GENERALIZED VARIATIONAL PRINCIPLE OF HERGLOTZ TYPE FOR NONCONSERVATIVE SYSTEM IN PHASE SPACE AND NOETHER'S THEOREM

Compared with the classical variational principle, the generalized variational principle of Herglotz based upon the action defined by differential equations gives a variational description of nonconservative dynamical system. The principle can describe all dynamical processes and nonconservative or dissipative systems. In the present study, the principle is extended to phase space, and the generalized variational principle of Herglotz type for non-conservative mechanical system in phase space is given and Noether's theorem and its inverse of the system are studied. Firstly, the generalized variational principle of Herglotz type in phase space is presented, a variational description of non-conservative system in phase space is given, and the corresponding Hamilton canonical equations are deduced. Secondly, based upon the relation between non-isochronal variation and isochronal variation, two basic formulae for the variation of Hamilton-Herglotz action in phase space are obtained. Thirdly, the definition and the criterion of Noether symmetry are given, and Noether's theorem and its inverse of nonconservative system for the variational problem of Herglotz type in phase space are proposed and proved, and the inner relation between the Noether symmetry and the conserved quantity for mechanical systems in phase space is revealed. The generalized variational principle of Herglotz type reduces to the classical variational principle under classical conditions, and Noether's theorem for the variational problem of Herglotz type reduces to the classical Noether's theorem of Hamilton system. In the end of the paper, we take the famous Emden equation and damping oscillator with second power as examples to illustrate the application of the results.