On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative
On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative
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DOI：
10.1007/s11071-007-9296-0
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发表时间：
2007-08
影响因子：
5.6
通讯作者：
D. Baleanu;S. Muslih;E. Rabei
中科院分区：
文献类型：
--
作者：
D. Baleanu;S. Muslih;E. Rabei
分数阶力学既描述保守系统又描述非保守系统。分数阶变分原理在研究分数阶力学中具有重要意义，并提出了几种不同的形式。在经典力学中，等效拉格朗日量起着重要的作用，因为它们承认相同的欧拉-拉格朗日方程。通过将适当函数的总时间导数加到给定的经典拉格朗日量或乘以常数，我们得到的拉格朗日量是相同的运动方程。在这项研究中，分数阶离散拉格朗日函数的分数阶导数不同的黎曼-刘维分数阶导数进行了分析。作为应用这一方法的结果，经典的结果作为一种特殊情况被重新得到，利用FaadiBruno公式的分数阶推广，得到了与给定分数阶Lagrange量不同的分数阶Lagrange量的具体表达式.研究了所得到的分数阶拉格朗日量所对应的分数阶欧拉-拉格朗日方程和汉密尔顿方程，并对两个算例进行了详细分析。
Fractional mechanics describe both conservative and nonconservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics, the equivalent Lagrangians play an important role because they admit the same Euler–Lagrange equations. By adding a total time derivative of a suitable function to a given classical Lagrangian or by multiplying with a constant, the Lagrangian we obtain are the same equations of motion. In this study, the fractional discrete Lagrangians which differs by a fractional derivative are analyzed within Riemann–Liouville fractional derivatives. As a consequence of applying this procedure, the classical results are reobtained as a special case.The fractional generalization ofFaàdi Bruno formula is used in order to obtain the concrete expression of the fractional Lagrangians which differs from a given fractional Lagrangian by adding a fractional derivative. The fractional Euler–Lagrange and Hamilton equations corresponding to the obtained fractional Lagrangians are investigated, and two examples are analyzed in detail.