Research paperConformal invariance and conserved quantities of mechanical system with unilateral constraints
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 and the system subjects to g unilateral ideal holonomic constraints The differential equation of motion of the system can be expressed as For every β, we have the relations . 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 where ε is infinitesimal parameter, ξ0, ξj are infinitesimal transformation generators. It has infinitesimal generator vector which 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 which subjects to non-potential forces . We make as the generalized coordinates, and for simplicity. The kinetic energy of the material point is and the potential energy is . The Lagrangian function of the mechanical system is The non-potential generalized forces are The constraint equation is
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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