Noether symmetry for fractional constrained Hamiltonian system within mixed derivatives

SONG Chuanjing

（School of Mathematical Sciences，SUST，Suzhou 215009，China）

Abstract：This study investigates the fractional singular system within mixed integer order and Riemann-Liouville fractional derivatives.The fractional singular Lagrange equation and the fractional constrained Hamilton equation are established.To find the solutions to the differential equations of motion for this singular system，Noether symmetry method has been studied and the corresponding conserved quantity has been investigated.Namely，Noether theorems of the fractional singular system within mixed integer order and Riemann-Liouville fractional derivatives are established.

Key words：fractional constrained Hamiltonian system；Noether symmetry；conserved quantity

1 Introduction

A system is called a singular system if it is described using a singular Lagrangian.A dynamic system can be described by Lagrangian as well as Hamiltonian.For a singular system，it is called a constrained Hamiltonian system when it is described by Hamiltonian，because the inherent constraints will exist when it is transformed from the Lagrangian to the Hamiltonian through the Legendre transformation[1-2].Many important systems in physics are singular systems or constrained Hamiltonian systems.

The problem of seeking conserved quantity of mechanical system is not only of great significance in mathematics，but also reflects the profound physical essence.Symmetry method is an important method to find conserved quantity，and there are three common symmetry methods：Noether symmetry method，Lie symmetry method and Mei symmetry method[3-5].Fractional calculus is a hot topic recently.There are four kinds of fractional derivatives that are commonly used：Riemann-Liouville fractional derivative，Caputo fractional derivative，Riesz-Riemann-Liouville fractional derivative and Riesz-Caputo fractional derivative（see Appendix）.Many results on fractional variational calculus and fractional Noether symmetry have been obtained[6-25].

This paper intends to find the conserved quantity of the constrained Hamiltonian system on the basis of mixed integer and Riemann-Liouville fractional derivatives using Noether symmetry method.Firstly，the constrained Hamilton equation is established.Then the infinitesimal transformations of the time，the generalized coordinates and the generalized momenta are given，as well as the definitions of Noether symmetry and conserved quantity.And finally the conserved quantity obtained from Noether symmetry is presented.

2 Fractional constrained Hamiltonian system

A dynamic system is described by generalized coordinates qi，i=1，2，…，n，then by finding the extremum of the functional

with the boundary conditionsq（t1）=q1，q（t2）=q2，we can get the following fractional Euler-Lagrange equation within mixed integer and Riemann-Liouville fractional derivatives

Define the generalized momenta and the Hamiltonian as

Equation（5）is called fractional primary constraint.

Introducing the Lagrange multiplier λa（t），a=1，2，…，n-R，0≤R＜n，from Eqs.（2）-（5），we get the following fractional constrained Hamilton equation within integer and Riemann-Liouville fractional derivatives

By introducing Poisson bracket，which is defined as

where F=F（t，q，p，pα），G=G（t，q，p，pα），we can get the following consistency condition of the fractional primary constraint

where a，b=1，2，…，n-R，0≤R＜n，i=1，2，…，n.Eq.（8）can be used to solve Lagrange multipliers.

3 Noether symmetry and conserved quantity

Noether symmetry means the invariance of the Hamilton action within integer and Riemann-Liouville fractional derivatives（Eq.（1））under the infinitesimal transformations.

The infinitesimal transformations are given as

and the expanded expressions are

where θ is a small parameter，ξ0，ξi，ηiand ηiαare called infinitesimal generators.

where ξ0（t1）=ξ0（t1，q（t1），pα（t1），p（t1））.

The definition of Noether symmetry gives ΔI=0，i.e.，

Eq.（12）is called Noether identity within mixed integer and Riemann-Liouville fractional derivatives.

Eq.（13）is called Noether quasi-identity within mixed integer and Riemann-Liouville fractional derivatives.

The Noether symmetry leads to a conserved quantity.A quantity C is called a conserved quantity if and only if dC/dt=0.Therefore，we have

Theorem 1For the fractional constrained Hamiltonian system Eq.（6），if the infinitesimalgenerators ξ0，ξi，ηiand ηiαsatisfy Eq.（12），then there exists a conserved quantity

ProofUsing Eqs.（5），（6）and（12），we have

This proof is completed.

Theorem 2For the fractional constrained Hamiltonian system Eq.（6），if there exists a gauge function G such that the infinitesimalgenerators ξ0，ξi，ηiand ηiαsatisfy Eq.（13），then there exists a conserved quantity

ProofUsing Eqs.（5），（6）and（13），we can get dCG/dt=0.

Remark 1It is obvious that when G=0，Theorem 2 reduces to Theorem 1.

4 An example

The Lagrangian is

try to study its Noether symmetry.

Firstly，we have

From Eq.（17），we have

Therefore，the Lagrangian in this example is singular.And we can verify that

satisfy Eq.（13）.Then Theorem 2 gives

5 Conclusion

Based on mixed integer and Riemann-Liouville fractional derivatives，the fractional constrained Hamilton equation is presented firstly.Then the Noether symmetry and the conserved quantity are studied.Theorem 1 and Theorem 2 are new work.

Appendix

Let f（t）be a function，t∈[t1，t2]，n is an integer,then the left and the right Riemann-Liouville fractional derivatives are defined,respectively，as

and the left and the right Caputo fractional derivatives are defined,respectively，as

Specially，when α→1，we have