Non-Noether conserved quantity for relativistic nonholonomic controllable mechanical system with variable mass

The non-Noether conserved quantity for relativistic nonholonomic controllable mechanical system with variable mass is discussed under special infinitesimal transformations in which the time is not a variable. The differential equations of motion of the system are established. The definition and criterion of the form invariance of the system under special infinitesimal transformations are discussed. The necessary and sufficient conditions under which the form invariance is a Lie symmetry is presented. The conditions under which a non-Noether conserved quantity can be induced by the form invariance and the form of the conserved quantity are obtained. An example is presented to illustrate the application of the result.