Studies Of Symmetry And Canonical Coordinate’s Method For The Constraint Mechanical Systems

Canonical coordinates are obtained by coordinates transformation which turnsone-parameter Lie group into the translation group, this property is widely used to solve thedifferential equations. Selecting canonical coordinates of system logically can not only reducethe order of the differential equations, but can reflect the physical nature of system on thespecific domain. In this paper, the method of canonical coordinates will be applied to constrainedmechanical system, including the Lagrange system, the non-conservative system and thenonholonomic system to seek for conserved quantities that provides a new approach for thesystem to conserved quantity. We had proved that the new and different conserved quantity canbe obtained by choosing different and reasonable canonical coordinates.Firstly, as the expanding application of canonical coordinates method to constrainedmechanical system, equations which can help to obtain the canonical coordinates of holonomicsystem including the single-degree-freedom system and n-degrees-freedom system are studiedbased on the relation between Lie symmetry theory and canonical coordinates theory. The basicforms of the first integral of the system are obtained by selecting the appropriate coordinatesbased on the definition and related properties of canonical coordinates. Several examples of thecanonical coordinates method applied in holomonic system are illustrated.Then, we studied the conserved quantity of the nonholonomic system with canonicalcoordinate’s methods. For nonholonomic constraint mechanics system, based on the differentialequations under the infinitesimal transformation of single-parameter Lie point group areinvariance, we get generator generated by determining equation and the limit equation.According to the canonical coordinate’s definition and related properties, we selected theappropriate canonical coordinates to establish the conserved quantity of the system. Theexamples will give to illustrate the application of canonical coordinates in nonholonomic system.Finally, we use Lagrange symmetry theory to seek and obtain the new kind conservedquantity of nonholonomic systems. On the basis of the equation of motion for constraint system,if the system has local Lie transformation, we can get generator generated by determiningequation and the limit equation. Using the definitions and qualities of Lagrange symmetry, the criterion of Lagrange symmetry of nonholonomic system are established and the conditions ofobtaining the conserved quantity which deduced by the Lagrange symmetry are given.There are four important results of this article: first, we obtain the relation between the Liegroups and canonical coordinates’ method. Second, the canonical coordinates equations ofconstrained mechanical system are expressed and canonical coordinates of system are obtainedthrough calculation; third, the form of conserved quantities for the single-degree-freedom andmulti-degree-freedom of constrained mechanical system had been expressed; four, Appling theLagrange symmetry theory to the field of the constraint system and getting new conservedquantity are completed.This paper has three innovation points: first, we expend the application of the canonicalcoordinate’s method to the field of constraints mechanical system. Second, the conservedquantity of the system by using the canonical coordinate transform is obtained. Third, thismethod about Lagrange symmetry is used to get the conserved quantity of the restraintmechanics system for the first time. Furthermore, the application of Lagrange symmetry isextended.