This paper concentrates on studying the symmetries and a new type of conserved quantities called Mei conserved quantities for Nonholonomic Systems with Servoconstraints. The criterions of the Mei symmetry, the Noether symmetry, and the Lie symmetry are given. The conditions and the forms of the Mei conserved quantities deduced from these three symmetries are obtained. An example is given to illustrate the application of the results.
We used the molecular dynamics simulation based on the Stillinger–Weber (SW) interatomic potential to calculate the high-index surface energies of surfaces containing any of the stereographic surfaces of silicon at zero temperature. An empirical formula based on the structural unit model was generalized for high-index surfaces. Our simulated results show that the generalized formula can give a good estimation of the surface energy and structural feature of the high-index surfaces not only on the edge of stereographic but also within it. Our simulation and empirical formula results reveal that the closest surface has the lowest energy, so the closest (101) surface has the lowest surface energy and the (101), (111) and (001) surfaces are the extremum on the curve of surface energy versus orientation angle. Both the theoretical simulation results and the empirical formula calculation results are consistent with the available first-principles theoretical data.
The non-Noether conserved quantities (the Hojman conserved quantity) for nonholonomic controllable mechanical systems with relativistic rotational variable mass are discussed. The differential equations of motion of the systems are established. The definition and criterion of the Mei symmetries and the Lie symmetries of the system are discussed respectively. The necessary and sufficient condition under which the Mei symmetry is Lie symmetry is presented. The condition under which a non-Noether conserved quantity can be induced by the Mei symmetries and the form of the conserved quantity are obtained. An example is presented to illustrate the application of the result.
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