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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.
Classical Mechanics and Field Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
where F is some function that may depend on t, qa and the derivatives of qa. Transformations of this sort are called quasi-symmetries or, when F = 0, symmetries, of the Lagrangian. The statement of Noether’s theorem is that if there exists a quasi-symmetry of the Lagrangian, then there exists a corresponding constant of motion, i.e., a function J that is generally an expression in terms of t, qa, and the derivatives of qa that does not change with time for any on-shell solution to the equations of motion.
On analytic series solutions and conserved fluxes of the time fractional (2+1)-dimensional Burger's system via invariant approach
Published in Waves in Random and Complex Media, 2021
Sait San, Pinki Kumari, Sachin Kumar
Conservation laws have a very important role in the study of nonlinear physical phenomena such as the total amount of a certain physical quantity (electric charge, energy, momentum, mass, angular momentum) [20]. Apart from physics, it also appears in many fields in mathematics. The theoretical aspect of differential equations, conserved vectors are frequently used for solving stability theorems, proving existence and uniqueness theorems, examination of the basic features of the solutions, finding exact solutions. In 1918, Emmy Noether stated that each physical system with the Lagrangian function can have a conservation law for the generator associated with it. However, this has been a major obstacle as many systems do not have a Lagrangian function in the classical sense. To overcome this restriction, Ibragimov proposed a new conservation theorem with a formal Lagrangian to construct conservation laws of PDEs that do not possess classical Lagrangian [21].
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].