Mei Symmetry for Constrained Mechanical System on Time Scales

1.School of Mathematics Science，Suzhou University of Science and Technology，Suzhou 215009，P.R.China；2.Department of Mechanical Engineering and Energy Processes，Southern Illinois University in Carbondale，IL 62901，USA

Abstract: Mei symmetry on time scales is investigated for Lagrangian system，Hamiltonian system，and Birkhoffian system. The main results are divided into three sections. In each section，the definition and the criterion of Mei symmetry are first presented. Then the conserved quantity deduced from Mei symmetry is obtained，and perturbation to Mei symmetry and adiabatic invariant are studied. Finally，an example is given to illustrate the methods and results in each section. The conserve quantity achieved here is a special case of adiabatic invariant. And the results obtained in this paper are more general because of the definition and property of time scale.

Key words：Mei symmetry；time scale；Lagrangian system；Hamiltonian system；Birkhoffian system

0 Introduction

Mei symmetry was first introduced by Mei［1］in 2000. Mei symmetry is a kind of invariance that the dynamical functions of system，under infinitesimal transformations of time and coordinates，still satisfy the original differential equations of motion. Con?served quantity，which helps find the solution to the differential equation，can be deduced from Mei sym?metry. Therefore，Mei symmetry and conserved quantity are important aspects deserved to be stud?ied in analytical mechanics. And lots of research on Mei symmetry can be found in Refs.［2-5］.

Time scale was first introduced by Stefan Hilg?er in 1988［6］. Time scale means an arbitrary nonemp?ty closed subset of the real numbers. Generally，re?search can be done on time scales first，then differ?ent results will be obtained from specific time scale.The real numbers R，the integers Z，the natural numbers N，the nonnegative integers N0，the Can?tor set，etc.are all specific time scales.

Constrained mechanical system on time scales has been studied recently. For example，calculus of variations on time scales［7-8］，Noether symmetry and conserved quantity on time scales［9-13］，Lie symme?try and conserved quantity on time scales［14-16］，and so on. In this paper，Mei symmetry and conserved quantity on time scales will be presented.The defini?tions and basic properties of time scale calculus used here can be read in Ref.［17］for details.

1 Mei Symmetry for Lagrangian System on Time Scales

1.1 Mei symmetry and conserved quantity

Lagrange equation on time scales has the form［7］wherei，j=1，2，…，nis the Lagrangian on time scales，qjthe coordinate，

Taking account of the LagrangianLafter the following infinitesimal transformations

we obtain

whereθLis an infinitesimal parameter andis the infinitesimal generator.

Definition 1If the form of Eq.（1）keeps in?variant when the original LagrangianLis replaced byL*，that is

holds，then this invariance is called the Mei symme?try of Lagrangian system on time scales.

Substituting Eqs.（1，3）into Eq.（4），and omit?ting the higher order ofθL，we obtain

Criterion 1 If the infinitesimal generatorξ0Ljsatisfies Eq.（5），the corresponding invariance is the Mei symmetry of the Lagrangian system on time scales.

Eq.（5）is called the criterion equation of the Mei symmetry for the Lagrangian system（Eq.（1））on time scales.

Generally speaking，additional conditions are necessary when conserved quantity is wanted to be deduced from the Mei symmetry.

Theorem 1For the Lagrangian system（Eq.（1）），if the infinitesimal generatorξ0Lj，which meets the requirement of the Mei symmetry（Eq.（5）），and a gauge functionsatisfies

then the Mei symmetry can deduce the following conserved quantity

ProofUsing Eqs.（1，6），we have

This proof is completed.

1.2 Perturbation to Mei symmetry and adiabat?ic invariant

When the Lagrangian system（Eq.（1））is dis?turbed，the conserved quantity may also change.

Assuming the Lagrangian system on time scales is disturbed as

If the disturbed infinitesimal generatorξLiand the disturbed gauge functionGLare

then the infinitesimal transformations can be ex?pressed as

From the Mei symmetry of the disturbed La?grangian system（Eq.（8）），that is

we obtain

Eq.（12）is called the criterion equation of the Mei symmetry for the disturbed Lagrangian system（Eq.（8））on time scales.

Definition 2If a quantityIz，withεone of its elements，satisfies that the highest power ofεiszandis in direct proportion toεz+1，thenIzis called thezth order adiabatic invariant on time scales.And we have the following theorem.

Theorem 2For the disturbed Lagrangian sys?tem（Eq.（8）），if the infinitesimal generatorξmLj，which meets the requirement of the Mei symmetry（Eq.（12）），and the gauge functionsatisfies

ProofUsing Eqs.（8，13），we have

This proof is completed.

Remark 1Whenz=0，the adiabatic invari?ant obtained from Theorem 2 has a special name，i.e.，exact invariant. Besides，Theorem 2 reduces to Theorem 1 whenz=0. Therefore，a conserved quantity is actually an exact invariant.

1.3 An example

The Lagrangian is

We try to find out its conserved quantity and adiabat?ic invariant deduced from the Mei symmetry on the time scaleT=hZ= {hk：k∈Z}，h＞0.

From Eq.（5）and Eq.（6），we have

It is easy to verify that

satisfy Eqs.（16—18）. Then from Theorem 1，a conserved quantity can be obtained，namely

When the system is disturbed byQL1=0，QL2=t3-2t，from Eq.（12）and Eq.（13），we have

Taking calculation，we obtain

Then

can be achieved as the first order adiabatic invariant from Theorem 2. Higher order adiabatic invariants can certainly be deduced.

2 Mei Symmetry for Hamiltonian System

2.1 Mei symmetry and conserved quantity

Hamilton equation on time scales has the form［12-13］

Taking account of the HamiltonianHafter the following infinitesimal transformations

we obtain

whereθHis an infinitesimal parameter，andare called the in?finitesimal generators.

Definition 3If the form of Eq.（26）keeps in?variant when the original HamiltonianHis replaced byH*，that is holds，this invariance is called the Mei symmetry of Hamiltonian system on time scales.

Substituting Eqs.（26，28）into Eq.（29），and omitting the higher order ofθH，we obtain

Criterion 2If the infinitesimal generatorssatisfy Eq.（30），the corresponding invari?ance is the Mei symmetry of the Hamiltonian sys?tem on time scales.

Eq.（30） is called the criterion equation of the Mei symmetry for the Hamiltonian system（Eq.（26））on time scales.Therefore，we have

Only the thirtieth little duck couldn t come to the land; it swam about close to the shore, and, giving out a piercing cry, it stretched its neck up timidly, gazed wildly around, and then dived under again

Theorem 3For the Hamiltonian system（Eq.（26）），if the infinitesimal generatorswhich meet the requirement of the Mei symmetry（Eq.（30）），and a gauge functionsatisfies

the Mei symmetry can deduce the following con?served quantity

ProofUsing Eqs.（26，31），we have

This proof is completed.

2.2 Perturbation to Mei symmetry and adiabat?ic invariant

When the Hamiltonian system（Eq.（26））is disturbed，the conserved quantity may also change.

Assuming the Hamiltonian system on time scales is disturbed as

If the disturbed infinitesimal generatorsξHi，ηHiand the disturbed gauge functionGHare

the infinitesimal transformations can be expressed as

we obtain

Eq.（37）is called the criterion equation of the Mei symmetry for the disturbed Hamiltonian system（Eq.（33））on time scales.Then we have

Theorem 4For the disturbed Hamiltonian system（Eq.（33）），if the infinitesimal generatorsmeet the requirement of the Mei symmetry（Eq.（37）），and the gauge functionsatisfies

ProofUsing Eqs.（33，38），we have

This proof is completed.

Remark 2Theorem 4 reduces to Theo?rem 3 whenz=0.

2.3 An example

The Hamiltonian is

We try to find out its conserved quantity and adiabat?ic invariant deduced from the Mei symmetry on the time scale

From Eqs.（30，31），we have

It is easy to verify that

satisfy Eqs.（41—43）. Then from Theorem 3，a conserved quantity can be obtained

When the system is disturbed byQH1=3t，QH2=0，from Eqs.（37，38），we have

Taking calculation，we obtain

Then

can be obtained as the first order adiabatic invariant from Theorem 4. Higher order adiabatic invariants can certainly be deduced.

3 Mei Symmetry for Birkhoffian System

3.1 Mei symmetry and conserved quantity

The Birkhoff equation on time scales has the form［11］

Taking account of the BirkhoffianBand the Birkhoff’s functionRνafter the following infinitesi?mal transformations

we have

whereθBis an infinitesimal parameter，called the infinitesimal generator.

Definition 4If the form of Eq.（51）keeps in?variant when the original BirkhoffianBand the Birk?hoff’s functionRνare replaced byB*and，that is，

holds，this invariance is called the Mei symmetry of Birkhoffian system on time scales.

Substituting Eqs.（51，53）into Eq.（54），and omitting the higher order ofθB，we obtain

Criterion 3If the infinitesimal generatorsatisfies Eq.（55），the corresponding invariance is the Mei symmetry of the Birkhoffian system on time scales.

Eq.（55）is called the criterion equation of Mei symmetry for the Birkhoffian system（Eq.（51））on time scales.Therefore，we have

Theorem 5For the Birkhoffian system（Eq. （51）），if the infinitesimal generator，which meets the requirement of the Mei symmetry（Eq.（55）），and a gauge functionsatisfies

the Mei symmetry can deduce the following con?served quantity

ProofUsing Eqs.（51，56），we have

This proof is completed.

3.2 Perturbation to Mei symmetry and adiabat?ic invariant

When the Birkhoffian system（Eq.（51））is dis?turbed，the conserved quantity may also change.

Assuming the Birkhoffian system on time scales is disturbed as

If the disturbed infinitesimal generatorξBμand the disturbed gauge functionGBare

the infinitesimal transformations can be expressed as

From the Mei symmetry of the disturbed Birk?hoffian system（Eq.（58）），that is

we obtain

Eq.（62）is called the criterion equation of the Mei symmetry for the disturbed Birkhoffian system（Eq.（58））on time scales. Then we have the fol?lowing theorem.

Theorem 6For the disturbed Birkhoffian sys?tem（Eq.（58）），if the infinitesimal generator，which meets the requirement of the Mei symmetry（Eq.（62）），and the gauge functionsatisfies

ProofUsing Eqs.（58，63），we have

This proof is completed.

Remark 3Theorem 6 reduces to Theorem 5 whenz=0.

3.3 An example

The Birkhoffian and Birkhoff’s functions are

try to find out its conserved quantity and adiabatic in?variant deduced from the Mei symmetry on the time scale

From Eqs.（55，56），we have

It is easy to verify that

satisfy Eqs.（66，70）. Then from Theorem 5，a con?served quantity can be obtained

When the system is disturbed byQB2=t2+1，，from Eqs.（62，63），we have

Taking calculation，we get

Then

can be obtained as the first order adiabatic invariant from Theorem 6. Higher order adiabatic invariants can certainly be deduced.

Remark 4When the time scale is the real numbers R，all the results obtained in this paper are consistent with those in Ref.［18］.

4 Conclusions

The Mei symmetry and perturbation to Mei symmetry are studied under special infinitesimal transformations in this paper. Theorems 1—6 are new work. However，further research on Mei sym?metry on time scales，for example，Mei symmetry under general infinitesimal transformations on time scales are to be further investigated.