关于分数阶Lagrange系统的Noether定理及经典Jost证明方法的推广
About the Noether's theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof
摘要：
最近，G.弗雷德里克和D.F.M. Torres在Appl.Math.Comp.217，3，2010中被R.A.C证明是错误的。Ferreira和A.B. Malinowska在JMAA 429，2，2015中使用了一个反例，并对其他Noether型定理的有效性提出了质疑，特别是（JMAA 334，2007，定理32）。然而，反例并没有解释为什么以及在哪里Appl.Math.Comp.217，3，2010中给出的证明不起作用。本文对G.弗雷德里克和D.F.M. Torres在JMAA 334，2007中，其基于J. Jost和X.Li-Jost在经典情况下提出的方法的分数推广。该方法也在Appl. Math. Comp. 217，3，2010中使用。我们首先详细介绍这种方法，然后它的分数版本。指出了几个难点，特别是分数阶情形下变分对称的定义和局部变换群的一些性质。这些困难出现在几个推广的约斯特的方法，特别是在离散设置。然后，我们推导出一个分数的诺特定理遵循这一策略，纠正了Frederico和托雷斯在JMAA 334，2007年的初始声明，并获得了Atanackovic等人的主要结果的替代证明非线性分析71，2009。
英文摘要：
Recently, the fractional Noether's theorem derived by G. Frederico and D.F.M. Torres in Appl. Math. Comp. 217,3,2010 was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in JMAA 429, 2, 2015 using a counterexample and doubts are stated about the validity of other Noether's type Theorem, in particular(JMAA 334, 2007,Theorem 32). However, the counterexample does not explain why and where the proof given in Appl. Math. Comp. 217,3,2010 does not work. In this paper, we make a detailed analysis of the proof proposed by G. Frederico and D.F.M. Torres in JMAA 334, 2007 which is based on a fractional generalization of a method proposed by J. Jost and X.Li-Jost in the classical case. This method is also used in Appl. Math. Comp. 217,3,2010. We first detail this method and then its fractional version. Several points leading to difficulties are put in evidence, in particular the definition of variational symmetries and some properties of local group of transformations in the fractional case. These difficulties arise in several generalization of the Jost's method, in particular in the discrete setting. We then derive a fractional Noether's Theorem following this strategy, correcting the initial statement of Frederico and Torres in JMAA 334, 2007 and obtaining an alternative proof of the main result of Atanackovic and al. in Nonlinear Analysis 71, 2009.
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