Birkhoff symmetry and Lagrange symmetry

MEI Feng-xiang(梅凤翔), WU Hui-bin(吴惠彬)

(1.School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China;2.School of Mathematics, Beijing Institute of Technology, Beijing 100081, China)



Birkhoff symmetry and Lagrange symmetry

MEI Feng-xiang(梅凤翔)1, WU Hui-bin(吴惠彬)

(1.School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China;2.School of Mathematics, Beijing Institute of Technology, Beijing 100081, China)

The purpose of this paper is to study the symmetry of first order Lagrangians and the corresponding conserved quantity. The relation between the Lagrangians and the Birkhoff’s functions and the Birkhoff symmetry of Birkhoffian systems are used to obtain the symmetry of first order Lagrangians and the corresponding conserved quantity. Two examples are given to illustrate the application of the result.

Birkhoffian system; first order Lagrangian; symmetry; conserved quantity

Researches on the symmetries of mechanical systems have important both physical and mathematical significance. Symmetries are mainly the Noether symmetry[1-4], the Lie symmetry[2-3], the form invariance[5], and the Lagrange symmetry[2,5-7], Birkhoff symmetry[8], etc. The main purpose of studying the symmetries of mechanical systems is to find conserved quantities. The Lagrangian which is linear to velocities is said to be a first order Lagrangian or singular one. The classical Lagrangian is in general order. But Lagrangians in both Hamiltonian and Birkhoffian formalisms are always in first order by introducing conjugate variables through Legendre transform. Even the Lagrangian for infinite-dimensional Hamiltonian system is also in first order[9]. So, it is very important to study the first order Lagrange symmetry. Many results have been obtained for the second order Lagrange symmetry, such as references[2,5-7], but there are very few works on the first order Lagrange symmetry. Ref.[10] pointed out the relation between the first order Lagrangian and the Birkhoffian. Ref.[8] presented the definition and criterion of Birkhoff symmetry, and the conserved quantity deduced by the symmetry for Birkhoffian systems. In this paper, basing on Refs.[8,10], we lead to the first order Lagrange symmetry from the Birkhoff symmetry, and present its corresponding conserved quantity. The main idea of this paper is to study the symmetry of first order Lagrangians by using Birkhoff symmetries and the relation between the first order Lagrangian and the Birkhoffian. It gives a new way to study the first order Lagrange symmetry.

1 Birkhoff symmetry of Birkhoffian system

The Birkhoff’s equations have the form[10]

(1)

Let

(2)

where

(3)

(4)

then the corresponding symmetry is a Birkhoff symmetry of system (1).

Proposition[8]For system (1), the Birkhoff symmetry leads to the conserved quantity

tr(Λm)=const.

(5)

wheremis any positive integer, and

(6)

2 First order Lagrangian and the Birkhoffian

Ref.[10] gave the relation between the first order Lagrangian and the Birkhoffian as follows:

(7)

then we have

(8)

(9)

3 First order Lagrange symmetry deduced by the Birkhoff symmetry

For the Lagrange symmetries discussed in Refs.[2, 5-7], their equations are second order, and expression (9) does not hold. For the first order Lagrangian, we can study its symmetry by relations (7) and (8), and using Birkhoff symmetries of Birkhoff system.

(10)

havetheLagrangesymmetry.TheconservedquantitiesdeducedbytheBirkhoffsymmetryarealsoonesdeducedbytheLagrangesymmetry.

4 Illustrative examples

Example 1 A fourth order Birkhoffian system is

(11)

The first order Lagrangian can be obtained from expression (7) as follows:

(12)

Take another group of Birkhoffian and Birkhoff’s functions to be

(13)

then the corresponding first order Lagrangian is

(14)

Expressions (11) and (13) have the Birkhoff symmetry[8], and the corresponding conserved quantity is

I=a3=const.

(15)

Expressions (12) and (14) possess the Lagrange symmetry, and the corresponding conserved quantity is also the expression (15).

Example 2 The Hojman-Urrutia example is

(16)

which has no second order Lagrangian expression, but can be expressed as a Birkhoffian system[10]. Let

then[10]

(17)

and the corresponding first order Lagrangian is

(18)

Take another group of Birkhoffian and Birkhoff’s functions to be

(19)

then the corresponding first order Lagrangian is

(20)

Expressions (17) and (19) have the Birkhoff symmetry. Since

the corresponding conserved quantity is

(21)

Expressions (18) and (20) possess the Lagrange symmetry, and the corresponding conserved quantity is also the expression (21).

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(Edited by Wang Yuxia)

10.15918/j.jbit1004-0579.201524.0101

O 316 Document code: A Article ID: 1004- 0579(2015)01- 0001- 03

Received 2013- 04- 07

Supported by the National Natural Science Foundation of China(10932002, 10972031, 11272050)

E-mail: huibinwu@bit.edu.cn