Symmetries For Dynamical Stystems With Non-standard Lagrangians And Non-standard Hamiltonians

Our nature almost admits no linear dynamics.However,for the description of non-linear dynamics,Friedmann-Robertson-Walker model and dissipaive systems,non-standard Lagrangians and non-standard Hamiltonians make the description more easily,compared with standard Lagrangians and standard Hamiltonians.Therefore,it is of great significance to make researches on non-standard Lagrangians and non-standard Hamiltonians.This paper focuses on the symmetries and conserved quantities for dynamical systems with non-standard Lagrangians and non-standard Hamiltonians.The contents are as follows:1.The definitions and the criteria of Noether-Mei symmetry for dynamical systems with non-standard Lagrangians are given.The conditions that Noether-Mei symmetry leads to Noether conserved quantities or Mei conserved quantities and the forms of conserved quantities are put forward.Theorems for Noether-Mei symmetry and conserved quantities are established.2.The Hamilton principles based on the actions for dynamical systems with non-standard Lagrangians on time scales are established,with which the corresponding differential equations of motion are deduced.Based on the invariance of Hamilton action in infinitesimal transformation and the technique of time-re-parameterization,Noether theorems are obtained and their proofs are given.3.The energy-like equations for dynamical systems with non-standard Lagrangians on time scales are established.And the Noether conserved quantities are proved with differential equations of motion,energy-like equations and Noether identities.4.The definitions of cyclic coordinates for dynamical systems with non-standard Lagrangians on time scales are given.Differential equations of motion of the system are reduced by the cyclic coordinates.And the Routh equations for the dynamical systems with non-standard Lagrangians on time scales are given.The reduced Routh equations hold the original form of differential equations of motion.5.The differential equations of motion for dynamical systems with non-standard Hamiltonians are deduced.The definitions and criteria of Noether symmetry and Noether quasi-symmetry are given.The Noether theorems are established and proved.