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Background
Published in L. Piccirillo, G. Coppi, A. May, Miniature Sorption Coolers, 2018
L. Piccirillo, G. Coppi, A. May
i.e., the variation of internal energy in a system is equal to the heat added to the system plus the work done on the system.4 The internal energy U is the total energy contained in the system without considering its whole kinetic energy (for example, due to the whole motion of the system) and potential energy (for example, due to an external force). The first law 1.3 tells us that the internal energy can be changed by transferring heat or doing work. For those systems where matter can be exchanged, then we need to consider the contribution of the matter transferred to/from the system. In analogy with the case of the heat death of the Universe discussed above, we need to be careful with the conservation of energy when the entire Universe is concerned. According to the Noether theorem, whenever we have a continuous symmetry of the Lagrangian describing the system, there is an associated conservation law. Energy conservation is connected to time shift invariance of physical laws.5 Einstein pointed out that his General Theory of Relativity did not imply a time shift invariance and therefore there is no law of conservation of energy on a large scale in the Universe. Energy is conserved only locally.
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Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[computational, general] A mathematician and scientist from the Prussian Empire, Germany. The work of Amalie Noether involved the definition of the following three concepts: associative law (mathematical grouping can be interchanged, in rudimentary format: (a + b) + c = a +(b + c) and (a * b) * c = a * (b * c)), commutative law (finding association between phenomena and functions characterized by substitution, interchange, or exchange: a + b = b + a and a * b = b * a), and the distributive law (captured in essence as a * ((b + c) = (a * b) + (a * c)). Her work was far reaching and influential and she was commended by Albert Einstein (1879–1955) for her mathematical contributions to his efforts. Noether created a theorem that united the symmetry in nature and the universal laws of conservation: Noether’s theorem, or Noether’s principle. Her mathematical efforts were influential in the uncovering of the Higgs Boson (see Figure N.43).
Symmetries and Group Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
The perhaps most iconic statement within classical physics is Noether’s theorem, which relates the existence of continuous symmetries in physical systems to conserved quantities. All of the conservation laws we are familiar with from elementary classical mechanics are related to a corresponding symmetry. To give some examples, energy conservation relates to a system being invariant under time translations, while conservation of momentum relates to spatial translations. We shall discuss this in greater detail in Section 10.2.4, when we have developed the necessary framework to prove the theorem in one of its simpler forms.
Lie point symmetries, conservation laws, and analytical solutions of a generalized time-fractional Sawada–Kotera equation
Published in Waves in Random and Complex Media, 2019
Li Zou, Zong-Bing Yu, Shou-Fu Tian, Xiu-Bin Wang, Jin Li
The famous Noether theorem [25] establishes a relationship between symmetries and conservation laws of differential equations. In order to construct the conservation laws of the FDEs, the generalized Noether theorem has been proposed [26–28]. However, there is a drawback for this method that it is not valid for the FDEs which do not satisfy a Lagrangian equation. In order to overcome this difficulty, Ibragimov [29] presented the new conservation laws in 2007. Based on the new conservation laws, Lukashchuk [30] proposed the fractional Noether operators and constructed the conservation laws of fractional diffusion wave and sub-diffusion equations, respectively. In addition, the conservation laws of some FDEs have been studied by making use of the fractional generalization of the Noether operators [4,9,31].
Lie symmetry analysis, conservation laws and analytical solutions for a generalized time-fractional modified KdV equation
Published in Waves in Random and Complex Media, 2019
Chun-Yan Qin, Shou-Fu Tian, Xiu-Bin Wang, Tian-Tian Zhang
The well-known Noether theorem establishes a connection between symmetries and conservation laws of differential equations provided that the equations are Euler–Lagrange equations [17]. In [18], fractional generalizations of the Noether operators were firstly proposed and conservation laws for time-fractional subdiffusion and diffusion-wave equations were obtained by using the new conservation laws theorem suggested by Ibragimov [19]. Lukashchuk has made great contributions to investigate conservation laws for FDEs that do not admit fractional Lagrangian. Conservation laws of some FDEs have been considered by making use of the fractional generalization of the noether operators [20].
The time fractional D(m,n) system: invariant analysis, explicit solution, conservation laws and optical soliton
Published in Waves in Random and Complex Media, 2022
Pinki Kumari, R.K. Gupta, Sachin Kumar
The mathematical solutions of differential equations play a vital role in envisaging and making predictions on the real-world problems [8–10]. Although no general theory is present in the literature that can solve all kinds of differential models, but symmetry reduction methods fit, to some extent, in the category of general theory. During the last decade of nineteenth century, a prominent mathematician, Sophus Lie superbly made use of continuous group transformations that leave a system of differential equations invariant. These transformations provide a widely applicable technique to seek explicit symmetry solutions and the method is called as Lie classical method [11,12]. Basically, the method reduces the number of independent variables by one, for instance, a partial differential equations with two independent variables reduces into ordinary differential equation, which is relatively less difficult to solve. The classical method also leads to local conservation laws and group invariant solutions of nonlinear partial differential equations (NLPDEs). In addition, nonlinear PDEs can be classified into equivalence classes and new solutions can be obtained from the known ones by exploiting their symmetries. The applications and effectiveness of the classical method can be seen in Refs. [13–18]. Later, the work has been extended to the system of fractional order partial differential equations (FOPDEs). In recent years, the classical method has been extensively used to deal with many kinds of complex FOPDEs [19–25]. Local conservation laws are one of the important aspects of symmetries associated with the fractional PDE. The popular Noether theorem, which establishes connection between symmetries and conservation laws, is applicable only to the systems with Lagrangian. To overcome this problem, Ibragimov [16] introduced new conservation theorem based on the self-adjoint equations for the nonlinear differential equations not having Lagrangians. The progress on the topic is evident in Refs. [22,26–28].