In this paper, we introduce a new definition of fractional derivative which contains a fractional factor, and its physical meanings are given. When studying the fractional Schrdinger equation(FSE) with this form of ...In this paper, we introduce a new definition of fractional derivative which contains a fractional factor, and its physical meanings are given. When studying the fractional Schrdinger equation(FSE) with this form of fractional derivative, the result shows that under the description of time FSE with fractional factor, the probability of finding a particle in the whole space is still conserved. By using this new definition to construct space FSE, we achieve a continuous transition from standard Schrdinger equation to the fractional one. When applying this form of Schrdinger equation to a particle in an infinite symmetrical square potential well, we find that the probability density distribution loses spatial symmetry and shows a kind of attenuation property. For the situation of a one-dimensional infinite δ potential well,the first derivative of time-independent wave function Φ to space coordinate x can be continuous everywhere when the particle is at some special discrete energy levels, which is much different from the standard Schrdinger equation.展开更多
In this paper, we extend Noether＇s theorem to nonholonomic constraints systems in optimal control. We present a systematic way to calculate conserved quantities along the Pontryagin extremals for optimal control prob...In this paper, we extend Noether＇s theorem to nonholonomic constraints systems in optimal control. We present a systematic way to calculate conserved quantities along the Pontryagin extremals for optimal control problems with nonholonomic constraints, which are invariant under the parameter groups of infinitesimal transformations that change all （time, state, control） variables. Meanwhile, the Noether equalities corresponding to the conservation laws are given. Then, we obtain a new version of Noether＇s theorem to optimal control systems. An example is given to illustrate the application of these results.展开更多
基金Supported by National Natural Science Foundation of China under Grant Nos.11472247 and 11872335
文摘In this paper, we introduce a new definition of fractional derivative which contains a fractional factor, and its physical meanings are given. When studying the fractional Schrdinger equation(FSE) with this form of fractional derivative, the result shows that under the description of time FSE with fractional factor, the probability of finding a particle in the whole space is still conserved. By using this new definition to construct space FSE, we achieve a continuous transition from standard Schrdinger equation to the fractional one. When applying this form of Schrdinger equation to a particle in an infinite symmetrical square potential well, we find that the probability density distribution loses spatial symmetry and shows a kind of attenuation property. For the situation of a one-dimensional infinite δ potential well,the first derivative of time-independent wave function Φ to space coordinate x can be continuous everywhere when the particle is at some special discrete energy levels, which is much different from the standard Schrdinger equation.
基金Supported by the National Natural Science Foundation of China(No.11272287,11472247)
文摘In this paper, we extend Noether＇s theorem to nonholonomic constraints systems in optimal control. We present a systematic way to calculate conserved quantities along the Pontryagin extremals for optimal control problems with nonholonomic constraints, which are invariant under the parameter groups of infinitesimal transformations that change all （time, state, control） variables. Meanwhile, the Noether equalities corresponding to the conservation laws are given. Then, we obtain a new version of Noether＇s theorem to optimal control systems. An example is given to illustrate the application of these results.
