Volume 42 Issue 11
Nov.  2021
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ZHENG Mingliang. Lie Symmetries and Conserved Quantities of Lagrangian Systems With Time Delays[J]. Applied Mathematics and Mechanics, 2021, 42(11): 1161-1168. doi: 10.21656/1000-0887.410184
Citation: ZHENG Mingliang. Lie Symmetries and Conserved Quantities of Lagrangian Systems With Time Delays[J]. Applied Mathematics and Mechanics, 2021, 42(11): 1161-1168. doi: 10.21656/1000-0887.410184

Lie Symmetries and Conserved Quantities of Lagrangian Systems With Time Delays

doi: 10.21656/1000-0887.410184
  • Received Date: 2020-06-23
  • Rev Recd Date: 2020-12-23
  • Available Online: 2021-12-07
  • Publish Date: 2021-11-30
  • The Lie symmetries and conserved quantities of non-conservative mechanical systems with time delays in configuration space were studied. Firstly, the piecewise Lagrangian equations for non-conservative systems with time delays were established according to the Hamiltonian principle of dynamics with time delay. Secondly, the determining equations of the Lie symmetry were obtained by means of the permissible Lie group theory for differential equations. Then, according to the relationship between symmetries and conserved quantities, the Lie theorem of non-conservative systems with time delays was obtained through construction of structural equations. Finally, 2 examples were given to illustrate the application of the method. The results show that, the time delay makes the Lagrangian equations of non-conservative systems piecewise, and the number of determining equations for Lie symmetry is twice of the number of degrees of freedom, which imposes higher restrictions on the generator functions. Meanwhile, the conserved quantity is also in a piecewise expression depending on the velocity term.
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