SYMMETRY AND HERGLOTZ TYPE CONSERVED QUANTITIES FOR NONHOLONOMIC SYSTEMS WITH TIME DELAY

Time delay is a common time delay phenomenon in nature and engineering practice, which has a profound impact on the dynamic behavior and basic properties of mechanical systems. The Herglotz type generalized variational principle extends the classical variational principle and can be used to study nonconservative systems. Therefore, using the Herglotz type generalized variational principle to study the symmetry and conserved quantity of nonholonomic systems with time delay is of great significance both in theory and application. In this paper, the Herglotz type Noether theorem is extended to nonholonomic systems with time delay. Firstly, the Herglotz type differential variational principle of the system with time delay is established. By means of the Lagrange multiplier method, the Routh-type differential equations of motion of the general nonholonomic system with time delay are derived. Secondly, based on the invariance of the Hamilton-Herglotz action with time delay under infinitesimal transformations, two basic formulas for the variation of the action are given, and then the Herglotz type Noether symmetry is defined and the Herglotz type Noether identity is given. Thirdly, the Noether theorem of Herglotz type for general nonholonomic systems with time delay is established. In addition, the Noether theorem in special cases is discussed. If all the constraints are holonomic, the theorem is reduced to the Herglotz type Noether theorem for holonomic systems with time delay. If the system is conservative, it is reduced to the Herglotz type Noether theorem for nonholonomic conservative systems with time delay. If the time delay is not considered, it is reduced to the Herglotz type Noether theorem for nonholonomic systems. Finally, the Noether inverse theorem of Herglotz type for nonholonomic systems with time delay is given. At the end of this paper, the theoretical analysis results are illustrated by solving an example with time delay, and the feasibility and correctness of the proposed method are verified by numerical simulation.