NOETHER'S THEOREM OF HERGLOTZ FORM FOR GENERALIZED CHAPLYGIN SYSTEMS WITH NONCONSERVATIVE FORCES

Herglotz principle is a variational principle for nonconservative systems, which is a generalization of Hamilton's principle. In the nonconservative case, Hamilton's principle, in its current generalized form, is no longer a variational principle, although it can also be used to derive equations of motion. Noether theorem is a basic principle of analytical mechanics, which reveals the intrinsic relationship between symmetry and conservation laws. More conservation laws can be found by using Noether theorem than Newtonian mechanics and analytical mechanics. Chaplygin system is a special and widely used nonholonomic system whose equations can be studied independently of the constraint equations. The nonconservative generalized Chaplygin system is a generalization of Chaplygin system to nonconservative mechanics, so it is of great meaning to explore and establish Herglotz type Noether theorem for this system. To aim at the nonholonomic mechanical systems, we proposed and established the dynamical equations of nonconservative generalized Chaplygin systems based on the Herglotz principle. By using variational relation and Chetaev condition of virtual displacement, the total variational equation of Hamilton-Herglotz action is derived from the derivation and solved. From this, the total variation of the action of nonconservative generalized Chaplygin system is derived. The definitions of Herglotz form Noether symmetric transformation and quasi-symmetric transformation are given, and their criterions (Noether identities) are derived by using total variational formula. Noether’s theorems of Herglotz form for generalized Chaplygin systems with nonconservative forces are proved, and a new type of conserved quantity is obtained. Finally, for two specific nonconservative nonholonomic systems, the generalized Chaplygin equations of Herglotz form, the criterion equations and Chetaev conditions are set up, and the generators are solved, and conserved quantities are found by the theorems obtained, and the validity of the results is verified.