Canonical equations of Hamilton for the nonlinear Schr?dinger equation

4th International Conference on Mathematical Modeling in Physical Sciences

物理学;数学

Liang, Guo^1 ; Guo, Qi^1 ; Ren, Zhanmei^1

Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou

510631, China^1

关键词: Canonical equations;    Classical mechanics;    Dinger equation;    First order differential system;    Generalized coordinates;    Mathematical physics;    Newton's second law;    Second-order differential systems;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/633/1/012041/pdf
DOI  :  10.1088/1742-6596/633/1/012041

来源: IOP

【 摘 要 】

We define two different systems of mathematical physics: the second order differential system (SODS) and the first order differential system (FODS). The Newton's second law of motion and the nonlinear Schrödinger equation (NLSE) are the exemplary SODS and FODS, respectively. We obtain a new kind of canonical equations of Hamilton (CEH), which exhibit some kind of symmetry in form and are formally different from the conventional CEH without symmetry [H. Goldstein, C. Poole, J. Safko, Classical Mechanics, third ed., Addison- Wesley, 2001]. We also prove that the number of the CEHs is equal to the number of the generalized coordinates for the FODS, but twice the number of the generalized coordinates for the SODS. We show that the FODS can only be expressed by the new CEH, but not introduced by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way.