Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations

IF 1 4区 数学 Q3 MATHEMATICS, APPLIED
J. Cresson, F. Jiménez, S. Ober-Blöbaum
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引用次数: 1

Abstract

We prove a Noether's theorem of the first kind for the so-called restricted fractional Euler-Lagrange equations and their discrete counterpart, introduced in [26,27], based in previous results [11,35]. Prior, we compare the restricted fractional calculus of variations to the asymmetric fractional calculus of variations, introduced in [14], and formulate the restricted calculus of variations using the discrete embedding approach [12,18]. The two theories are designed to provide a variational formulation of dissipative systems, and are based on modeling irreversbility by means of fractional derivatives. We explicit the role of time-reversed solutions and causality in the restricted fractional calculus of variations and we propose an alternative formulation. Finally, we implement our results for a particular example and provide simulations, actually showing the constant behaviour in time of the discrete conserved quantities outcoming the Noether's theorems.

有限变分法中的连续和离散Noether分数守恒量
基于先前的结果[11,35],我们证明了在[26,27]中引入的所谓受限分数欧拉-拉格朗日方程及其离散对应物的第一类Noether定理。在此之前,我们将受限分数阶变分演算与[14]中介绍的非对称分数阶变分演算进行了比较,并使用离散嵌入方法制定了受限变分演算[12,18]。这两种理论旨在提供耗散系统的变分公式,并基于通过分数阶导数建模的不可逆性。我们明确了时间反转解和因果关系在有限分数变分中的作用,并提出了一种替代公式。最后,我们为一个特定的例子实现了我们的结果,并提供了模拟,实际显示了由诺特定理得出的离散守恒量在时间上的恒定行为。
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来源期刊
Journal of Geometric Mechanics
Journal of Geometric Mechanics MATHEMATICS, APPLIED-PHYSICS, MATHEMATICAL
CiteScore
1.70
自引率
12.50%
发文量
23
审稿时长
>12 weeks
期刊介绍: The Journal of Geometric Mechanics (JGM) aims to publish research articles devoted to geometric methods (in a broad sense) in mechanics and control theory, and intends to facilitate interaction between theory and applications. Advances in the following topics are welcomed by the journal: 1. Lagrangian and Hamiltonian mechanics 2. Symplectic and Poisson geometry and their applications to mechanics 3. Geometric and optimal control theory 4. Geometric and variational integration 5. Geometry of stochastic systems 6. Geometric methods in dynamical systems 7. Continuum mechanics 8. Classical field theory 9. Fluid mechanics 10. Infinite-dimensional dynamical systems 11. Quantum mechanics and quantum information theory 12. Applications in physics, technology, engineering and the biological sciences.
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