Noether symmetries and their inverse problems of the nonholonomic systems with the fractional derivatives are studied. Based on the quasi-invariance of fractional Hamilton action under the infinitesimal transformations without the time and the general transcoordinates of time-reparametrization, the fractional Noether theorems are established for the nonholonomic constraint systems. Further, the fractional Noether inverse problems are firstly presented for the nonholonomic systems. An example is designed to illustrate the applications of the results.

Based on constraint Euler-Lagrange equations, a general dynamic model is presented for power converters with multiple or single switches. From the research on the energy of power converters, the generalized charges and generalized currents coordinates of dynamic elements are chosen, switching function is introduced, and a set of differential algebraic equations with constraint can be obtained based on Euler-Lagrange equations with the Kirchhoff’s current constraints. The inductor currents and the capacitor voltages are chosen as the state variables, and then the state equations of the converters can be got ultimately. The proposed method is applied to model an ?uk converter circuit with an ideal switch and a three phase PWM rectifier with multiple switches. From the modeling process, it can be seen that this method is unified, clear in physical meaning and strong commonality. Finally, with the obtained state equations, the working process of ?uk converter is simulated in MATLAB, the simulation results coincide with the operation of the power converter, and it verify the effectiveness of the proposed method, which has higher application value to model the more complex power converters.