Velocity-dependent symmetries and non-Noether conserved quantities of electromechanical systems

The theory of velocity-dependent symmetries(or Lie symmetry) and non-Noether conserved quantities are presented corresponding to both the continuous and discrete electromechanical systems.Firstly,based on the invariance of Lagrange-Maxwell equations under infinitesimal transformations with respect to generalized coordinates and generalized charge quantities,the definition and the determining equations of velocity-dependent symmetry are obtained for continuous electromechanical systems;the Lie's theorem and the non-Noether conserved quantity of this symmetry are produced associated with continuous electromechanical systems.Secondly,the operators of transformation and the operators of differentiation are introduced in the space of discrete variables;a series of commuting relations of discrete vector operators are defined.Thirdly,based on the invariance of discrete Lagrange-Maxwell equations under infinitesimal transformations with respect to generalized coordinates and generalized charge quantities,the definition and the determining equations of velocity-dependent symmetry are obtained associated with discrete electromechanical systems;the Lie's theorem and the non-Noether conserved quantity are proved for the discrete electromechanical systems.This paper has shown that the discrete analogue of conserved quantity can be directly demonstrated by the commuting relation of discrete vector operators.Finally,an example is discussed to illustrate the results.