基于分数阶模型的非保守系统的 Noether 准对称性

摘要

提出并研究了非保守力学系统的分数阶 Noether 对称性及其守恒量。基于非保守系统的 Hamilton 原理，导出了分数阶模型下非保守系统的运动微分方程；根据分数阶 Hamilton 作用量在时间，广义坐标和广义速度的无限小群变换下的不变性，给出了非保守力学系统的分数阶 Noether 准对称性的定义和判据，建立了分数阶Noether 准对称性与守恒量之间的联系，得到了分数阶 Noether 守恒量；最后，讨论了不存在非势广义力或规范函数等于零的特例，并举例说明结果的应用。%The Noether symmetries and conserved quantities for non-conservative systems are proposed and studied with fractional model.Based on the Hamilton principle for the non-conservative systems,the fractional differential equations of motion are derived.With using the invariance of the fractional Hamilton action under the infinitesimal transformations of group which depends on the time,the generalized coordi-nates and velocities,the definition and the criterion of the fractional Noether generalized quasi-symmetry for the non-conservative systems are given.The relation between the fractional Noether quasi-symmetry and the conserved quantity is established,and the fractional conserved quantities are obtained.The spe-cial cases,which the generalized nonpotential forces do not exit or the gauge function is equal to zero,are discussed.At the end,two examples are given to illustrate the application of the results.