale. For t ∈ T we define the forward and backward jump operators σ,ρ : T → T by

σt:=inf{sT:s>t}andρt:=sup{sT:s<t}for alltT,

(supplemented by inf ϕ = sup T and sup ϕ = inf T) and the graininess function μ : T → [0, ∞) is defined by μ (t) = σ (t) – t for each t ∈ T.

If T =, then σ (t) = t = ρ (t) and μ (t) = 0 for any t ∈ T. If T =, then σ (t) = t + 1, ρ (t) = t – 1 and μ (t) ≡ 1 for every t ∈. A point t ∈ T is called right scattered, right dense, left scattered and left dense if σ (t) > t, σ (t) = t, ρ (t) < t, ρ (t) = t, respectively. We can consider that t is isolated if ρ(t) < t < σ(t), then t is dense if ρ (t) = t = σ (t). If sup T is finite and left-scattered, we set Tκ = T\{sup T}. Otherwise,.

Definition 2.2

Assume f : T → is a function and let t ∈ Tκ. Then we define fΔ(t) to be the number (provided it exists) with the property that given any ε > 0, there is a neighborhood U of (i.e., U = (tδ, t + δ) ∩ T for some δ > 0 ) such that

f(σ(t))f(s)fΔ(t)(σ(t)s)εσ(t)sfor allsU

we call fΔ(t) the delta (or Hilger) derivative of f at t.

For differentiable f, the formula

fσt=f+μfΔandfσt=ft+μtfΔt.

Definition 2.3

A function f : T → is called re-continuous if it is continuous at the right- dense points in T and its left-sided limits exist at all left-dense points in T. A function f : T →n is re-continuous if all its components are re-continuous.

The set of all re-continuous is denoted by Crd. Similarly, Crd1 will denote the set of functions from Crd whose delta derivative belongs to Crd.

Theorem 2.1

Let f be regulated. Then there exists a function F : T → is called an pre-antiderivative of f : T → if it satisfies FΔ (t) = f(t), for all t ∈ Tκ.

Definition 2.4

Assume f : T → is a regulated function. Any function F as in Theorem 2.1. is called a pre-antiderivative of f. We define the indefinite integral of a regulated function f by

ftΔt=Ft+C

where C is an arbitrary constant and F is a pre-antiderivative of f. We define the Cauchy integral by

abftΔt=FbFaforalla,bT.

We shall need the following properties of delta derivatives and integrals:

fgΔ=fΔgσ+fgΔ,

f+gΔt=fΔt+gΔt,

abfαtαΔtΔt=αaαbftΔt,

where α : [a, b] ∩ T → is an increasing Crd1 function and image is a new time scale.

Lemma 2.1

(Dubois-Reymond) Let gCrd, g : [a, b] →n, then

abgTtηΔtΔt=0

for all ηCrd1 with η (a) = η (b) = 0, holds if and only if g(t) ≡ c on [a, b]κ for some c ∈.

3 Hamilton’s principle and Lagrange equations for variable mass systems with delta derivatives

Consider a mechanical system consisting of N variable mass particles. Suppose at time t, the mass of the particle i is supposed to be mi(i = 1, 2, ⋯, N). At the moment t + Δt the mass of a small particle separated from the particle i or combined with the particle i is supposed to be Δmi. The configuration of the system is determined by n generalized coordinates qs(s = 1, 2, ⋯, n) and the mass of the particle depends on time, generalized coordinates and generalized velocity

mi=mit,qsσ,qsΔ.

Assuming that the kinetic energy function of the variable mass system on time scales is T=T(t,qsσ,qsΔ), Hamilton’s principle states that the actual pace exists when the Hamiltonian action has determining value. Thus the Hamilton’s principle for variable mass systems with delta derivatives can be written in the following form:

ab(δT+Qsδqsσ+Psδqsσ)Δt=0

where Qs δqsσ is the virtual work of generalized force, Ps δqsσ is the virtual work of generalized counter thrust, δqsσ=εξστσqsΔσ.

Ps=Ri+miΔriΔriqs12riΔriΔmiqs+ΔΔt12riΔriΔmiqsΔ

where ri and riΔ are respectively the position vector and the velocity vector of the i-th particle and the velocity vector of the i-th particle and Ri=ΔmiΔtui, where ui is the corpuscle’s velocity relative to the i-th particle.

The exchanging relationships with respect to the derivatives on time scales and isochronous variation on time scales [5]:

δqsΔ=δqsΔ,

and following eq. (1) we can find qsΔσ=qsΔ+μtqsΔΔ.

Taking total variation for function T, we have

δT=Tqsσδqsσ+TqsΔδqsΔ.

substituting eq. (11) into eq. (8), we have

abTqsσδqsσ+TqsΔδqsΔ+Qsδqsσ+PsδqsσΔt=abQs+Ps+Tqsσδqsσ+TqsΔδqsΔΔt=abQs+PsTqsσ(qs)σ+TqsΔ(δqs)ΔΔTatQs+Ps+Tτ,qστ,qΔτqsσΔτδqsΔ+TqsΔδqsΔΔt=abTqsΔatQs+Ps+Tτ,qστ,qΔτqsσΔτδqsΔΔt=0.

Therefore, by Lemma 2.1, we can derive

TqsΔabQs+Ps+Tτ,qστ,qΔτqsσΔτconst,

hence

ΔΔtTqsΔTqsσQsPs=0.

When contains conservative force and nonconservative force Qs,andQs satisfies the following conditions:

If is potential, that is, there exists a function such that

Qs=Vqsσ.

substituting eq.(14) into eq.(13), we have

ΔΔtTqsΔTqsσ+VqσsQsPs=0.

as the function V=Vqsσ,t only depends on the generalized coordinates, therefore

VqsΔ=0.

Then eq.(13) can be written in the form

ΔΔtLqsΔLqsσ=Qs+Ps.

where L = TV.

If Qs has generalized potential, that is, there exists a function U=Ut,qsσ,qsΔ such that

Qs=UqsσΔΔtUqsΔ

then eq.(13) can be written as:

ΔΔtLqsΔLqsσ=Qs+Ps.

where L = T + U = TV is the Lagrangian of the variable mass systems with derivatives on time scales.

4 Noether’s theorem of variable mass systems on time scales

In order to simplify expressions, we write Lt,qsσ,qsΔ instead of Lt,qsσ(t),qsΔ(t), similarly for the partial derivatives of L.

We consider the fundamental problem of the calculus of variations on time scales as defined by Bohner [3, 20]

Sqs=abLt,qsσ(t),qsΔ(t)Δtmin

qsσ (t) = (qsσ)(t), qsΔ (t) is the delta derivative of qs, t ∈ T, and the Lagrangian L : ×n ×n → is a C1 function with respect to its arguments. By iL we will denote the partial derivative of L with respects to the ith variable, i = 1, 2, 3. Admissible functions qs(⋅) are assumed to be Crd1.

The relationship between the isochronous variation and the total variation on time scale T:

Δqs=δqs+qsΔΔt.

Let us consider now the following infinitesimal transformations with respect to the time and the state variable:

t=Ht,qs,ε=t+ετt,qs+oεqs=Ft,qs,ε=qs+εξt,qs+oε

Let as before U be a set of Crd1 functions qs : [a, b] →n and we assume that the map tα (t) : T(t, qs, ε) ∈ is an increasing Crd1 function for every qsU, every ε, and any t ∈ [a, b], and its image is a new time scale with the forward jump operator σ and the delta derivative Δ. We need to employ the following property:

σα=ασ.

Definition 4.1

(Invariance for variable mass systems) Function I is said to be quasi-invariant on U under the infinitesimal transformations (21) if and only if for any subinterval [ta,tb] ∈ [a, b], any ε, any qsU:

tatbLt,qsσ,qsΔΔt=Tta,qstaTtb,qstbLt,qsσt,qsΔtΔt+tatbΔΔtΔG+Qs+PsδqsσΔt

Theorem.4.1

If infinitesimal transformations (21) satisfy

Ltτ+Lqsσξsσ+LqsΔξsΔ+LτΔLqsΔτΔqsΔ+Qs+PsξστσqsΔ+μtqsΔΔ=ΔΔtG

then transformations (21) is the Noether generalized quasi-symmetric transformation of variable mass system on time scales.

Proof

Substituting formula δqsσ=εξστσqsΔσ=εξστσqsΔ+μtqsΔΔ into eq. (19), we obtain

εΔΔtLqsΔ+Lqsσ+Qs+PsξsστσqsΔσ=0

adding and subtracting a function Ltτ from eq. (24), we obtain

εLqsσξsσ+LqsΔξsΔ+Qs+PsξsστσqsΔσLt+LqsσqsΔ+LqsΔqsΔΔτ+LtτLqsΔqsΔτΔΔΔtLqsΔξsΔτσqsΔσ=0

adding and subtracting a gauge function ΔΔtG(t,qsσ,qsΔ) from eq. (25), we obtain

εLqsσξsσ+LqsΔξsΔ+Qs+PsξsστσqsΔσ+LLqsΔqsΔτΔ+Ltτ+ΔΔtGt,qsσ,qsΔΔΔtLqsΔξsσ+LLqsΔqsΔτ+Gt,qsσ,qsΔ=0

eq. (26) is the condition of infinitesimal transformations of the variable mass system on time scales.

If generators τ, ξs of infinitesimal transformations and gauge function G(t,qsσ,qsΔ) satisfy

Ltτ+Lqsσξsσ+LqsΔξsΔ+LLqsΔqsΔτΔ+Qs+PsξστσqsΔ+μtqsΔΔ=ΔΔtG

eq.(27) is called Noether’s identity of the variable mass system on time scales.

Theorem 4.2

If functional I is quasi-invariant on U under the infinitesimal transformations (21), then

I=LqsΔξs+LLqsΔqsΔLtμtτ+G

is a conservational law for variable mass dynamical systems on time scales.

Proof

Let (t;s,qs;r,v) := L(sμ(t)r,qs,vr)r for qs, v ∈ ℝn, t ∈ [a, b] and s, r, ∈ ℝ, r ≠ 0.

It is readily apparent that for s(t) = t and any qs : [a, b] → ℝn

L(t,qsσ(t),ssΔ(t))=L~(t;sσ(t),qsσ(t);sΔ(t),qΔ(t)),

so for the functional:

S[qs()]=S~[s(),qs()]

and

S~[s(),qs()]:=abL~(t;sσ(t),qsσ(t);sΔ,qΔ(t))Δt

Consider the infinitesimal transformation eq.(21) given by (Hϵ, Fϵ) and let qsU. For s(t) = t, making use of eq. (23) we can obtain

S~[s(),qs()]=αtaαtbLt,qsσt,qsΔtΔt+tatbΔΔtΔG+Qs+PsδqsσΔt=tatbL(α(t),(qsσα)(t),qsΔ(α(t)))αΔ(t)Δt+tatbΔΔt(ΔG)+(Qs+Ps)δqsσΔttatbLασ(t)μ(t)αΔ(t),(qsσα)(t),(qsα)Δ(t)αΔ(t)αΔ(t)Δt+tatbΔΔt(ΔG)+(Qs+Ps)δqsσΔt=tatbL~(t;ασ(t),(qsα)σ(t);αΔ(t),(qsα)Δ(t))Δt+tatbΔΔt(ΔG)+(Qs+Ps)δqsσΔt=S~[α(),(qsα)()]+tatbΔΔt(ΔG)+(Qs+Ps)δqsσΔt

so for s(t) = t we can obtain

(α(),(qsα)(t))=(Hϵ(t,qs(t)),Fϵ(t,qs(t)))=(Hϵ(s(t),qs(t)),Fϵ(s(t),qs(t)))

we observe that is an invariant on = {(s, qs)|s(t) = t, qsU} under the infinitesimal transformations:

(s,qs)=(Hϵ(s,qs),Fϵ(s,qs))

then

ΔΔtLqsΔξs+LLqsΔqsΔLtμtτ+G=ΔΔtLqsΔξsσ+LqsΔξsΔ+ΔΔtLLqsΔqsΔLtμtτσ+LLqsΔqsΔLtμtτΔ+ΔΔtG=ΔΔtLqsΔξsσ+LqsΔξsΔ+Lt+LqsσqsΔσ+LqsΔqsΔΔΔΔtLqsΔqsΔLqsΔqsΔΔΔΔtLtμtτσ+LLqsΔqsΔLtμtτΔ+ΔΔtG=Lqsσ+Qs+Psξsσ+LqsΔξsΔ+Lt+LqsσqsΔσΔΔtLqsΔqsΔτσ+LLqsΔqsΔLtμtτΔ+ΔΔtG=Lqsσ+Qs+Psξsσ+LqsΔξsΔ+Lt+LqsσqsΔσLqsσ+Qs+PqsΔτσ+LLqsΔqsΔLtμtτΔ+ΔΔtG=Ltτ+Lqsσξsσ+LqsΔξsΔ+LτΔLqsΔτΔqsΔ+Qs+PξsστσqsΔσ+ΔΔtG=0.

The calculus on time scales unifies and extends variable mass system continuous model and discrete model into a single theory.

Remark 1

If T = ℝ, then σ(t) = t,μ(t) = 0, therefore eq(27) give classical variable mass system Noether equation:

Ltτ+Lqsξs+Lqsξs+LLqsqsτ+Qs+Psξsτqs=ΔΔtG

and the conservational law become the classical variable mass system Noether conservational law

I=Lqsξs+LLqsqsτt,qs+G

Remark 2

If T = ℤ, h > 0, then σ(t) = t + h,μ(t) = h, therefore eq(27) give

Ltτ+Lqsξst+h+Lqst+hqthξst+hξsth

+LLqst+hqthqst+hqthτt+hτth

+Qs+Psξsτqst+h=ΔΔtG

and the conservational law give

I=Lqsξs+LLqsqshLtτt,qs+G

eq.(34)and eq.(35) are the discrete variable mass system Noether identity and Noether conservational law.

5 Noether’s inverse theorem of variable mass system on time scales

Suppose that a first integral of the variable mass system on time scales has been given as

I=I(t,qsσ,qsΔ)=const.

then we have

ΔIΔt=It+IqsσqsΔ+IqsΔqsΔΔ=0

multiply ξsσ~=ξστσqsΔσ both sides of eq.(19), we obtain

ξsσ~ΔΔtLqsΔLqsσQsPs=0

according to eq. (9), Ps is generally a linear function of qsΔΔ and can be written as

Ps=Wsk(t,qsσ,qsΔ)qskΔΔ+Ws(t,qsσ,qsΔ),

where Wsk=miqkΔ(ui+riΔ)riqs+12riΔriΔ2miqsΔqkΔ+miqsΔriΔriqk.

Adding eq.(38) to eq.(39), and putting the coefficients of qsΔΔ equal to zero, we obtain

2LqsΔqkΔWskξsσ~IqkΔ=0

suppose

dethsk=det2LqsΔqkΔWsk0

then from eq.(41) we obtain

ξsσ~=hsk~IqkΔ

in which

hsk~hkr=δsr,hsk=2LqsΔqskΔWsk

Now let the integral (37) equal conserved quantity (28), i.e.

I=LqsΔξs+LLqsΔqsΔLtμtτt,qs+G

Thus, from eq.(42) and (43), generators τ, ξ of infinitesimal transformation can be found.

Theorem 5.1

If the integral of the variable mass holonomic system has been given, then the infinitesimal transformations determined by eq.(21), (43) and (44) are the system’s transformation satisfying Noetheris identity (27).

Theorem 5.1 is called the generalized Noether’s inverse theorem of the variable mass holonomic system.

6 Examples

Example 1

The time scale and the Lagrangian of the variable mass system are given as:

L=12mq1Δ2+q2Δ212mq1σ2+q2σ2.

The generalized force is Q1 = Q2 = 0, and the generalized counter thrust is P1 = 0, Ps = mΔqsΔ following the eq.(19), we find

ΔΔtmq1Δ+mq1σ=0,ΔΔtmq2Δ+mq2σmΔq2Δ=0,

then the Noether identity (27) becomes

12mΔq1Δ2+q2Δ212mΔq1σ2+q2σ2τmq1σξ1σ+q2σξ2σ+mq1Δξ1Δ+q2Δξ2Δ

we can find solution of eq.(47) as follows

τ=0,ξ1=1,ξ2=1,

substituting the generator (48) into the structure eq. (47) yields

G=mq1Δmq2Δ

According to Theorem 4.2, substituting the generator (47) and the gauge function (48) into the formula (28), we get the following conserved quantity

I=mq1Δmq2Δ.

Example 2

The time scale and the Lagrangian of the variable mass system are given as:

L=12m(t)(q1Δ)2+(q2Δ)2.

The generalized force is Q1=0,Qs=q2Δ+q1q1Δ, the generalized counter thrust is P1 = P2 = 0.

Firstly, following the eq. (19), we find

ΔΔt[m(t)q1Δ]=0,ΔΔt[m(t)q2Δ]=q2Δ+q1q1Δ

then the Noether identity (27) becomes

12mΔ(q1Δ)2+(q2Δ)2τ+m(q1Δξ1Δ+q2Δξ2Δ)12m(q1Δ)2+(q2Δ)2τΔ+(q2Δ+q1q1Δ)(ξ1στq1Δσ)=ΔΔtG.

We can find solution of eq. (53) as follows

τ=0,ξ1=0,ξ2=1,

substituting the generator (54) into the structure eq(53) yields

G=q212q12

According to Theorem 4.2, substituting the generator (53) and the gauge function (54) into the formula (28), we get the following conserved quantity

I=mq2Δq212q12

Secondly, let us find the corresponding infinitesimal transformations from a known integral. Suppose there is an integral in the form

I=m2Δq212q12

According to eq.(42) and eq.(44) we can find

ξ1σ~=0ξ2σ~=1

Lτ+mq1Δδq1σ+mq2Δδq2σ+G=mq2Δq212q12

following the ξsσ~=ξστσqsΔσ we obtain

τ=1LG+q2+12q12,ξ1=q1Δ,ξ2=1+q2Δτ

we have G=q212q12, then τ = 0,ξ1 = 0,ξ2 = 1.

7 Summary

In this manuscript, the Noether’s theorems of variable mass systems on time scales have been studied. We established the Hamilton principle and derived the Lagrange equations for the variable mass system on time scales. Under the kind of infinitesimal transformations, we gave the definitions and criteria of Noether symmetries. And the Noether theorems and its inverse theorem of variable mass system on time scales are established. This paper considered the continuous case and the discrete case, so the results of this paper are of universal meaning. Besides, further study could include Lie symmetry. The approach of this paper can be furthermore generalized to other systems such as relative motion system; Birkhoffian systems and electromectro mechanial coupling system are equally worth studying on time scales.

Communicated by Juan L.G. Guirao

Acknowledgement

Project supported by the National Natural Science Foundation of China (Grant No. 11472247), the Program for Changjiang Scholars and Innovative Research Team in University, China (Grant No. IRT13097), and the Key Science and Technology Innovation Team Project of Zhejiang Province, China (Grant No. 2013TD18).

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