Conserved quantities have always played an important role in mathematics,mechanics and physics.In recent years,finding conserved quantities of mechanical systems is an important aspect of analytical mechanics.Time scale is an arbitrary non-empty closed subset of real number set.This theory integrates continuous dynamics with discrete dynamics system and provides an effective mathematical tool for scholars.Compared with integer-order model,fractional-order model is more accurate in describing complex dynamic behavior in nature.In order to find the conserved quantity of mechanical system further,the integral factor method is used to study the conserved quantities of mechanical systems on time scale and fractional order in this paper.The contents are as follows:1.The integral factors and conserved quantities of Lagrange,Hamilton and Birkhoff systems on time scale are studied.The definition of integral factors and the energy equation of the systems are established.The integral factor method is used to solve the conservation theorem of the conserved quantity of the system.2.Integral factors and conserved quantities of nonholonomic systems on time scale are studied.The definition of integral factor and energy equation for nonholonomic systems on time scale are established.The conservation theorem for solving the conserved quantities of nonholonomic systems on time scale is established and degraded to the general case.3.The integrating factors and conservation theorems for a fractional Birkhoffian system are studied.Based on the Riemann-Liouville derivative,the integral factor definition of fractional Birkhoffian system differential equation of motion is given by the expression of fractional Birkhoffian system differential equation of motion.Conservation theorem of fractional Birkhoffian system is established,and the generalized Killing equation of fractional Birkhoffian system is proposed.4.Integral factors and conservation theorems for a class of nonholonomic systems are studied.Based on El-Nabulsi model of fractional order integral extended by periodic law,the definition of integral factor for differential equation of motion of the system is given.Conservation theorem and inverse theorem of the system are established,and the generalized Killing equation of the system is proposed. |