Variational Problems And Symmetry With Time Delay

In this manuscript, the Noether symmetries and conserved quantities for Lagrangesystem and Hamilton system with time delay are proposed and studied based on theinteger model and fractional model. Firstly, the Hamilton principle for Lagrangesystem and Hamilton system with time delay are established, the differentialequations of motion for the corresponding dynamics system with time delay areobtained. Based upon the invariance of the Hamilton action with time delay under thegroup of infinitesimal transformations which depends on the generalized velocities,the generalized coordinates and the time, the Noether symmetric transformations, theNoether quasi-symmetric transformations and the Noether generalizedquasi-symmetric transformations of the corresponding system are defined, and thecriteria of the Noether symmetries of the system are established. The relationshipbetween the Noether symmetries and conserved quantities with time delay are studied,and the Noether theories of the corresponding system with time delay are established.Secondly, under the definitions of Riemann-Liouville derivative, the fractionalHamilton principle for the corresponding non-conservative Lagrange system withtime delay are established, the fractional differential equations of motion for thecorresponding non-conservative system with time delay are obtained. Based upon theinvariance of the fractional Hamilton action with time delay under the group ofinfinitesimal transformations which depends on the generalized velocities, thegeneralized coordinates and the time, the definitions and criteria of the fractionalNoether quasi-symmetric transformations and the fractional generalized Noetherquasi-symmetric transformations for the corresponding mom-conservative systemwith time delay are given; The relationship between the fractional Noethersymmetries and conserved quantities with time delay are studied, and the fractionalNoether theories of the corresponding non-conservative system with time delay isestablished.Finally we give a conclusion and a forecast to the future.