Noether’s theorems for the relative motion systems on time scales

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Abstract

This paper propose Noether symmetries and the conserved quantities of the relative motion systems on time scales. The Lagrange equations with delta derivatives on time scales are presented for the system. Based upon the invariance of Hamilton action on time scales, under the infinitesimal transformations with respect to the time and generalized coordinates, the Hamilton’s principle, the Noether theorems and conservation quantities are given for the systems on time scales. Lastly, an example is given to show the application the conclusion.

Abstract

This paper propose Noether symmetries and the conserved quantities of the relative motion systems on time scales. The Lagrange equations with delta derivatives on time scales are presented for the system. Based upon the invariance of Hamilton action on time scales, under the infinitesimal transformations with respect to the time and generalized coordinates, the Hamilton’s principle, the Noether theorems and conservation quantities are given for the systems on time scales. Lastly, an example is given to show the application the conclusion.

1 Introduction

A time scale is an arbitrary nonempty closed subset of the real numbers.The calculus of time scales were initiated by B.Aulbach and S.Hilger [1] in order to create a theory which can unify discrete and continuous analysis. A time scale is a model of time, and the new theory has found important applications in several fields which require simultaneous modeling of discrete and continuous data, in the calculus of variations, control theory, and optimal control [2,3,4,5]. The calculus of variations on time scales was initiated with the presentation of Euler-Lagrange equations on time scales was presented in 2004 [6]. But, Torres put forward the second Euler-Lagrange equations and researched the higher-order calculus of variations on time scales [7,8]. The calculus of variations and control theory are disciplines in which there appears to be many opportunities for application of time scales [9,10,11].

In 1918, Noether proposed famous Noether symmetry theorems which could be used to 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. Bartosiewicz and Torres showed that there existed a conserved quantity in Lagrangian system for each Noether symmetry [12] on time scales by the technique of time-re-parameterization. Using this technique, Cai and Fu studied the theories of Noether symmetries of the nonconservative and nonholonomic systems on time scales [13,14]. Noether theory of the Hamilton systems on time scales was given by Zhang [15,16]. It is worth mentioning that the dynamic systems on time scales with delta derivative have just started to originate.

With the development of modern science and technology, people pay attention to the dynamic of relative motion. Jet aircrafts, rockets, satellite, spacecraft and so on generally involve application of the relative motion systems. We also know that the movement of Mechanical systems is researched in either absolute coordinate system or moving coordinate system. The dynamic systems in the moving coordinate system is called the relative motion dynamic. In 1961, Lur’e introduced the equation of the relative motion systems for conservative systems [17]. In 1993, The Lagrange equation of relative motion dynamics for the general holonomic system was first studied by Liu [18]. In recent decades, a series of innovative research results about dynamics of relative motion have been obtained [19,20,22].

In this letter, we study the Noether symmetry of relative motion systems on time scales. The structure of this letter as follows: In Section 2, we review preparatory knowledge and properties of time scales. In Section 3, we establish the equations of the relative motion systems with delta derivatives. In Section 4, Noether theorems and conserved quantities for the relative motion systems are founded. Lastly, an example is used to illustrate the results.

2 Previous results of time scales

To begin with, we briefly present some main definitions and properties about times scales. More detailed theory of time scales can refer to [23,24,25,].

Definition 1

A time scale T is an arbitrary nonempty closed subset of the set R of real number. For tT, we define the forward jump operator σ : TT by

σ(t)=inf{sT:s>t},

and the backward jump operator ρ : TT by

ρ(t)=sup{sT:s<t}.

The graininess function μ : T → [0, ∞] is defined by μ(t) = σ(t) − t for each tT.

A point is called right-dense, right-scattered, left-dense or left-scattered if σ(t) = t, σ(t) > t, ρ(t) = t, ρ(t) < t, respectively. We say Tk = T − {M} if T has a left-scattered maximum M, otherwise Tk = T.

Definition 2

Assume f : TR is a function and tTk, we define fΔ(t) to be the real number with the property with given any ε, there is neighborhood U = (tδ, t + δ) ⋂ T of such that

|[f(σ(t))f(s)]fΔ(t)[σ(t)s]|ε|σ(t)s|,

is true for all sU, we call fΔ(t) the delta derivative of f at t.

Definition 3

A function f : TR is continuous at tTk and t is right-scattered, then f is differentiable at t with

fΔ(t)=fσ(t)f(t)μ(t).

Furthermore, if f is differentiable at tTk, then

f(σ(t))=f(t)+μ(t)fΔ(t).

Definition 4

Assume f : TR is a regulated function, existing a function F with FΔ(t) = f(t) is called a pre-antiderivative of f and in this case an integral of f from a to b(a, bT) is defined by

abf(t)Δt=F(b)F(a),

We define the indefinite integral of f by

f(t)Δt=F(t)+C.

Where C is an arbitrary constant.

We shall often note fΔ(t) by ΔΔt f(t) if f is a composition of other functions. Furthermore, if f and g are both differentiable, the next formulate hold

(f+g)Δ(t)=fΔ(t)+gΔ(t),(kf)Δ(t)=kfΔ(t),(fg)Δ(t)=fΔ(t)g(t)+fσ(t)gΔ(t)+fΔ(t)gσ(t),

where we abbreviate fσ = f(σ(t)) by fσ.

Remark 1

we consider the two cases T = R and T = Z.

  • If T = R, f : RR is delta differentiable at tR, then fΔ(t) = f′(t).

  • If T = Z, f : ZR is delta derivative at every tZ with fΔ(t) = f(t + 1) − f(t).

Lemma 1

(Dubois-Reymond) [6] Let gCrd, g : [a, b] → Rn, if

abgTηΔ(t)Δt=0,

for all ηCrd1 with η(a) = η(b) = 0, then,

g(t)c,t[a,b]kforsomecRn,

where Crd1 means the set of differentiable functions with rd-continuous derivative.

We can also obtain the following conclusions about the time scales [25].

Assume that α : TR is strictly increasing and T: = α(T) is a time scale, Let β : TR, then there exists t in the real interval [t, σ(t)] with

(αβ)Δ(t)=βΔ(α(t))αΔ(t),1αΔ=(α1)Δα.

If f : TR is an rd-continuous function and α is differentiable with rd-continuous derivative, then for a, bT,

abf(t)αΔ(t)Δt=α(a)α(b)fα1(t)Δt=α(a)α(b)f(t)Δt.

Let γ = α−1, then q(t) := Qε(γ(t), q(γ(t))).

3 Lagrange equations for the relative motion systems on time scales

3.1 Equation of Chetaev constraint the relative motion systems

We know that the motion of a complex system may include the motion of a carrier, as well as the motion of a carried system relative to the carrier.

Suppose that the velocity of the base point in a carrier v0 and its angular velocity is ω. We assume that the motion of N particles wouldn’t change the motion rule of the carrier which is predetermined. N generalized coordinates qs (s = 1, ⋯, n) determine the configuration of systems. If the movement of the systems are constrained by the double-sided ideal Chetaev nonholonomic constraints,

fβ(qs,qs˙,t)=0(β=1,,g;s=1,,n).

The equations of the relative motion systems are[18]

ddtTqs˙Tqs=Qsqs(V0+Vω)+Qsω+Γs+Λs,Λs=λβfβqs˙(s=1,,n),

where T is the kinetic energy of the relative motion function, λβ is Lagrange multiplier, Qs, V0, Vω, Qsω , Γs are respectively the generalized forces, the potential energy of uniform force field, the potential energy of inertial centrifugal force field, generalized rotary inertia force, generalized gyroscopic force.

The Qs can be divided into parts of potential and nonpotential

Qs=Qs+Qs,Qs=Vqs

We construct Lagrangian of the relative motion system

L=TVV0Vω.

Equations can be written as follows

ddtLqs˙Lqs=Qs+Qsω+Γs+Λs.

3.2 Lagrange equation for the relative motion systems with delta derivatives

Firstly, we introduce the following relationships [13]

ΔΔt(δqs)=δΔΔtqs=δqsΔ,

(δqs)σ=δqsσ.

The Hamilton principle for the relative motion system on time scales is written by

tatb[δL+(Qs+Qsω+Γs)δqsσ]Δt=0,

where (Qs+Qsω+Γs)δqsσ is the virtual work of generalized force.

We take total variation for Lagrange function L, then

δL=Lqsσδqsσ+LqsΔδqsΔ.

By using of Eq. (15) and Eq. (14), we can obtain

tatbLqsσδqsσ+LqsΔδqsΔ+(Qs+Qsω+Γs)δqsσΔt=tatbLqsσ+Qs+Qsω+Γsδqsσ+LqsΔδqsΔΔt=tatbLqsσ+Qs+Qsω+Γs(δqs)σ+LqsΔ(δqs)ΔΔt=tatbtatQs+Qsω+Γs+L(τ,qsσ(τ),qsΔ(τ))qsσ(τ)ΔτδqsΔΔttatbtatQs+Qsω+Γs+L(τ,qsσ(τ),qsΔ(τ))qsσ(τ)Δτ(δqs)ΔΔt+tatbLqsΔ(δqs)ΔΔt=tatbLqsΔtatQs+Qsω+Γs+L(τ,qsσ(τ),qsΔ(τ))qsσ(τ)Δτ(δqs)ΔΔt=0

Therefore, by Lemma 1

LqsΔtatQs+Qsω+Γs+L(τ,qsσ(τ),qsΔ(τ))qsσ(τ)Δτconst,t[ta,tb].

Hence

ΔΔtLqsΔLqsσ(Qs+Qsω+Γs)=0

Assuming the movement of the system is constrained by the double-sided ideal nonholonomic of Chetaev type with delta derivatives

fβ(t,qsσ,qsΔ)=0,(β=1,2,,g).

Suppose the restrictions that constraints impose on the virtual displacements are

fβqsΔδqsσ=0,(s=1,2,,n;β=1,2,,g).

Multiplying δqsσ on both sides of Eq. (16)

(Qs+Qsω+Γs)ΔΔtLqsΔ+Lqsσδqsσ=0.

Introducing the Lagrange multiplier λ and multiplying λ on both sides of Eq. (18),

λfβqsΔδqsσ=0.

Form Eq. (20) and Eq. (19), we get

(Qs+Qsω+Γs)ΔΔtLqsΔ+Lqsσ+λfβqsΔδqsσ=0.

Differential Eq. (21), we obtain the equation of the relative motion systems with Chetaev type constraints on time scales

ΔΔtLqsΔLqsσ=Qs+Qsω+Γs+Λs.(Λs=λfβqsΔ)

4 Noether theorem of the relative motion systems

4.1 Noether’s theorem without transforming time

The Hamilton action with the delta derivative on time scales can be expressed as

S(γ)=tatbL(t,qsσ,qsΔ)Δt,

where γ is a curve.

Introducing the following single parameter infinitesimal transformations without transforming time:

t=t,qs=qs+εξs(t,q)+o(ε),

if and only if

tatbL(t,qsσ,qsΔ)Δt=tatbL(t,qsσ,qsΔ)Δt.

Where ε is the infinitesimal parameter, ξs : [a, b] × RnR is delta differentiable functions.

The relationship between the isochronous δ and the total variation Δ on time scale is as follows

Δqsσ=δqsσ+qsΔΔt.

According to Eq. (23), we have

δqsσ=ΔqsσqsΔΔt=εξsσ.

Substituting Eq. (25) into Eq. (18) has

fβqsΔξsσ=0,(s=1,2,,n;β=1,2,,g).

Definition 5

The action S is said to be quasi invariant on U under the transformation groups (23), if and only if for any subinterval [ta, tb] ∈ [a, b], any ε, qU

tatbL(t,qsσ,qsΔ)Δt=tatbL(t,qsσ,qsΔ)Δt+tatbΔΔt(ΔG)+(Qs+Qsω+Γs+Λs)δqsσΔt,

where the transformations satisfy the condition Eq. (26). we say that the invariance is called Noether generalized quasi-symmetry of the relative motion systems on time scales.

Theorem 1

If the action S is quasi-invariant on the infinitesimal transformations Eq. (23) then

Lqsσξsσ+LqsΔξsΔ+(Qs+Qsω+Γs+Λs)ξsσ=ΔΔtG,

where ξsσ(t,q)=ξs(σ(t),q(σ(t))),ξsΔ(t,q)=ΔΔtξs(t,q).

Proof

Consider the infinitesimal transformations (t, qs ) given by group (23) and Definition 5, we can obtain

tatbL(t,qsσ,qsΔ)Δt=tatbL(t,qsσ,qsΔ)Δt+tatbΔΔt(ΔG)+(Qs+Qsω+Γs+Λs)δqsσΔt=tatbL(t,qsσ+εξsσ+o(ε),qsΔ+εξsΔ+o(ε))Δt+tatbΔΔt(ΔG)+(Qs+Qsω+Γs+Λs)δqsσΔt

Taking into account any subinterval [ta, tb] ∈ [a, b], we can get the following equivalent equation:

L(t,qsσ,qsΔ)=L(t,qsσ+εξsσ+o(ε),qsΔ+εξsΔ+o(ε))+(Qs+Qsω+Γs+Λs)Δqsσ+ΔΔt(ΔG).

Differentiation both sides of Eq. (28) with respect to ε,

Lttε+Lqsσqsσε+LqsΔqsΔε+(Qs+Qsω+Γs+Λs)δqsσε+ΔΔtG=0

Since

qsσε|ε=0=ξsσ,qsΔε|ε=0=ξsΔ.

Eq. (25) shows that

δqsσε=ξsσ.

Then, setting ε = 0, we can obtain the Eq. (27).

Theorem 2

For Chetaev constraint the relative motion systems on time scales, if the infinitesimal transformations Eq. (23) satisfy the conditions Eq. (26), then the system Eq. (22) has conserved quantities of the form

I=LqsΔξs+G.

Proof

It proves that Eq. (29) is equivalent to the proof of I = const, takeing the derivative of I with respect to t, then

ΔΔtI=ΔΔtLqsΔξs+G=ΔΔtLqsΔξsσ+LqsΔξsΔ+ΔΔtG.

Multiplying ξsσ on both sides of Eq.(22)

ΔΔtLqsΔξsσ=Lqsσξsσ+(Qs+Qsω+Γs+Λs)ξsσ.

Observe that Eq. (27)

ΔΔtI=LqsΔξsΔ+Qs+Qsω+Γs+Λs+Lqsσξsσ+ΔΔtG=0.

Namely

I=const.

4.2 Noether theorem with transforming time

Considering the following infinitesimal transformations with the time and the state variables:

t=t+εξ0(t,q)+o(ε),qs=qs+εξs(t,q)+o(ε).

Where ξ0, ξs : [a, b] × RnR are delta differentiable functions.

In this case, we assume the map t ∈ [a, b] ↦ α (t) = tR is a strictly increasing Crd1 function and its image is a new time scale t = α(t), whose forward jump operator and delta derivative are denote by σ and Δ. Following the arguments provided above,

σα=ασ.

According to groups (30), we have

δqsσ=ΔqsσqsΔΔt=ε(ξsσqsσΔξ0σ).

Substituting Eq. (31) into Eq. (18)

fβs(ξsσqsσΔξ0σ)=0

Definition 6

The action S is called a generalized quasi invariant in the transformation groups (30) and if and only if for any subinterval [ta, tb] ∈ [a, b]

tatbL(t,qsσ,qsΔ)Δt=α(ta)α(tb)Lt,qsσ(t),qsΔ(t)Δt+tatbΔΔt(ΔG)+(Qs+Qsω+Γs+Λs)δqsσΔt.

Theorem 3

Noether theory points out that ξ0, ξs satisfy the Noether identity if it exists the gauge G = G(t, qsσ,qsΔ ). If the action S is generalized quasi-invariant in the infinitesimal transformations Eq. (30), then

Ltξ0+Lqsσξsσ+LqsΔξsΔ+Lξ0ΔLqsΔξ0ΔqsΔ+(Qs+Qsω+Γs+Λs)×[ξsσξ0σ(qsΔ+μ(t)qsΔΔ)]=ΔΔtG.

Proof

Consider the Definition 6, we have

tatbL(t,qsσ,qsΔ)Δt=α(ta)α(tb)Lt,qsσ,qsΔΔt+tatbΔΔt(ΔG)+(Qs+Qsω+Γs+Λs)δqsσΔt=tatbLα(t),(qsσα)(t),qsΔ(α(t))αΔ(t)Δt+tatbΔΔt(ΔG)+(Qs+Qsω+Γs+Λs)δqsσΔt.

Since σ is the new forward jump operator and Δ is the new delta derivative, we can have by Eq. (10)

(qsα)Δ(t)=qsΔ(α(t))αΔ(t).

Then

qsΔ(α(t))=(qsα)Δ(t)αΔ(t).

Gathering Eq. (34) and Eq. (35), we can derive

tatbL(t,qsσ,qsΔ)Δt=tatbLα(t),(qsσα)(t),(qsα)Δ(t)αΔ(t)αΔ(t)Δt+tatbΔΔt(ΔG)+(Qs+Qsω+Γs+Λs)δqsσΔt.

Since [ta, tb] is arbitrary subinterval

L(t,qsσ,qsΔ)=Lα(t),(qsσα)(t),(qsα)Δ(t)αΔ(t)αΔ(t)+(Qs+Qsω+Γs+Λs)δqsσ+ΔΔt(ΔG).

According to group (30), differentiating both sides of Eq. (36) with respect to ε, setting ε = 0, we get

0=Ltα(t)ε+Lqsσ(qsασ)ε+LqsΔε(qsα)ΔαΔαΔ+LαΔε+(Qs+Qsω+Γs+Λs)δqsσε+ΔΔtG.

For the map t ∈ [a, b] ↦ α(t) ≡ tR, and qsασ(t) = qs (α(σ(t)))

tε|ε=0=ξ0,[(qsασ)]ε|ε=0=ξsσ.

For f(t) = t, then tΔ = 1 by Remark 1, we can derive

ε(qsα)ΔαΔ|ε=0=εqsΔ+εξsΔ+o(ε)tΔ+εξ0Δ+o(ε)=ξsΔ(tΔ+εξ0Δ)ξ0Δ(qsΔ+εξsΔ)(tΔ+εξ0Δ)2|ε=0=ξsΔqsΔξ0Δ.

and

tΔε|ε=0=ξ0Δ,

δqsσε|ε=0=ΔqsσqsσΔ(Δt)σ|ε=0=ξsσξ0σqsσΔ.

Considering Definition 3, we have

qsσΔ=(qsΔ)σ=qsΔ+μ(t)qsΔΔ.

Then,

ξsσξ0σqsΔσ=ξsσξ0σ(qsΔ+μ(t)qsΔΔ).

All of the above equation, we obtain the Noether identity Eq. (33) of the relative motion systems with Chetaev type constraints on time scales.

Theorem 4

For Chetaev constraint the relative motion systems on time scales, if the infinitesimal transformations Eq. (30) satisfy the conditions Eq. (32), then the system Eq. (22) has conserved quantities of the form

I=LqsΔ+LLqsΔqsΔLtμ(t)ξ0σ+G.

Proof

It proves that Eq. (39) is equivalent to the proof of I = const, takeing the derivative of I with respect to t, then

ΔΔtI=ΔΔtLqsΔ+LLqsΔqsΔLtμ(t)ξ0σ+G=ΔΔtLqsΔξsσ+LqsΔξsΔ+ΔΔtLLqsΔqsΔLtμ(t)ξ0σ+LLqsΔqsΔLtμ(t)ξ0σ+ΔΔtG

From Eq. (22) and (33), we can obtain

ΔΔtI=Lqsσ+(Qs+Qsω+Γs+Λs)ξsσ+LqsΔξsΔ+Lt(Qs+Qsω+Γs+Λs)qsσΔξ0σ+Lξ0ΔLqsΔξsΔqsΔLtμ(t)ξ0Δ+ΔΔtG=Lqsσ+(Qs+Qsω+Γs+Λs)ξsσ+LqsΔξsΔ+Lt(Qs+Qsω+Γs+Λs)qsσΔ(ξ0σ+μ(t)ξ0Δ)+Lξ0ΔLqsΔξsΔqsΔLtμ(t)ξ0Δ+ΔΔtG=Ltξ0+Lqsσξsσ+Lqsσξsσ+Lξ0ΔLqsΔξsΔqsΔ+(Qs+Qsω+Γs+Λs)(ξsσξ0σqsσΔ)+ΔΔtG=0

According to the above proof, we can learn that Eq. (39) is called the Noether’ s conserved quantities for the relative motion systems with Chetaev type constraints on time scales.

5 Examples

We first consider an example of the relative motion systems, the time scale is:

T=2n:nN0

Suppose lagrangian equation of the system is:

L=12m(q1Δ)2+(q2Δ)2+(q3Δ)2+12mω2(q1σ)2+(q2σ)212k(q1σ)2+(q2σ)2+(q3σ)2.

The generalized rotary inertia force and nonconservative force are respectively

Qs=0,Qsω=0(s=1,2,3).

The generalized gyroscopic force are

Γ1=2mωq2Δ,Γ2=2mωq1Δ,Γ3=0.

The Chetaev constraint is

f=q2Δtq1Δ=0

From Eq. (22), we can obtain

mq1ΔΔ=kq1σ+mω2q1σ+2mωq2Δλt,mq2ΔΔ=kq2σ+mω2q2σ+2mωq1Δ+λ,mq3ΔΔ=kq3σ.

Using Eq. (42) and Eq. (43), we have

λ=2mωq1Δ+mq1Δ+(kmω2)(q2σq1σt)1+t2.

The differential equation of the relative motion systems

mq1ΔΔ=(kmω2)q1σ+tq2σ1+t2mq1Δt1+t2,mq2ΔΔ=(kmω2)q1σt+t2q2σ1+t2+mq1Δ1+t2,mq3ΔΔ=kq3σ.

From the Theorem 3, we obtain

ξ1σ(mω2q1σkq1σ)+ξ2σ(mω2q2σkq2σ)ξ3σkq3σ+mq1Δ(ξ1Δξ0Δq1Δ)+mq2Δ(ξ2Δξ0Δq2Δ)+mq3Δ(ξ1Δξ0Δq3Δ)+12m(q1Δ)2+(q2Δ)2+(q3Δ)2+12mω2(q1σ)2+(q2σ)212k(q1σ)2+(q2σ)2+(q3σ)2ξ0Δ+2mωq2Δ2mωq1Δtmq1Δ+(kmω2)(q2σq1σt)1+t2t×(ξ1σξ0σq1σΔ)+2mωq1Δ2mωq1Δ+mq1Δ+(kmω2)(q2σq1σt)1+t2×(ξ2σξ0σq2σΔ)=ΔΔtG.

Choosing the infinitesimal generators as:

ξ0=ξ1=ξ2=0,ξ3=q3Δ.

Using Eqs. (47-48), we have

G=k2q3σ12m(q3Δ)2.

According to Theorem 4, the system has the Noether conserved quantity on time scale as

I=12m(q3Δ)2+12k(q3σ)2=const.

6 Conclusion

In this paper, based on the theory of calculus on time scale and variational principle, we studied Noether symmetries and the conservation laws of relative motion systems on time scales. The results have shown significant approaches to seek conservation laws for these systems and provide a good method for solving the practical problems such asbiology, thermodynamics, engineering and so on.

Communicated by Juan L.G. Guirao

Acknowledgement

This work is supported by the National Natural Science Foundation of China (Grant No. 11472247)

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