Conserved Quantities And Symmetry Theories Of The Relative Motion Dynamics On Time Scales

In this paper,conserved quantities and symmetry theories of the relative motion dynamics on time scales are studied.According to the invariance of differential equations under infinitiesimal transformations,Noether theorems of the relative motion systems on the time scale and Lie symmetries of the relative motion systems in phase space,the relative motion systems of Chetaev type on time scales are studied respectively.Firstly,according to the time scales calculus theory,differential equations of motion of relative motion systems on time scale are established.The motion equations of the relative motion systems with Chetaev-type constraints on the time scale are derived from the D'Alembert principle on the time scale.Based on the invariance principle of the Hamiltonian action on the time scale under the infinitesimal transformation,Noether theorem of the relative motion systems on the time scale is obtained under the condition of time coordinate invariant and time coordinate change.Second,we study Lie symmetries theory of the relative motion systems on time scales.For the relative motion systems on the time scale,the invariance of the differential equations based on the relative motion system under the infinitesimal transformation proves Lie symmetries of the relative motion systems on the time scales,and the deterministic equations of Lie symmetries are obtained.Structural equations and Noether-type conserved quantities.Finally,the classification discusses the symmetries of continuous mechanical systems and discrete mechanical systems.Thirdly,Noether theorem of the relative motion systems in phase space on time scales are established.We extend the symmetries range of the relative motion systems on time scales to the phase space.The differential equations of the relative systems in the phase space are derived from the Hamilton principle on the time scales.We also explore symmetries of the system when the time parameters are constant and changed and Noether theorem of the relative motion systems are derived by using the time parameter recombination method.Fourthly,we study Lie symmetries of the relative motion systems in phase spaceon the time scale.According to the Lagrange equation with triangular derivatives on the time scale,the differential equations of the relative motion systems in phase space on time scale are derived.Based on the generailzed quasi-invariance of the Hamilton action on time scales,the determining equations,structural equations and conserved quantities of the relative motion systems are given.