Research paper
Conformal invariance and conserved quantities of mechanical system with unilateral constraints

https://doi.org/10.1016/j.cnsns.2017.12.005Get rights and content

Highlights

  • The definition and determining equations of conformal invariance in mechanical system with unilateral constraint are given.

  • The sufficient and necessary conditions for the conformal invariance to be Lie symmetry are given.

  • The conformal factors are got.

  • The whole course and piecewise conserved quantities for unilateral constraint system are discussed.

Abstract

By distinguishing the different constraint cases, the whole course and piecewise conserved quantities, which deduced from conformal invariance of mechanical systems with unilateral constraints, are given. The determining equation of conformal invariance of the system is obtained. The sufficient and necessary conditions for the conformal invariance must be Lie symmetry of the system are given. The forms of conformal factors are obtained. An example is given to illustrate the results in this paper.

Introduction

The research on symmetries and conserved quantities of mechanical systems possesses important theoretical and practical significance. The well known Noether symmetry has broadly applications in mathematics, dynamics and physics [1], [2], [3], [4], [5], [6], [7], [8], [9], it always can lead to conserved quantities. In fact, it is also named variational symmetry [4]. Besides Noether symmetry, there are Lie symmetry, Mei symmetry, and so on [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. Above symmetries are all basing on the Lie continuous transformation group. In 1997, Galiullin et al. [21] discussed the conformal invariance of Birkhoff system and deduced Noether conserved quantities from this symmetry. Mei et al. [22] extended the conformal invariance to generalized Birkhoff equations and gave the Noether conserved quantities. The key question is to find out the conformal factor to the conformal invariance of dynamics. Considerable progress has been made on the application of conformal invariance to mechanical systems in decades [23], [24], [25], [26], [27], [28].

The unilateral constraints exist in many mechanical systems [29], [30], [31], [32], [33], the motions of these systems can be represented by a set of differential equations with unilateral constraints. In mathematics the unilateral constraint can be represented by inequalities. The symmetries and conserved quantities of mechanical system with unilateral constraints had been extensively investigated [34], [35], [36], [37], [38]. However, the conserved quantities they got are all whole course invariants. In fact, there is a transition from free motion to constraint motion for the mechanical systems with unilateral constraints, so the conserved quantities should be piecewise for these motions. In this paper, we will distinguish this case and give the whole course and piecewise conserved quantities for mechanical systems with unilateral constraints. These results will be helpful for the study of the dynamic behaviour of systems.

In the present paper, we study the conformal invariance and conserved quantities of mechanical system with unilateral constraints. We first give the Lagrange equation of mechanical system with unilateral constraints. Secondly, basing on the Lie point transformation group, we give the mathematical definition form of conformal invariance of system. Thirdly, the sufficient and necessary conditions are proposed to ensure that conformal invariance of mechanical with unilateral constraints is Lie symmetry. Fourthly, we give the propositions that conformal invariance leads to whole course and piecewise conserved quantities. Finally, we give an example to illustrate the application of the results.

Section snippets

The differential equations of motion of mechanical system with unilateral constraints

We consider a mechanical system whose configuration is determined by n generalized coordinates qs(s=1,,n), and the system subjects to g unilateral ideal holonomic constraints fβ(t,q)0(β=1,,g)The differential equation of motion of the system can be expressed as ddtLq˙sLqs=Qs+λβfβqs(s=1,,n)For every β, we have the relations fβ0,λβ0,fβλβ=0. where L is Lagrangian function of the mechanical system, and Qs are nonconservative forces, λβ are constraint multipliers. Because the constraints

Conformal invariance of mechanical system with unilateral constraints

In order to get the conform invariance of mechanical system with unilateral constraints, we need to explore the transformation sets of independent and non-independent variables corresponding to Eqs. (1) and (2). We introduce the one-parameter Lie group of point transformations t*=t+ɛξ0(t,q,q˙),qs*=qs+ɛξs(s,q,q˙),s=1,,nwhere ε is infinitesimal parameter, ξ0, ξj are infinitesimal transformation generators. It has infinitesimal generator vector X(0)=ξ0t+ξsqswhich is the operator for the

Conformal factor and Lie symmetry

We can deduce conserved quantities by conformal invariance of system. Before doing this, we firstly should determine the conformal transformation and find out the conformal factors. In order to get the conformal factors, one of the methods is that demands the mechanical system with unilateral constraints has both conformal invariance and Lie symmetry simultaneously under the infinitesimal transformations.

Propostion 1

For the mechanical system with unilateral constraints that determined by Eqs. (1) and (2),

Conserved quantities and conformal invariance

As we know, a symmetry may correspond to some conserved quantities. There always exists conserved quantities corresponding to Noether symmetry. The Lie symmetry may directly lead to Hojman conserved quantities or indirectly lead to Noether conserved quantities. The conformal invariance of mechanical system can also lead to conserved quantities when certain condition satisfied. In this section, we will give the Noether conserved quantities that deduced from the conformal invariance of unilateral

Illustrative example

Suppose a material point with mass m moves in a vertical plane not below a smooth curve y=x, which subjects to non-potential forces Q1=Ftmx˙,Q2=mx˙. We make q1=x,q2=y as the generalized coordinates, and m=g=F=1 for simplicity. The kinetic energy of the material point is T=12m(q˙12+q˙22), and the potential energy is V=mgq2. The Lagrangian function of the mechanical system is L=12(q˙12+q˙22)q2The non-potential generalized forces are Q1=tq˙1,Q2=q˙1,The constraint equation is fβ=q2q10

When

Conclusion

The conformal invariance of mechanical system with unilateral constraints is studied. By the definition of conformal invariance of mechanical system with unilateral constraints, the determining equation of conformal invariance that defined by Eqs. (13)–(15) is obtained . The sufficient and necessary conditions for the conformal invariance must be Lie symmetry of the system which defined by Eqs. (16)–(18) are given. The forms of conformal factors are obtained through these conditions. The

Acknowledgments

This work is partly supported by National Natural Science Foundation of China (grant nos. 11772141, 11262019).

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