Available online at www.HighTechJournal.org HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 187 Relativistic Extended Thermodynamics of Polyatomic Gases with Rotational and Vibrational Modes Sebastiano Pennisi a* a Department of Mathematics and Informatics, University of Cagliari, Cagliari, Italy. Received 17 December 2020; Revised 07 April 2021; Accepted 14 May 2021; Published 01 September 2021 Abstract In a recent article, an infinite set of balance equations has been proposed to modelize polyatomic gases with rotational and vibrational modes in a non-relativistic context. To obtain particular cases, it has been truncated to obtain a model with 7 or 15 moments. Here the following objectives are pursued: 1) to obtain the relativistic counterpart of this model, which, at the non-relativistic limit, gives the same balance equations as in the known classical case; 2) to obtain the previous result for the model with an arbitrary but fixed number of moments; and 3) to obtain the closure of the resulting relativistic model so that all the functions appearing in the balance equations are expressed in terms of the independent variables. To achieve these goals, the following methods are used: 1) the principle of entropy is imposed. As a result, it is obtained that the closure is determined up to a single 4-vectorial function, usually called a 4-potential. 2) To determine this last function, a more restrictive principle is imposed, namely the Maximum Entropy Principle (MEP). 3) Since all the functions involved must be expressed in the covariant form so as not to depend on the observer, the Representation Theorems are used. The findings of this article exactly match the goals outlined earlier. They are clearly novel because they have never been achieved before. They can also be considered improvements because, if the aforementioned arbitrary number of moments is restricted to 16, the present work coincides with that already known in literature. Keywords: Moments Equations; Extended Thermodynamics; Non-equilibrium Thermodynamics.. 1. Introduction Based on Arima et al. (2018) [1] study (recently improved to describe dense polyatomic gases in Arima et al. (2020) [2]), the following balance equations have been introduced: 𝜕𝑡𝐹 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐹 𝑘𝑖1⋯𝑖𝑟 = 𝑃𝑖1⋯𝑖𝑟 , 𝜕𝑡𝐹𝑉 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐹𝑉 𝑘𝑖1⋯𝑖𝑟 = 𝑃𝑉 𝑖1⋯𝑖𝑟 , 𝜕𝑡𝐹𝑅 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐹𝑅 𝑘𝑖1⋯𝑖𝑟 = 𝑃𝑅 𝑖1⋯𝑖𝑟 , (1) Where 𝑟 goes from 0 to +∞, 𝐹𝑖1⋯𝑖𝑟 = 𝑚∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓 𝜉𝑖1⋯𝜉𝑖𝑟𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 𝜉 , 𝐹𝑉 𝑖1⋯𝑖𝑟 = 𝑚∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓 𝜉𝑖1⋯𝜉𝑖𝑟 2 ℐ𝑉 𝑚 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 𝜉 , 𝐹𝑅 𝑖1⋯𝑖𝑟 = 𝑚∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓 𝜉𝑖1⋯𝜉𝑖𝑟 2 ℐ𝑅 𝑚 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 𝜉 . (2) * Corresponding author: spennisi@unica.it http://dx.doi.org/10.28991/HIJ-2021-02-03-04  This is an open access article under the CC-BY license (https://creativecommons.org/licenses/by/4.0/). © Authors retain all copyrights. https://creativecommons.org/licenses/by/4.0/ https://orcid.org/0000-0002-2495-9107 HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 188 The definitions of 𝐹𝑘𝑖1⋯𝑖𝑟 , 𝐹𝑉 𝑘𝑖1⋯𝑖𝑟 , 𝐹𝑅 𝑘𝑖1⋯𝑖𝑟 are similar, with an extra factor 𝜉𝑘 inside the integrals. Moreover, they have 𝑃 = 0, 𝑃𝑖 = 0, 𝑃𝑉 + 𝑃𝑅 + 𝑃 𝑙𝑙 = 0 in order to ensure the conservation laws of mass, momentum and total energy. After that, the truncated systems with 7 and 15 moments have been considered and fully investigated. The Equations (1) 1 have been called the mass block, while (1) 2,3 constitute the vibrational and rotational blocks respectively. In the previous models for monoatomic gases, only the mass block (1) 1 was considered (see for example Liu and MÃŒller (1983) [3], MÃŒller, T. Ruggeri (1998) [4]). Its extension to the polyatomic gases began with Arima et al. (2012) [5] and gave inspiration, as the previous one, to many other articles part of which are cited in Ruggeri and Sugiyama (2015) [6]. In these articles, the two blocks of Equations (1) 1,2 were considered. The subsequent generalization [1], considers all the three blocks of Equations (1) 1−3 by distinguishing the contribute of rotational and vibrational modes. The sum of (1) 2, (1) 3 and of the trace of (1) 1 can substitute (1) 3 and reads: 𝜕𝑡𝐻0 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐻0 𝑘𝑖1⋯𝑖𝑟 = 𝐜0 𝑖1⋯𝑖𝑟 , (3) With: 𝐻0 𝑖1⋯𝑖𝑟 = 𝐹𝑉 𝑖1⋯𝑖𝑟 + 𝐹𝑅 𝑖1⋯𝑖𝑟 + 𝐹𝑖1⋯𝑖𝑟+2𝛿𝑖𝑟+1𝑖𝑟+2 , 𝐻0 𝑘𝑖1⋯𝑖𝑟 = 𝐹𝑉 𝑘𝑖1⋯𝑖𝑟 + 𝐹𝑅 𝑘𝑖1⋯𝑖𝑟 + 𝐹𝑘𝑖1⋯𝑖𝑟+2𝛿𝑖𝑟+1𝑖𝑟+2 , 𝐜0 𝑖1⋯𝑖𝑟 = 𝑃𝑉 𝑖1⋯𝑖𝑟 + 𝑃𝑅 𝑖1⋯𝑖𝑟 + 𝑃𝑖1⋯𝑖𝑟+2𝛿𝑖𝑟+1𝑖𝑟+2 . In the sequel we will use also the quantities; 𝐻𝑞 𝑖1⋯𝑖𝑟 = 𝐻0 𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑞−1𝑙𝑞−1 , 𝐻𝑞 𝑘𝑖1⋯𝑖𝑟 = 𝐻0 𝑘𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑞−1𝑙𝑞−1 𝐻𝑞 𝑖1⋯𝑖𝑟 = 𝐹𝑉 𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑞−1𝑙𝑞−1 + 𝐹𝑅 𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑞−1𝑙𝑞−1 , 𝐻𝑞 𝑘𝑖1⋯𝑖𝑟 = 𝐹𝑉 𝑘𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑞−1𝑙𝑞−1 + 𝐹𝑅 𝑘𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑞−1𝑙𝑞−1 . (4) Obviously, from Equations (3) and (1) 2,3 it follows: 𝜕𝑡𝐻𝑞 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐻𝑞 𝑘𝑖1⋯𝑖𝑟 = 𝑃 𝑖1⋯𝑖𝑟 , 𝜕𝑡ￜ̃ￜ𝑞 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐻𝑞 𝑘𝑖1⋯𝑖𝑟 = ￜ̃ￜ𝑖1⋯𝑖𝑟 , (5) With obvious meaning of 𝑃 𝑖1⋯𝑖𝑟 and ￜ̃ￜ𝑖1⋯𝑖𝑟 . Here we propose to find the relativistic counterpart of (1); since this non relativistic approach started from the classical Boltzman-Chernikov equation, we do the same starting from the generalized relativistic Boltzman-Chernikov equation; 𝑝𝛌 𝜕𝛌𝛌 𝑓 = 𝑄 , where 𝑓 is the distribution function. By multipying it by polynomials 𝑝 in the 4-momentum 𝑝𝛌, by a function 𝑓1 of the rotational energy ℐ𝑅, by a function 𝑓2 of the vibrational energy ℐ𝑣, by the product of their measures 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) and integrating in 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ, one obtains a field equations. So the problem is now how to determine these quantities 𝑝, 𝑓1 and 𝑓2 such that the resulting relativistic field equations have (1) as non reltivistic limit. We have found the result expressed by the following set of balance equations as relativistic counterpart of (1) truncated in a convenient way in terms of an arbitrary but fixed integer non negative number 𝑆: 𝜕𝛌𝐎 𝛌𝛌1⋯𝛌𝑟 = 𝐌𝛌1⋯𝛌𝑟 , 𝜕𝛌𝐎𝑉 𝛌𝛌1⋯𝛌𝑠 = 𝐌𝑉 𝛌1⋯𝛌𝑠 , (6) With 𝑟 = 0 ,⋯ , 𝑆 + 2, 𝑠 = 0 ,⋯ , 𝑆 and where; 𝐎𝛌𝛌1⋯𝛌𝑟 = 𝑐 𝑚𝑟−1 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓 𝑝𝛌𝑝𝛌1⋯𝑝𝛌𝑟 (1 + 𝑟 ℐ 𝑚 𝑐2 )𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ , 𝐎𝑉 𝛌𝛌1⋯𝛌𝑠 = 𝑐 𝑚𝑠−1 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓 𝑝𝛌𝑝𝛌1⋯𝑝𝛌𝑠 2 ℐ𝑉 𝑚 𝑐2 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ , (7) and ℐ = ℐ𝑅 + ℐ𝑉. Despite the appearances, in the system (6) there is a complete symmetry between the rotational and vibrational modes. In fact, for 𝑟 = 0 ,⋯ , 𝑆 we can add to each equation of (6) 1 the trace of that with 𝑟 + 2 instead of 𝑟 multiplied by − 𝑐−2; after that, we sum the corresponding equation in (6) 2 so obtaining; 𝜕𝛌𝐎𝑅 𝛌𝛌1⋯𝛌𝑟 = 𝐌𝑅 𝛌1⋯𝛌𝑟 , 𝑀𝑖𝑡ℎ 𝐌𝑅 𝛌1⋯𝛌𝑟 = −𝐌𝛌1⋯𝛌𝑟 + 𝑐−2𝐌𝑅 𝛌1⋯𝛌𝑟+2𝑔𝛌𝑟+1𝛌𝑟+2 − 𝐌𝑉 𝛌1⋯𝛌𝑟 , And 𝐎𝑅 𝛌𝛌1⋯𝛌𝑟 defined as (7) 2 with 𝑅 instead of 𝑉 except that in 𝜙(ℐ𝑅) 𝜓(ℐ𝑉). This is a consequence of the property 𝑝𝛌𝑝𝛌 = 𝑚 2𝑐2. It follows that the system (6) can be written also as; 𝜕𝛌𝐎𝑅 𝛌𝛌1⋯𝛌𝑟 = 𝐌𝑅 𝛌1⋯𝛌𝑟 , 𝜕𝛌𝐎𝑉 𝛌𝛌1⋯𝛌𝑟 = 𝐌𝑉 𝛌1⋯𝛌𝑟 , 𝑀𝑖𝑡ℎ 𝑟 = 0 ,⋯ , 𝑆 , 𝜕𝛌𝐎 𝛌𝛌1⋯𝛌𝑆+1 = 𝐌𝛌1⋯𝛌𝑆+1 , 𝜕𝛌𝐎 𝛌𝛌1⋯𝛌𝑆+2 = 𝐌𝛌1⋯𝛌𝑆+2 . (8) HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 189 We note that, from the definition (7) 2 it follows that the trace conditions hold; 𝐎𝑅 𝛌𝛌1⋯𝛌𝑟𝑔𝛌𝑟−1𝛌𝑟 = 𝑐 2𝐎𝑅 𝛌𝛌1⋯𝛌𝑟−2 , 𝐎𝑉 𝛌𝛌1⋯𝛌𝑟𝑔𝛌𝑟−1𝛌𝑟 = 𝑐 2𝐎𝑉 𝛌𝛌1⋯𝛌𝑟−2 . (9) In the next section we will find the non relativistic limit of Equations (6) and the resulting field equations are reported in the subsequent Equations (10). By comparing them with the above Equations (1), the following facts become evident: • The mass block (1) 1 has to be considered for 0 ≀ 𝑟 ≀ 𝑆 + 2. • The vibrational and rotational blocks appear for 0 ≀ 𝑟 ≀ 𝑆; also the subsequent orders 𝑆 + 1, 𝑆 + 2, ⋯, 2𝑆 have to be considered (here "order" of a tensor is the number of its indexes) but only by means of their traces and this is according to the law: "If 𝑟 is the number of its free indexes, then 𝑆 − 𝑟 traces have to be taken, for 0 ≀ 𝑟 ≀ 𝑆 − 1". • A number of traces, less than 𝑆 − 𝑟, have also to be considered but only by means of the sum of the tensors in the rotational and that in the vibrational mode. In fact, from (10) 7 we see that 𝑞 ≀ 𝑆 − 𝑟 and consequently, from (13) we see that the number of traces there occurring is 𝑞 − 1 ≀ 𝑆 − 𝑟 − 1. • Equations involving terms of the mass block (1) 1 of order 𝑆 + 3, 𝑆 + 4, ⋯, 2𝑆 + 4 have to be considered but only by means of the sum of them and that belonging to the rotational and vibrational blocks. In fact, from (4) 1 and (3) 3 we see that 𝐻𝑞 𝑖1⋯𝑖𝑟 involves the tensor of the mass block of order 2𝑞 + 𝑟; from (10) 6 we see that 𝐻𝑞 𝑖1⋯𝑖𝑆−𝑞+2 involves a tensor of order 𝑆 + 𝑞 + 2 and we see also that 𝑆 + 3 ≀ 𝑆 + 𝑞 + 2 ≀ 2𝑆 + 4. However, this tensor appears only after having taken its trace 𝑞 times. We note that the model introduced in Pennisi and Ruggeri (2020) [7] is a particular case of the present one; in fact, with the theory of subsystems developed in Boillat and Ruggeri (1997) [8], by dropping out (6) 2, what remains gives the model of Pennisi and Ruggeri (2020) [7] were there was considered no distinction between the rotational and vibrational modes. Moreover, the present model has been tested in the simpler case 𝑆 = 0 and the results have been published in [9]; this correspondence will be verified in Sect. 3. So also the model in Carrisi and Pennisi (2019) [9] is a subsystem of the present one when 𝑆 = 0. In section 5 we will find the closure of the new field Equations (6). It is expressed by the Equations (33) jointly with (23) reported below in that section. In this way the first parts of the following flowchart have been described. In particular, in this introduction its first step was obtained, i.e., the field Equations (6) as relativistic counterpart of the classical model (1) proposed in Pennisi and Ruggeri (2017) [11]. Flowchart of the research methodology is presented by Figure 1. Figure 1. Flowchart of the research methodology HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 190 2. The Non-relativistic Limit We prove now that the non relativistic limit of Equations (6) leads to the following hierarchy of balance equations for the classical case: 𝜕𝑡𝐹 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐹 𝑘𝑖1⋯𝑖𝑟 = 𝑃𝑖1⋯𝑖𝑟 , 𝑓𝑜𝑟 0 ≀ 𝑟 ≀ 𝑆 + 2 , (10) 𝜕𝑡𝐹𝑉 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐹𝑉 𝑘𝑖1⋯𝑖𝑟 = 𝑃𝑉 𝑖1⋯𝑖𝑟 , 𝜕𝑡𝐹𝑅 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐹𝑅 𝑘𝑖1⋯𝑖𝑟 = 𝑃𝑅 𝑖1⋯𝑖𝑟 , 𝑓𝑜𝑟 0 ≀ 𝑟 ≀ 𝑆 , 𝜕𝑡𝐹𝑉 𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆−𝑟𝑙𝑆−𝑟 + 𝜕𝑘𝐹𝑉 𝑘𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆−𝑟𝑙𝑆−𝑟 = 𝑄𝑉 𝑖1⋯𝑖𝑟 , 𝜕𝑡𝐹𝑅 𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆−𝑟𝑙𝑆−𝑟 + 𝜕𝑘𝐹𝑅 𝑘𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆−𝑟𝑙𝑆−𝑟 = 𝑄𝑅 𝑖1⋯𝑖𝑟 , 𝑓𝑜𝑟 0 ≀ 𝑟 ≀ 𝑆 − 1 , 𝜕𝑡𝐻𝑞 𝑖1⋯𝑖𝑆−𝑞+2 + 𝜕𝑘𝐻𝑞 𝑘𝑖1⋯𝑖𝑆−𝑞+2 = 𝐜𝑞 𝑖1⋯𝑖𝑆−𝑞+2 , 𝑓𝑜𝑟 1 ≀ 𝑞 ≀ 𝑆 + 2 , 𝜕𝑡𝐻𝑞 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐻𝑞 𝑘𝑖1⋯𝑖𝑟 = 𝐜𝑞 𝑖1⋯𝑖𝑟 , 𝑓𝑜𝑟 { 2 ≀ 𝑞 ≀ 𝑆 , 0 ≀ 𝑟 ≀ 𝑆 − 𝑞 , With 𝐻𝑞 𝑖1⋯𝑖𝑟 and 𝐻𝑞 𝑖1⋯𝑖𝑟 defined below in Equations (12) and (13). This result have already been described at the end of the previous section. So there remains here to prove it. Thanks to the trace condition (9) 2 and to (6) 2, we see that we can applicate the results of Borghero et al. (2005) [10]. We have only to observe that in this article there are 2 arbitrary numbers 𝑀 < 𝑁. We observe also that the free indexes appearing in Equation (1) of Borghero et al. (2005) [10] starts from 𝛌2 instead of 𝛌1 as in the present article. So by comparing these equations with the present (6) 2, we see that 𝑁 = 𝑆 + 1, 𝑀 = 𝑆. After that, we can use Equation (2) of Borghero et al. (2005) [10] and see that the non relativistic limit of the present Equation (6) 2 gives the above reported Equations (10) 2,4. Let us consider now the present Equation (6) 1; there isn’t a trace condition on it, so that we cannot apply the results of Borghero et al. (2005) [10]. But we can apply those in Equation (11) of Pennisi and Ruggeri (2020) [7] and have that its non relativistic limit is: 𝜕𝑡𝐻𝑞 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐻𝑞 𝑘𝑖1⋯𝑖𝑟 = 𝐜𝑞 𝑖1⋯𝑖𝑟 , 𝑓𝑜𝑟 { 0 ≀ 𝑞 ≀ 𝑆 + 2 , 0 ≀ 𝑟 ≀ 𝑆 − 𝑞 + 2 , (11) 𝐻𝑞 𝑖1⋯𝑖𝑟 = 𝑚 ∫ ℝ3 ∫ +∞ 0 ∫ +∞ 0 𝑓𝐶 𝜉𝑖1⋯𝜉𝑖𝑟 𝜉2(𝑞−1) (𝜉2 + 2𝑞 ℐ 𝑚 ) 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 𝜉 , 𝐻𝑞 𝑘𝑖1⋯𝑖𝑟 = 𝑚 ∫ ℝ3 ∫ +∞ 0 ∫ +∞ 0 𝑓𝐶 𝜉𝑘𝜉𝑖1⋯𝜉𝑖𝑟 𝜉2(𝑞−1) (𝜉2 + 2𝑞 ℐ 𝑚 ) 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 𝜉 . (12) We now further elaborate these last Equations (11) and (12). First Step: For { 1 ≀ 𝑞 ≀ 𝑆 + 2 , 0 ≀ 𝑟 ≀ 𝑆 − 𝑞 + 1 , we substitute 𝐻𝑞 𝑖1⋯𝑖𝑟 with; 𝐻𝑞 𝑖1⋯𝑖𝑟 = 𝐻𝑞 𝑖1⋯𝑖𝑟 − 𝐻𝑞−1 𝑖1⋯𝑖𝑟+2𝛿𝑖𝑟+1𝑖𝑟+2 = (13) = 𝑚 ∫ ℝ3 ∫ +∞ 0 ∫ +∞ 0 𝑓𝐶 𝜉𝑖1⋯𝜉𝑖𝑟 𝜉2(𝑞−1) 2 ℐ 𝑚 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 𝜉 . We can do this because both (𝑟 , 𝑞) and (𝑟 + 2 , 𝑞 − 1) satisfy the condition (11) 2. Of course, we do this starting from the highest value of 𝑞 (i.e. 𝑆 + 2) to go down to 𝑞 = 1. So, now the equations are (10) 2,4, (11) for 𝑞 = 0, 0 ≀ 𝑟 ≀ 𝑆 + 2, (11) for { 1 ≀ 𝑞 ≀ 𝑆 + 2 , 𝑟 = 𝑆 − 𝑞 + 2 , and: 𝜕𝑡𝐻𝑞 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐻𝑞 𝑘𝑖1⋯𝑖𝑟 = 𝐜𝑞 𝑖1⋯𝑖𝑟 , 𝑓𝑜𝑟 { 1 ≀ 𝑞 ≀ 𝑆 + 2 , 0 ≀ 𝑟 ≀ 𝑆 − 𝑞 + 1 , (14) We note that (11) for 𝑞 = 0, 0 ≀ 𝑟 ≀ 𝑆 + 2 are exactly the Equations (10) 1 of the mass block. Second Step: Let us explicitate the subset of Equation (14) with 𝑞 = 1, i.e., HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 191 𝜕𝑡ￜ̃ￜ1 𝑖1⋯𝑖𝑟 + 𝜕𝑘𝐻1 𝑘𝑖1⋯𝑖𝑟 = 𝐜1 𝑖1⋯𝑖𝑟 , 𝑓𝑜𝑟 0 ≀ 𝑟 ≀ 𝑆 . It is easy to recognize that these equations are equivalent to (10) 3 because from Equation (13) we desume that 𝐻1 𝑖1⋯𝑖𝑟 = 𝐹𝑅 𝑖1⋯𝑖𝑟 + 𝐹𝑉 𝑖1⋯𝑖𝑟 . After that, the first condition in (14) 2 must be replaced by 2 ≀ 𝑞 ≀ 𝑆 + 2. But when 𝑞 = 𝑆 + 2, the second condition in (14) 2 becomes 0 ≀ 𝑟 ≀ −1. It follows that the first condition in (14) 2 must be replaced by 2 ≀ 𝑞 ≀ 𝑆 + 1. Let us explicitate now the subset of Equation (14) with 𝑞 = 𝑆 − 𝑟 + 1; from the condition (14) 2 we see that this can be done only when { 2 ≀ 𝑆 − 𝑟 + 1 ≀ 𝑆 + 1 , 0 ≀ 𝑟 ≀ 𝑟 , , i.e., 0 ≀ 𝑟 ≀ 𝑆 − 1. But, for 0 ≀ 𝑟 ≀ 𝑆 − 1 we have also; 𝐻𝑆−𝑟+1 𝑖1⋯𝑖𝑟 = 𝐹𝑉 𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆−𝑟𝑙𝑆−𝑟 + 𝐹𝑅 𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆−𝑟𝑙𝑆−𝑟 . It follows that, from each equation of the subset of Equation (14) with 𝑞 = 𝑆 − 𝑟 + 1 we can subtract Equations (10) 4 and obtain (10) 5. So, now the equations are (10) 1−5, (11) for 𝑟 = 𝑆 − 𝑞 + 2, 1 ≀ 𝑞 ≀ 𝑆 + 2, (14) for { 2 ≀ 𝑞 ≀ 𝑆 + 1 , 0 ≀ 𝑟 ≀ 𝑆 − 𝑞 , . But, for 𝑞 = 𝑆 + 1 the second of these conditions becomes 0 ≀ 𝑟 ≀ −1; so the first condition must be replaced by 2 ≀ 𝑞 ≀ 𝑆. The corresponding equations are the above reported (10) 7, while (11) for 𝑟 = 𝑆 − 𝑞 + 2, 1 ≀ 𝑞 ≀ 𝑆 + 2 is the above reported (10) 6. This completes the proof. We conclude this section noting that, by changing index in (10) 6 according to the law 𝑞 = 𝑆 + 2 − 𝑟 and by taking into account (4) 1, it becomes; 𝜕𝑡𝐻0 𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆+1−𝑟𝑙𝑆+1−𝑟 + 𝜕𝑘𝐻0 𝑘𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆+1−𝑟𝑙𝑆+1−𝑟 = 𝐜𝑆+2−𝑟 𝑖1⋯𝑖𝑟 , (15) for 0 ≀ 𝑟 ≀ 𝑆 + 1. From (3) 2 we see that 𝐻0 𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆+1−𝑟𝑙𝑆+1−𝑟, 𝐻0 𝑘𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆+1−𝑟𝑙𝑆+1−𝑟 are respectively equal to 𝐹𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆+2−𝑟𝑙𝑆+2−𝑟 , 𝐹𝑘𝑖1⋯𝑖𝑟𝑙1𝑙1⋯𝑙𝑆+2−𝑟𝑙𝑆+2−𝑟 plus terms of the rotational and vibrational modes. So, in the subcase without these rotational and vibrational modes, Equation (15) is the counterpart of (10) 4,5 for the mass block, obviously with 𝑆 + 2 instead of 𝑆. In this way the second step of the above flowchart has been obtained, i.e., that the non relativistic limit of the field Equations (6) gives exactly those of the classical model (1) proposed in Pennisi and Ruggeri (2017) [11]; moreover, a further information has been achieved, i.e., how the classical Equations (1) must be interrupted to obtain a model with a finite set of equations. 3. The Particular Case S=0 In this case the Equations (6) and (7) become: 𝜕𝛌𝐎 𝛌 = 0 , 𝜕𝛌𝐎 𝛌𝛌1 = 0 , 𝜕𝛌𝐎 𝛌𝛌1𝛌2 = 𝐌𝛌1𝛌2 , 𝜕𝛌𝐎𝑉 𝛌 = 𝐌𝑉 , (16) 𝐎𝛌 = 𝑚 𝑐 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓 𝑝𝛌𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ , 𝐎𝛌𝛌1 = 𝑐 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓 𝑝𝛌𝑝𝛌1 (1 + ℐ 𝑚 𝑐2 )𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ , 𝐎𝛌𝛌1𝛌2 = 𝑐 𝑚 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓 𝑝𝛌𝑝𝛌1𝑝𝛌2 (1 + 2 ℐ 𝑚 𝑐2 )𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ , 𝐎𝑉 𝛌 = 𝑚 𝑐 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓 𝑝𝛌 2 ℐ𝑉 𝑚 𝑐2 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ . (17) This is the 16 moments model which is present in Carrisi and Pennisi (2019) [9]. If we take off the trace of the third equation, as it was done in Pennisi and Ruggeri (2017) [11], we obtain a 15 moments model which is the relativistic counterpart of Equations (12) in Arima et al. (2018) [1]. Its non relativistic limit can be desumed from the above Equations (10). • In particular, (10) 1 gives 10 equations of the mass block, i.e., (12) 1−3 of Arima et al. (2018) [1]. • Equations (10) 2,3 are to be considered only for 𝑟 = 0 and give Equations (12) 4−5 of Arima et al. (2018) [1], i.e., HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 192 the balance equations of the vibrational and rotational energies respectively. • Equations (10) 4,5 aren’t to be considered because they hold only for the empty set 0 ≀ 𝑟 ≀ −1. • Equations (10) 6 have to be considered only for 𝑞 = 1 and 𝑞 = 2. With the first of these values we obtain (12) 6 of Arima et al. (2018) [1]; with 𝑞 = 2 we obtain the hybrid equation which is present in Equations (3) 6 of Carrisi and Pennisi (2019) [9]. • Equation (10) 7 must not to be considered because it holds only for the empty set 2 ≀ 𝑞 ≀ 0, 0 ≀ 𝑟 ≀ −𝑞. So the third step of the above flowchart has been obtained, i.e., the subsystem with 16 moments and it is exactly the model considered in Carrisi and Pennisi (2019) [9]; moreover, if we take off the trace of the third equation, as it was done in Pennisi and Ruggeri (2017) [11], we obtain a further subsytem, i.e., the 15 moments model which is the relativistic counterpart of Equations (12) in Arima et al. (2018) [1], one of the two model with a finite number of equations proposed there. 4. The 7 Moments Model This case is presented in subsection 3.5 of Arima et al. (2018) [1] and is described by the balance equations; 𝜕𝑡𝐹 + 𝜕𝑘𝐹 𝑘 = 0 , 𝜕𝑡𝐹 𝑖1 + 𝜕𝑘𝐹 𝑘𝑖1 = 0 , 𝜕𝑡𝐹 𝑙𝑙 + 𝜕𝑘𝐹 𝑘𝑙𝑙 = −𝑃𝑉 𝑙𝑙 − 𝑃𝑅 𝑙𝑙 , 𝜕𝑡𝐹𝑉 + 𝜕𝑘𝐹𝑉 𝑘 = 𝑃𝑉 𝑙𝑙 , 𝜕𝑡𝐹𝑅 + 𝜕𝑘𝐹𝑅 𝑘 = 𝑃𝑅 𝑙𝑙 . (18) Its relativistic counterpart cannot be written as (6) but as; 𝜕𝛌𝐎 𝛌 = 0 , 𝜕𝛌𝐎 𝛌𝛌1 = 0 , 𝜕𝛌𝐎 𝛌𝛌1𝛌2𝑔𝛌1𝛌2 = 𝐌 𝛌1𝛌2𝑔𝛌1𝛌2 , 𝜕𝛌𝐎𝑉 𝛌 = 𝐌𝑉 . (19) In fact, 1 𝑐2 𝐎𝛌𝛌1𝛌2𝑔𝛌1𝛌2 = 𝐎 𝛌 + 𝐎𝑅 𝛌 + 𝐎𝑉 𝛌 so that the third equation can be substituted by; 𝜕𝛌𝐎𝑅 𝛌 = 𝐌𝑅 = 𝑑𝑒𝑓 1 𝑐2 𝐌𝛌1𝛌2𝑔𝛌1𝛌2 − 𝐌𝑉 , (20) which is the counterpart of Equation (19) 4 with the rotational mode instead of the vibrational one. The non relativistic limit of Equations (19) 1,2 has been calculated in Equation (17) of Pennisi and Ruggeri (2017) [11] and is; 𝜕𝑡𝐹 + 𝜕𝑘𝐹 𝑘 = 0 , 𝜕𝑡𝐹 𝑖1 + 𝜕𝑘𝐹 𝑘𝑖1 = 0 , 𝜕𝑡𝐺 𝑙𝑙 + 𝜕𝑘𝐺 𝑘𝑙𝑙 = 0 , (21) with 𝐺𝑙𝑙 = 𝐹𝑙𝑙 + 𝐹𝑉 𝑙𝑙 + 𝐹𝑅 𝑙𝑙 , 𝐺𝑘𝑙𝑙 = 𝐹𝑘𝑙𝑙 + 𝐹𝑉 𝑘𝑙𝑙 + 𝐹𝑅 𝑘𝑙𝑙 . Now, the non relativistic limits of (19) 3 and (20) are respectively (18) 4,5. By subtracting them from (21) 3, we see that (21) become (18) 1−3. This completes our proof. So the last step of the above flowchart has been obtained, i.e., the subsystem with 7 moments (19) which is the relativistic counterpart of Equations (18) which describe the second and last example with a finite number of equations proposed in Arima et al. (2018) [1]. 5. The Closure of the New Relativistic Field Equations By using the Maximum Entropy Pinciple, as in Pennisi and Ruggeri (2017) [11] and recently in Mentrelli and Ruggeri (2021) [12], we find that the distribution funcion 𝑓 has the form; 𝑓 = 𝑒 −1− 1 𝑘𝐵 𝜒 𝑀𝑖𝑡ℎ 𝜒 = 1 𝑚𝑟−1 (1 + 𝑟 ℐ 𝑚 𝑐2 ) 𝑝𝛌1⋯𝑝𝛌𝑟 𝜆𝛌1⋯𝛌𝑟 + 1 𝑚𝑠−1 ( 2 ℐ𝒱 𝑚 𝑐2 ) 𝑝𝛌1⋯𝑝𝛌𝑠 𝜇𝛌1⋯𝛌𝑠 , Where summation over the indexes 𝑟 and 𝑠 is implied and 𝑘𝐵 is the Boltzmann constant. So, if we define the 4- potential; ℎ′𝛌 = − 𝑘𝐵𝑐 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓 𝑝𝛌𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ , We have; 𝐎𝛌𝛌1⋯𝛌𝑟 = 𝜕 ℎ′𝛌 𝜕 𝜆𝛌1⋯𝛌𝑟 , 𝐎𝑉 𝛌𝛌1⋯𝛌𝑠 = 𝜕 ℎ′𝛌 𝜕 𝜇𝛌1⋯𝛌𝑠 . Since ℎ′𝛌 𝜉𝛌 is a convex function of the Lagrange multipliers for whatever time-like 4-vector 𝜉𝛌, it follows that HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 193 the field Equations (6) are symmetric hyperbolic. This result is guaranteed also if 𝑓 is substituted by its Taylor expansion up to whatever fixed order greater than 2 but with the Lagrange multipliers as independent variables. Moreover, as usual in Rational Extended Thermodynamics, the closure is determined except for the 4-potentials ℎ′𝛌 . Now people doesn’t like these variables and requires that the cloure is expressed in terms of variables with a clear physical meaning. In reality this is not reasonable: It is like saying, in the geometric framework that the parametric equation of a curve or a surface are not significant. Another objection is made that, not knowing these variables, we cannot know their boundary values necessary to solve the field equations. This objection is also unfounded because, knowing the law that binds the physical variables to the Lagrange multipliers, from the boundary conditions for the phyisical variables we can deduce those for the Lagrange multipliers and then solve the field equations. However, in order to meet the commonly accepted taste, we will make the change of variables in the next subsections, from the Lagrange multipliers to the phyisical variables. Obviouly, hyperbolicity is not compromised by an invertible change of independent variables. Unfortunately, up to now nobody was able to do this exactly and we too will be content to do it in an approximate way, at first order with respect to equilibrium. Due to this approximation, the hyperbolicity requirement will be satisfied only in a neighborhood of equilibrium called "the hyperbolicity zone" (See [13-16]). But this cannot be adduced as proof of the weakness of the model; it is only a proof of our mathematical inability to perform this transformation without introducing approximations. We certainly cannot expect Nature to bow to our mathematical weakness. 5.1. The Variables at Equilibrium Equilibrium is defined as the state governed only by (6) 1 with 𝑟 = 0,1, i.e., the conservation laws of mass and of momentum-energy (Euler Equations), i.e., the subsystem of (6) with 𝑆 = −1 in the sense of Boillat and Ruggeri (1997) [8]. It follows that 𝜆𝛌1⋯𝛌𝑟 𝐞 = 0, 𝜇𝛌1⋯𝛌𝑠 𝐞 = 0 for 𝑟 = 2,⋯ 𝑆 + 2, 𝑠 = 0,⋯ 𝑆. Moreover, from the Representation Theorems we have 𝐎𝛌 = 𝑚 𝑛 𝑈𝛌 , 𝐎𝛌𝛜 = 𝑒 𝑐2 𝑈𝛌𝑈𝛜 + 𝑝 ℎ𝛌𝛜 , 𝑈𝛌𝑈𝛌 = 𝑐 2 , ℎ𝛌𝛜 = −𝑔𝛌𝛜 + 1 𝑐2 𝑈𝛌𝑈𝛜 , whose phyisical meaning is obvious: 𝑛 is the number density, 𝑝 is the pressure and 𝑒 the energy. From (6) 1 with 𝑟 = 0 it follows that 𝜆𝛌 𝐞 is parallel to 𝑈𝛌; so, by calling 1 𝑇 the coefficient, we have that; 𝜆𝛌 𝐞 = 𝑈𝛌 𝑇 . The physical meaning of 𝑇 is evident; it is the absolute temperature. Of (6) 1 with 𝑟 = 0,1 there remain; 𝑚 𝑛 𝑈𝛌 = 𝑚 𝑐 𝑒 −1− 𝑚 𝑘𝐵 𝜆𝐞 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑒 − 𝑝𝜇𝑈𝜇 𝑚 𝑐2 (1+ ℐ 𝑚 𝑐2 ) 𝑝𝛌𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ 𝑒 𝑐2 𝑈𝛌𝑈𝛜 + 𝑝 ℎ𝛌𝛜 = 𝑐 𝑒 −1− 𝑚 𝑘𝐵 𝜆𝐞 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑒 − 𝑝𝜇𝑈𝜇 𝑚 𝑐2 (1+ ℐ 𝑚 𝑐2 ) 𝑝𝛌𝑝𝛜 (1 + ℐ 𝑚 𝑐2 ) ⋅ ⋅ 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ . The first one of these equations must be imposed only through its contraction with 𝑈𝛌 and the second one through its contractions with 𝑈𝛌𝑈𝛜 and ℎ𝛌𝛜. So we obtain; 𝑛 = 4 𝜋𝑚3 𝑒 −1− 𝑚 𝑘𝐵 𝜆𝐞 𝐜2,1 ∗ , 𝑝 = 𝑛 𝑚 𝑐2 𝛟 = 𝑛 𝑘𝐵 𝑇 , 𝑒 = 𝑛 𝑚 𝑐2 𝐜2,2 ∗ (1+ ℐ 𝑚 𝑐2 ) 𝐜2,1 ∗ , (22) Where the integration in 𝑑 ￜ⃗⃗ￜ has been performed and overlined terms denote that they are multiplied by 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) and then integrated in 𝑑 ℐ𝑅 𝑑 ℐ𝑉. Moreover, in (22) 2,3 the term 𝜆𝐞 has been eliminated by use of (22) 1, 𝛟 = 𝑚 𝑐2 𝑘𝐵𝑇 . 𝐜𝑚,𝑛(𝛟) = ∫ + ∞ 0 𝑒− 𝛟 cosh 𝑠sinh𝑚 𝑠 cosh𝑛 𝑠 𝑑 𝑠 𝛟∗ = 𝛟 (1 + ℐ 𝑚 𝑐2 ) , 𝐜𝑚,𝑛 ∗ = 𝐜𝑚,𝑛(𝛟 ∗) . The Equation (22) 3 is the generalization of the Synge energy to the case of polyatomic gases with rotational and vibrational modes; in the case with only one mode it is the same of Equation (42) of Pennisi and Ruggeri (2017) [11]. The other functions in Equation (7) don’t play a role at equilibrium but nothing prevent us from calculating them and HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 194 they will be useful in the sequel. They are: 𝐎𝐞 𝛌1⋯𝛌𝑟+1 = ∑ [ 𝑟+1 2 ] 𝑞=0 𝑎𝑞,𝑟(𝛟)ℎ (𝛌1𝛌2⋯ℎ𝛌2𝑞−1𝛌2𝑞𝑈𝛌2𝑞+1⋯𝑈𝛌𝑟+1) , 𝐎𝑉𝐞 𝛌1⋯𝛌𝑠+1 = ∑ [ 𝑠+1 2 ] 𝑞=0 𝑎𝑞,𝑠 𝑉 (𝛟)ℎ(𝛌1𝛌2⋯ℎ𝛌2𝑞−1𝛌2𝑞𝑈𝛌2𝑞+1⋯𝑈𝛌𝑠+1) , (23) Where; 𝑎𝑞,𝑟 = ( 𝑟 + 1 2𝑞 ) 𝑛 𝑚 2𝑞 + 1 𝑐2𝑞 𝐜2,1 ∗ 𝐜2𝑞+2 ,𝑟+1−2𝑞 ∗ (1 + 𝑟 ℐ 𝑚 𝑐2 ) , 𝑎𝑞,𝑠 𝑉 = ( 𝑠 + 1 2𝑞 ) 𝑛 𝑚 2𝑞 + 1 𝑐2𝑞 𝐜2,1 ∗ 𝐜2𝑞+2 ,𝑠+1−2𝑞 ∗ ( 2 ℐ𝒱 𝑚 𝑐2 ) . These expressions have been found by using the techniques exposed in Carrisi and Pennisi (2013) [17]. We note that (23) 1 for 𝑟 = 0,1 gives the above reported 𝐎𝐞 𝛌1 = 𝑉𝛌1, 𝐎𝐞 𝛌1𝛌2 = 𝑇𝐞 𝛌1𝛌2 because:: 𝑎0,0 = 𝑛 𝑚 , 𝑎0,1 = 𝑒 𝑐2 , 𝑎1,1 = 𝑛 𝑚 𝑐2 3 𝐜4,0 ∗ (1 + ℐ 𝑚 𝑐2 ) 𝐜2,1 ∗ = 𝑝 , Where for the last one we have used the property 𝛟𝐜4,0(𝛟) = 3 𝐜2,1(𝛟) from which it follows 𝛟 (1 + ℐ 𝑚 𝑐2 ) 𝐜4,0 ∗ = 3 𝐜2,1 ∗ . Moreover, (23) 1 for 𝑟 = 2 and (23) 2 for 𝑠 = 0 give: 𝐎𝐞 𝛌1𝛌2𝛌3 = 𝑎0,2𝑈 𝛌1𝑈𝛌2𝑈𝛌3 + 𝑎1,2ℎ (𝛌1𝛌2𝑈𝛌3⋯𝑈𝛌3) , 𝐎𝑉𝐞 𝛌1 = 𝑎0,0 𝑉 (𝛟)𝑈𝛌1 . These expressions are the same found in Equation (13) of Carrisi and Pennisi (2019) [9] and the first one of these, in the particular case with only one mode, is the same of Equation (48) in Pennisi and Ruggeri (2017) [11] because; 𝑎0,2 = 𝐎1 0 , 𝑎1,2 = 3 𝐎11 0 , 𝑎0,0 𝑉 = 𝑛 𝑚 𝐜2,1 ∗ (1 + 2 ℐ𝒱 𝑚 𝑐2 ) 𝐜2,1 ∗ = 𝑐3 𝑚 𝑛 𝐻𝑉 = 𝛟 𝑚 𝑛 𝑐 𝐵10 . 5.2. The Linear Deviation from Equilibrium At a first step we will consider as first order deviations from equilibrium the variables 𝜋 (dynamic pressure), 𝑞𝛌 (heat flux), 𝑡<𝛌𝛜>3 (viscous deviatoric stress), 𝜆𝛌1⋯𝛌𝑟 for 𝑟 = 2 ,⋯ , 𝑆 + 2, and 𝜇𝛌1⋯𝛌𝑠 for 𝑠 = 0 ,⋯ , 𝑆. These variables are constrained by 𝑈𝛌𝑞 𝛌 = 0, 𝑈𝛌𝑡 <𝛌𝛜>3 = 0, 𝑔𝛌𝛜𝑡 <𝛌𝛜>3 = 0. Also the remaining Lagrange multipliers can be found in terms of a corresponding set of components of 𝐎𝛌𝛌1⋯𝛌𝑟 and 𝐎𝑉 𝛌𝛌1⋯𝛌𝑠 (for example, those components whose non relativistic limit gives the variables that, in the classical model, are derivated with respect to time; or, more precisely, their deviations from equilibrium); but this further change can be done in a second step, if it will be considered necessary. Since this amounts only in some complicated systems, we will refrain to report them here. The change of variables is performed in the following way: We will use; 𝑉𝛌 − 𝑉𝐞 𝛌 = 0 , 𝑇𝛌𝛜 − 𝑇𝐞 𝛌𝛜 = 𝜋 ℎ𝛌𝛜 + 2 𝑐2 𝑞(𝛌𝑈𝛜) + 𝑡<𝛌𝛜>3 (24) to determine 𝜆 − 𝜆𝐞 , 𝜆𝛜 − 𝜆𝛜 𝐞 , 𝜆<𝛜𝛟> in terms of 𝑛, 𝛟, 𝑈𝛌 , 𝜋, 𝑞𝛌, 𝑡<𝛌𝛜>3 , 𝜇 = 1 4 𝑔𝛌𝛜𝜆𝛌𝛜, 𝜆𝛌1⋯𝛌𝑟 for 𝑟 = 3 ,⋯ , 𝑆 + 2 and 𝜇𝛌1⋯𝛌𝑠 for 𝑠 = 0 ,⋯ , 𝑆. After that, we will substitute these values in 𝐎𝛌𝛌1⋯𝛌𝑟 − 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟 and 𝐎𝑉 𝛌𝛌1⋯𝛌𝑠 − 𝐎𝐞𝑉 𝛌𝛌1⋯𝛌𝑠 so determining the closure, as a consequence, the mass and energy-momentum conservation laws (6) 1 with 𝑟 = 0,1 will be the usual equations with; 𝑉𝛌 = 𝑚 𝑛 𝑈𝛌 , 𝐎𝛌𝛜 = 𝑒 𝑐2 𝑈𝛌𝑈𝛜 + (𝑝 + 𝜋) ℎ𝛌𝛜 + 2 𝑐2 𝑞(𝛌𝑈𝛜) + 𝑡<𝛌𝛜>3 . Now Equations (24) become; 𝐎𝐞 𝛌 (𝜆 − 𝜆𝐞) + 𝐎𝐞 𝛌𝜈 (𝜆𝜈 − 𝜆𝜈 𝐞) + 𝐎𝐞 𝛌𝛟𝛿 𝜆<𝛟𝛿> + (𝐎𝐞 𝛌𝛟𝛿 𝑔𝛟𝛿) 𝜇 + +∑𝑆+2𝑟′=3 𝐎𝐞 𝛌𝛜1⋯𝛜𝑟′ 𝜆𝛜1⋯𝛜𝑟′ + ∑ 𝑆 𝑠′=0 𝐎𝑉𝐞 𝛌𝛜1⋯𝛜𝑠′ 𝜇𝛜1⋯𝛜𝑠′ = 0 , 𝐎𝐞 𝛌𝛜 (𝜆 − 𝜆𝐞) + 𝐎11 𝛌𝛜𝜈 𝑚 (𝜆𝜈 − 𝜆𝜈 𝐞) + 𝐎12 𝛌𝛜𝛟𝛿 𝑚 𝜆<𝛟𝛿> + ( 𝐎12 𝛌𝛜𝛟𝛿 𝑚 𝑔𝛟𝛿) 𝜇 + +∑𝑆+2𝑟′=3 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ + ∑ 𝑆 𝑠′=0 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′ = = − 𝑘𝐵 𝑚 (𝜋 ℎ𝛌𝛜 + 2 𝑐2 𝑞(𝛌𝑈𝛜) + 𝑡<𝛌𝛜>3) . (25) HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 195 Similarly, Equations (7) give; 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟 (𝜆 − 𝜆𝐞) + 1 𝑚 𝐎𝑟1 𝛌𝛌1⋯𝛌𝑟𝜈 (𝜆𝜈 − 𝜆𝜈 𝐞) + 1 𝑚 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 𝜆<𝛟𝛿> + 1 𝑚 (𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 𝑔𝛟𝛿) 𝜇 + + 1 𝑚 ∑𝑆+2𝑟′=3 𝐎𝑟𝑟′ 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑟′ 𝜆𝛜1⋯𝛜𝑟′ + 1 𝑚 ∑𝑆𝑠′=0 𝐵𝑟𝑠′ 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑠′ 𝜇𝛜1⋯𝛜𝑠′ = − 𝑘𝐵 𝑚 (𝐎𝛌𝛌1⋯𝛌𝑟 − 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟) , 𝐎𝑉𝐞 𝛌𝛌1⋯𝛌𝑠 (𝜆 − 𝜆𝐞) + 1 𝑚 𝐵𝑠1 𝛌𝛌1⋯𝛌𝑠𝜈 (𝜆𝜈 − 𝜆𝜈 𝐞) + 1 𝑚 𝐵𝑠2 𝛌𝛌1⋯𝛌𝑠𝛟𝛿 𝜆<𝛟𝛿> + 1 𝑚 (𝐵𝑠2 𝛌𝛌1⋯𝛌𝑠𝛟𝛿 𝑔𝛟𝛿) 𝜇 + + 1 𝑚 ∑𝑆+2𝑟′=3 𝐵𝑠𝑟′ 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑟′ 𝜆𝛜1⋯𝛜𝑟′ + 1 𝑚 ∑𝑆𝑠′=0 𝐶𝑠𝑠′ 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑠′ 𝜇𝛜1⋯𝛜𝑠′ = − 𝑘𝐵 𝑚 (𝐎𝑉 𝛌𝛌1⋯𝛌𝑠 − 𝐎𝑉𝐞 𝛌𝛌1⋯𝛌𝑠) . (26) In Equations (25) and (26) the new tensors appear: 𝐎𝑟𝑟′ 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑟′ = 𝑐 𝑚𝑟+𝑟′−2 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓𝐞 𝑝 𝛌𝑝𝛌1⋯𝑝𝛌𝑟𝑝𝛜1⋯𝑝𝛜𝑟′ ⋅ ⋅ (1 + 𝑟 ℐ 𝑚 𝑐2 ) (1 + 𝑟′ ℐ 𝑚 𝑐2 )𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ , 𝐵𝑟𝑠 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑠 = 𝑐 𝑚𝑟+𝑠−2 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓𝐞 𝑝 𝛌𝑝𝛌1⋯𝑝𝛌𝑟𝑝𝛜1⋯𝑝𝛜𝑠 (1 + 𝑟 ℐ 𝑚 𝑐2 ) 2 ℐ𝑉 𝑚 𝑐2 ⋅ ⋅ 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ , 𝐶𝑠𝑠′ 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑠′ = 𝑐 𝑚𝑠+𝑠′−2 ∫ ℜ3 ∫ + ∞ 0 ∫ + ∞ 0 𝑓𝐞 𝑝 𝛌𝑝𝛌1⋯𝑝𝛌𝑠𝑝𝛜1⋯𝑝𝛜𝑠′ ( 2 ℐ𝑉 𝑚 𝑐2 ) 2 ⋅ ⋅ 𝜙(ℐ𝑅) 𝜓(ℐ𝑉) 𝑑 ℐ𝑅 𝑑 ℐ𝑉 𝑑 ￜ⃗⃗ￜ , (27) Their expressions can be found with the procedure used above or, simply by comparing the definitons of 𝐎𝑟𝑟′ 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑟′ and 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟 and noting that the former can somehow be obtained from the latter by replacing 𝑟 with 𝑟 + 𝑟′ and multiplying by 𝑚 (1 + 𝑟′ ℐ 𝑚 𝑐2 ). So we obtain the first one of the following relations, with coefficients given by Equation (29) 1: 𝐎𝑟𝑟′ 𝛌1⋯𝛌𝑟+𝑟′+1 = ∑ [ 𝑟+𝑟′+1 2 ] 𝑞=0 𝑎𝑞,𝑟,𝑟′(𝛟)ℎ (𝛌1𝛌2⋯ℎ𝛌2𝑞−1𝛌2𝑞𝑈𝛌2𝑞+1⋯𝑈𝛌𝑟+𝑟′+1) , 𝐵𝑟𝑠′ 𝛌1⋯𝛌𝑟+𝑠′+1 = ∑ [ 𝑟+𝑠′+1 2 ] 𝑞=0 𝑏𝑞,𝑟,𝑠′(𝛟)ℎ (𝛌1𝛌2⋯ℎ𝛌2𝑞−1𝛌2𝑞𝑈𝛌2𝑞+1⋯𝑈𝛌𝑟+𝑠′+1) 𝐶𝑠𝑠′ 𝛌1⋯𝛌𝑠+𝑠′+1 = ∑ [ 𝑠+𝑠′+1 2 ] 𝑞=0 𝑐𝑞,𝑠,𝑠′(𝛟)ℎ (𝛌1𝛌2⋯ℎ𝛌2𝑞−1𝛌2𝑞𝑈𝛌2𝑞+1⋯𝑈𝛌𝑠+𝑠′+1) . (28) 𝑎𝑞,𝑟,𝑟′ = ( 𝑟 + 𝑟′ + 1 2𝑞 ) 𝑚2𝑛 2𝑞+1 𝑐2𝑞 𝐜2,1 ∗ 𝐜2𝑞+2 ,𝑟+𝑟′+1−2𝑞 ∗ (1 + 𝑟 ℐ 𝑚 𝑐2 ) (1 + 𝑟′ ℐ 𝑚 𝑐2 ) , 𝑏𝑞,𝑟,𝑠′ = ( 𝑟 + 𝑠′ + 1 2𝑞 ) 𝑚2𝑛 2𝑞+1 𝑐2𝑞 𝐜2,1 ∗ 𝐜2𝑞+2 ,𝑟+𝑠′+1−2𝑞 ∗ (1 + 𝑟 ℐ 𝑚 𝑐2 ) 2 ℐ𝒱 𝑚 𝑐2 , 𝑐𝑞,𝑠,𝑠′ = ( 𝑠 + 𝑠′ + 1 2𝑞 ) 𝑚2𝑛 2𝑞+1 𝑐2𝑞 𝐜2,1 ∗ 𝐜2𝑞+2 ,𝑠+𝑠′+1−2𝑞 ∗ ( 2 ℐ𝒱 𝑚 𝑐2 ) 2 . (29) Similarly, by comparing the definitons of 𝐵𝑟𝑠 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑠 and 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟, we note that the former can somehow be obtained from the latter by replacing 𝑟 with 𝑟 + 𝑠 and multiplying by 2 ℐ𝒱 𝑚 𝑐2 . So we obtain (28) 2 with coefficients given by Equation (29) 2. Finaly, by comparing the definitons of 𝐶𝑠𝑠′ 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑠′ and 𝐎𝐞 𝛌𝛌1⋯𝛌𝑠, we note that the former can somehow be obtained from the latter by replacing 𝑠 with 𝑠 + 𝑠′ and multiplying by 1 𝑚 2 ℐ𝒱 𝑚 𝑐2 . So we obtain (28) 3 with coefficients given by (29) 3. We note that (27) 1 with 𝑟 = 1, 𝑟′ = 1 gives the expression (15) 1 of Carrisi and Pennisi (2019) [9] multiplied by 𝑚2; (27) 1 with 𝑟 = 1, 𝑟′ = 2 gives the expression (15) 2 of Carrisi and Pennisi (2019) [9] multiplied by 𝑚2; (27) 1 with 𝑟 = 2, 𝑟′ = 2 gives the expression (15) 3 of Carrisi and Pennisi (2019) [9] multiplied by 𝑚2. Similarly, (27) 2 with 𝑟 = 1, 𝑠 = 0 gives 𝐵10 𝛌𝛌1 = 𝑚 𝑐 𝑇𝑉 𝛌𝛌1 , where 𝑇𝑉 𝛌𝛌1 is the expression (15) 4 of Carrisi and Pennisi (2019) [9]; (27) 2 with 𝑟 = 2, 𝑠 = 0 gives 𝐵20 𝛌𝛌1𝛌2 = 𝑚 𝑐 𝐎𝑉 𝛌𝛌1𝛌2, where 𝐎𝑉 𝛌𝛌1𝛌2 is the expression (15) 5 of Carrisi and Pennisi (2019) [9]. Finally, (27) 3 with 𝑠 = 0, 𝑠′ = 0 gives 𝐶00 𝛌 = 𝑚 𝑐 𝑉𝑉𝑉 𝛌 with 𝑉𝑉𝑉 𝛌 defined in (15) 6 of Carrisi and Pennisi (2019) [9]. By comparing the correspondent decompositions (28) with (16) of Carrisi and Pennisi (2019) [9], we see that we must have: HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 196 𝑎0,1,1 = 𝑚 2𝐵5 ; 𝑎1,1,1 = 𝑚 2𝐵4 ; 𝑎0,1,2 = 𝑚 2𝐵3 ; 𝑎1,1,2 = 2 𝑚 2𝐵2 ; 𝑎2,1,2 = 1 5 𝑚2𝐵1 ; 𝑎0,2,2 = 𝑚 2𝐵8 ; 𝑎1,2,2 = 10 3 𝑚2𝐵7 ; 𝑎2,2,2 = 𝑚 2𝐵6 ; 𝑏0,1,0 = 𝑚 𝑐 𝐵9 ; 𝑏1,1,0 = 𝑚 𝑐 𝐵10 ; 𝑏0,2,0 = 𝑚 𝑐 𝐎1𝑉 0 ; 𝑏1,2,0 = 3 𝑚 𝑐 𝐎11𝑉 0 ; 𝑐0,0,0 = 𝑚 𝑐 𝐵11 , with 𝐵1-𝐵11, 𝐎1𝑉 0 , 𝐎11𝑉 0 given by (17) of Carrisi and Pennisi (2019) [9]. This is confirmed by the present Equation (29). We are now ready to determine 𝜆 − 𝜆𝐞 , 𝜆𝛜 − 𝜆𝛜 𝐞 , 𝜆<𝛜𝛟> from (26) in terms of 𝑛. 𝛟, 𝑈𝛌 , 𝜋, 𝑞𝛌, 𝑡<𝛌𝛜>3 , 𝜇 = 1 4 𝑔𝛌𝛜𝜆𝛌𝛜, 𝜆𝛌1⋯𝛌𝑟 for 𝑟 = 3 ,⋯ , 𝑆 + 2 and 𝜇𝛌1⋯𝛌𝑠 for 𝑠 = 0 ,⋯ , 𝑆. After that, we will substitute them in (27) and obtain the requested closure. To this end, let us contract Equation (25) 1 with 𝑈𝛌 and Equation (25) 2 a first time with 𝑈𝛌𝑈𝛜 and a second time with ℎ𝛌𝛜; so we obtain 𝑛𝑐2(𝜆 − 𝜆𝐞) + 𝑒 𝑚 𝑈𝜇 (𝜆𝜇 − 𝑈𝜇 𝑇 ) + 1 𝑚 (𝐎1 0𝑐2 + 𝐎11 0 ) 𝑈𝜇𝑈𝜈𝜆<𝜇𝜈< = = − 𝑐2 𝑚 (𝐎1 0𝑐2 − 3𝐎11 0 ) 𝜇 − ∑𝑆+2𝑟′=3 𝑈𝛌 𝐎𝐞 𝛌𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 𝑈𝛌 𝐎𝑉𝐞 𝛌𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′ , 𝑒 𝑚 𝑐2(𝜆 − 𝜆𝐞) + 𝑐 4𝐵5 𝑈 𝜇 (𝜆𝜇 − 𝑈𝜇 𝑇 ) + ( 1 3 𝐵2𝑐 2 + 𝐵3𝑐 4)𝑈𝜇𝑈𝜈𝜆<𝜇𝜈> = = (𝐵2 − 𝐵3𝑐 2)𝑐4𝜇 − ∑𝑆+2𝑟′=3 𝑈𝛌𝑈𝛜 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 𝑈𝛌𝑈𝛜 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′ , (30) 𝑝 𝑚 (𝜆 − 𝜆𝐞) + 1 3 𝐵4𝑈 𝜇 (𝜆𝜇 − 𝑈𝜇 𝑇 ) + ( 1 3 𝐵2 + 1 9 𝐵1 𝑐2 )𝑈𝜇𝑈𝜈𝜆<𝜇𝜈> = = − 𝑘𝐵 𝑚2 𝜋 + 1 3 (𝐵1 − 𝐵2𝑐 2) 𝜇 − 1 3 ∑ 𝑆+2 𝑟′=3 ℎ𝛌𝛜 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − 1 3 ∑ 𝑆 𝑠′=0 ℎ𝛌𝛜 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′ . If we calculate this system in 𝜇 = 0, 𝜆𝛜1⋯𝛜𝑟′ = 0, 𝜇𝛜1⋯𝛜𝑠′ = 0 we obtain exactly the system (A.10) 1−3 of Pennisi and Ruggeri (2017) [11]. Obviously, the matrix of coefficients is the same of that reported in (A.11) 1 of Pennisi and Ruggeri (2017) [11], i.e., ￜ̃ￜ𝜋 = ( 𝑛𝑐2 𝑒 𝑚 1 𝑚 (𝐎1 0𝑐2 + 𝐎11 0 ) 𝑒 𝑚 𝑐2 𝑐4 𝐵5 1 3 𝐵2𝑐 2 + 𝐵3𝑐 4 𝑝 𝑚 1 3 𝐵4 1 3 𝐵2 + 1 9 𝐵1 𝑐2 ) . (31) So, we can define ￜ̃ￜ𝑖𝑗 𝜋 the algebraic complement of its element in the line 𝑖, coulumn 𝑗 and, by using the Kramer’ s theorem, we find; 𝜆 − 𝜆𝐞 = ￜ̃ￜ31 𝜋 |ￜ̃ￜ𝜋| (− 𝑘𝐵 𝑚2 𝜋 + 1 3 (𝐵1 − 𝐵2𝑐 2) 𝜇 − (32) 1 3 ∑ 𝑆+2 𝑟′=3 ℎ𝛌𝛜 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − 1 3 ∑ 𝑆 𝑠′=0 ℎ𝛌𝛜 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) + + ￜ̃ￜ21 𝜋 |ￜ̃ￜ𝜋| ((𝐵2 − 𝐵3𝑐 2)𝑐4𝜇 − ∑ 𝑆+2 𝑟′=3 𝑈𝛌𝑈𝛜 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 𝑈𝛌𝑈𝛜 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) + + ￜ̃ￜ11 𝜋 |ￜ̃ￜ𝜋| (− 𝑐2 𝑚 (𝐎1 0𝑐2 − 3𝐎11 0 ) 𝜇 − ∑ 𝑆+2 𝑟′=3 𝑈𝛌 𝐎𝐞 𝛌𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 𝑈𝛌 𝐎𝑉𝐞 𝛌𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) , 𝑈𝜇(𝜆𝜇 − 𝜆𝐞𝜇) = ￜ̃ￜ32 𝜋 |ￜ̃ￜ𝜋| (− 𝑘𝐵 𝑚2 𝜋 + 1 3 (𝐵1 − 𝐵2𝑐 2) 𝜇 − 1 3 ∑ 𝑆+2 𝑟′=3 ℎ𝛌𝛜 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − 1 3 ∑ 𝑆 𝑠′=0 ℎ𝛌𝛜 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) + HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 197 + ￜ̃ￜ22 𝜋 |ￜ̃ￜ𝜋| ((𝐵2 − 𝐵3𝑐 2)𝑐4𝜇 − ∑ 𝑆+2 𝑟′=3 𝑈𝛌𝑈𝛜 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 𝑈𝛌𝑈𝛜 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) + + ￜ̃ￜ21 𝜋 |ￜ̃ￜ𝜋| (− 𝑐2 𝑚 (𝐎1 0𝑐2 − 3𝐎11 0 ) 𝜇 − ∑ 𝑆+2 𝑟′=3 𝑈𝛌 𝐎𝐞 𝛌𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 𝑈𝛌 𝐎𝑉𝐞 𝛌𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) , 𝑈𝜇𝑈𝜈𝜆<𝜇𝜈> = ￜ̃ￜ33 𝜋 |ￜ̃ￜ𝜋| (− 𝑘𝐵 𝑚2 𝜋 + 1 3 (𝐵1 − 𝐵2𝑐 2) 𝜇 − 1 3 ∑ 𝑆+2 𝑟′=3 ℎ𝛌𝛜 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − 1 3 ∑ 𝑆 𝑠′=0 ℎ𝛌𝛜 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) + + ￜ̃ￜ23 𝜋 |ￜ̃ￜ𝜋| ((𝐵2 − 𝐵3𝑐 2)𝑐4𝜇 − ∑ 𝑆+2 𝑟′=3 𝑈𝛌𝑈𝛜 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 𝑈𝛌𝑈𝛜 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) + + ￜ̃ￜ13 𝜋 |ￜ̃ￜ𝜋| (− 𝑐2 𝑚 (𝐎1 0𝑐2 − 3𝐎11 0 ) 𝜇 − ∑ 𝑆+2 𝑟′=3 𝑈𝛌𝐎𝐞 𝛌𝛜1⋯𝛜𝑟′ 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 𝑈𝛌𝐎𝑉𝐞 𝛌𝛜1⋯𝛜𝑠′ 𝜇𝛜1⋯𝛜𝑠′) , If we calculate these expressions in 𝜇 = 0, 𝜆𝛜1⋯𝛜𝑟′ = 0, 𝜇𝛜1⋯𝛜𝑠′ = 0, we obtain exactly those reported in the equations subsequent to (61) of Pennisi and Ruggeri (2017) [11]. We consider now Equation (25) 1 contracted by ℎ𝛌 𝛿 and Equation (25) 2 contracted ℎ𝛌 𝛿 𝑈𝛜. So we obtain the system: ( 𝑝 𝑚 2 𝐎11 0 𝑚 1 3 𝐵4𝑐 2 2 3 𝐵2 𝑐 2 ) ( ℎ𝛿𝜇 (𝜆𝜇 − 𝑈𝜇 𝑇 ) ℎ𝛿𝜇𝑈𝜈𝜆<𝜇𝜈> ) = = ( − ℎ𝛌 𝛿 ∑𝑆+2𝑟′=3 𝐎𝐞 𝛌𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ℎ𝛌 𝛿 ∑𝑆𝑠′=0 𝐎𝑉𝐞 𝛌𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′ − 𝑘𝐵 𝑚2 𝑞𝛿 − ∑𝑆+2𝑟′=3 ℎ𝛌 𝛿 𝑈𝛜 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 ℎ𝛌 𝛿 𝑈𝛜 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′ . ) By calling ￜ̃ￜ𝑞 the determinant of the coefficients we can use the Kramer’ s theorem and find; ℎ𝛿𝜇 (𝜆𝜇 − 𝑈𝜇 𝑇 ) = = − 2 𝐎11 0 𝑚 ￜ̃ￜ𝑞 (− 𝑘𝐵 𝑚2 𝑞𝛿 − ∑ 𝑆+2 𝑟′=3 ℎ𝛌 𝛿 𝑈𝛜 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 ℎ𝛌 𝛿 𝑈𝛜 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) + + 2 3 ￜ̃ￜ𝑞 𝐵2 𝑐 2 (− ℎ𝛌 𝛿 ∑ 𝑆+2 𝑟′=3 𝐎𝐞 𝛌𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ℎ𝛌 𝛿 ∑ 𝑆 𝑠′=0 𝐎𝑉𝐞 𝛌𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) , ℎ𝛿𝜇𝑈𝜈𝜆<𝜇𝜈> = 𝑝 𝑚 ￜ̃ￜ𝑞 (− 𝑘𝐵 𝑚2 𝑞𝛿 − ∑ 𝑆+2 𝑟′=3 ℎ𝛌 𝛿𝑈𝛜 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 ℎ𝛌 𝛿𝑈𝛜 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) , + 1 3 ￜ̃ￜ𝑞 𝐵4 𝑐 2 (− ℎ𝛌 𝛿 ∑ 𝑆+2 𝑟′=3 𝐎𝐞 𝛌𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ℎ𝛌 𝛿 ∑ 𝑆 𝑠′=0 𝐎𝑉𝐞 𝛌𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) . If we calculate these expressions in 𝜇 = 0, 𝜆𝛜1⋯𝛜𝑟′ = 0, 𝜇𝛜1⋯𝛜𝑠′ = 0, we obtain exactly Equation (A.14) 1,2 of Pennisi and Ruggeri (2017) [11]. Finally, Equation (25) 2 contracted ℎ𝛌 <𝛿 ℎ𝛜 𝜃>3 gives ℎ𝜇 <𝛿 ℎ𝜈 𝜃>3 𝜆<𝜇𝜈> = HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 198 = 15 2 𝐵1 (− 𝑘𝐵 𝑚2 𝑡<𝛿𝜃>3 − ∑ 𝑆+2 𝑟′=3 ℎ𝛌 <𝛿 ℎ𝛜 𝜃>3 𝐎1𝑟′ 𝛌𝛜𝛜1⋯𝛜𝑟′ 𝑚 𝜆𝛜1⋯𝛜𝑟′ − ∑ 𝑆 𝑠′=0 ℎ𝛌 <𝛿 ℎ𝛜 𝜃>3 𝐵1𝑠′ 𝛌𝛜𝛜1⋯𝛜𝑠′ 𝑚 𝜇𝛜1⋯𝛜𝑠′) . If we calculate these expressions in 𝜇 = 0, 𝜆𝛜1⋯𝛜𝑟′ = 0, 𝜇𝛜1⋯𝛜𝑠′ = 0, we obtain exactly Equation (A.15) 1 of Pennisi and Ruggeri (2017) [11]. Now we have to substitute all these results in (26) and, to thi end, we will use the identies: 𝜆𝜈 − 𝜆𝜈 𝐞 = − (𝜆𝜇 − 𝜆𝜇 𝐞)ℎ𝜈 𝜇 + [(𝜆𝜇 − 𝜆𝜇 𝐞)𝑈𝜇] 𝑈𝜈 𝑐2 , 𝜆<𝜇𝜈> = 𝜆<𝛌𝛜> ℎ<𝜇 𝛌 ℎ𝜈>3 𝛜 − 2 𝑐2 𝜆<𝛌𝛜> 𝑈 𝛌 ℎ(𝜇 𝛜 𝑈𝜈) + 𝜆<𝛌𝛜>𝑈 𝛌𝑈𝛜 𝑐2 (𝑈𝜇𝑈𝜈 + 1 3 ℎ𝜇𝜈) . So we obtain: 𝐎𝛌𝛌1⋯𝛌𝑟 − 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟 = 𝒜𝜋 𝛌𝛌1⋯𝛌𝑟 𝜋 + 𝒜𝜇 𝛌𝛌1⋯𝛌𝑟 𝜇 + 𝒜𝑞 𝛌𝛌1⋯𝛌𝑟𝛿 𝑞𝛿 + 𝒜𝑡 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 𝑡<𝛟𝛿>3 + (33) +∑ 𝑆+2 𝑟′=3 𝒜𝜆 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑟′ 𝜆𝛜1⋯𝛜𝑟′ + ∑ 𝑆 𝑠′=0 𝒜𝜇 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑠′ 𝜇𝛜1⋯𝛜𝑠′ , 𝐎𝑉 𝛌𝛌1⋯𝛌𝑠 − 𝐎𝑉𝐞 𝛌𝛌1⋯𝛌𝑠 = 𝒜𝑉𝜋 𝛌𝛌1⋯𝛌𝑠 𝜋 + 𝒜𝑉𝜇 𝛌𝛌1⋯𝛌𝑠 𝜇 + 𝒜𝑉𝑞 𝛌𝛌1⋯𝛌𝑠𝛿 𝑞𝛿 + 𝒜𝑉𝑡 𝛌𝛌1⋯𝛌𝑠𝛟𝛿 𝑡<𝛟𝛿>3 + +∑ 𝑆+2 𝑟′=3 𝒜𝑉𝜆 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑟′ 𝜆𝛜1⋯𝛜𝑟′ + ∑ 𝑆 𝑠′=0 𝒜𝑉𝜇 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑠′ 𝜇𝛜1⋯𝛜𝑠′ , Where; 𝒜𝜋 𝛌𝛌1⋯𝛌𝑟 = 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟 ￜ̃ￜ31 𝜋 𝑚 |ￜ̃ￜ𝜋| + 𝐎𝑟1 𝛌𝛌1⋯𝛌𝑟𝜈 𝑈𝜈 𝑐2 ￜ̃ￜ32 𝜋 𝑚2 |ￜ̃ￜ𝜋| + 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 (𝑈𝛟𝑈𝛿 + 1 3 ℎ𝛟𝛿) ￜ̃ￜ33 𝜋 𝑚2 𝑐2|ￜ̃ￜ𝜋| , 𝒜𝜇 𝛌𝛌1⋯𝛌𝑟 = = − 𝑚 𝑘𝐵 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟 [ ￜ̃ￜ31 𝜋 3 |ￜ̃ￜ𝜋| (𝐵1 − 𝐵2𝑐 2) + ￜ̃ￜ21 𝜋 |ￜ̃ￜ𝜋| (𝐵2 − 𝐵3𝑐 2)𝑐4 − ￜ̃ￜ11 𝜋 𝑚 |ￜ̃ￜ𝜋| (𝐎1 0𝑐2 − 3𝐎11 0 )𝑐2] − 1 𝑘𝐵 𝐎𝑟1 𝛌𝛌1⋯𝛌𝑟𝜈 𝑈𝜈 𝑐2 [ ￜ̃ￜ32 𝜋 3 |ￜ̃ￜ𝜋| (𝐵1 − 𝐵2𝑐 2) + ￜ̃ￜ22 𝜋 |ￜ̃ￜ𝜋| (𝐵2 − 𝐵3𝑐 2)𝑐4 − ￜ̃ￜ21 𝜋 𝑚 |ￜ̃ￜ𝜋| (𝐎1 0𝑐2 − 3𝐎11 0 )𝑐2] − 1 𝑘𝐵 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 (𝑈𝛟𝑈𝛿 + 1 3 ℎ𝛟𝛿) [ ￜ̃ￜ33 𝜋 3 𝑐2|ￜ̃ￜ𝜋| (𝐵1 − 𝐵2𝑐 2) + ￜ̃ￜ23 𝜋 |ￜ̃ￜ𝜋| (𝐵2 − 𝐵3𝑐 2)𝑐2 − ￜ̃ￜ13 𝜋 𝑚 |ￜ̃ￜ𝜋| (𝐎1 0𝑐2 − 3𝐎11 0 )] − 1 𝑘𝐵 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 𝑔𝛟𝛿 , 𝒜𝑞 𝛌𝛌1⋯𝛌𝑟𝛿 = 𝐎𝑟1 𝛌𝛌1⋯𝛌𝑟𝛿 2 𝐎11 0 𝑚3 ￜ̃ￜ𝑞 − 2 𝑝 𝑚3𝑐2ￜ̃ￜ𝑞 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 𝑈𝛟 , 𝒜𝑡 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 = 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 15 2 𝑚2 𝐵1 , 𝒜𝜆 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑟′ = − 1 𝑘𝐵 𝐎𝑟𝑟′ 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑟′ + + 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟 𝑚 [ ￜ̃ￜ31 𝜋 3 |ￜ̃ￜ𝜋| ℎ𝜇𝜈 𝐎1𝑟′ 𝜇𝜈𝛜1⋯𝛜𝑟′ + ￜ̃ￜ21 𝜋 |ￜ̃ￜ𝜋| 𝑈𝜇𝑈𝜈 𝐎1𝑟′ 𝜇𝜈𝛜1⋯𝛜𝑟′ + ￜ̃ￜ11 𝜋 |ￜ̃ￜ𝜋| 𝑈𝜇 𝐎𝐞 𝜇𝛜1⋯𝛜𝑟′] + + 𝐎𝑟1 𝛌𝛌1⋯𝛌𝑟𝜈 𝑚 𝑘𝐵 𝑈𝜈 𝑐2 [ ￜ̃ￜ32 𝜋 3 |ￜ̃ￜ𝜋| ℎ𝜇𝜗 𝐎1𝑟′ 𝜇𝜗𝛜1⋯𝛜𝑟′ + ￜ̃ￜ22 𝜋 |ￜ̃ￜ𝜋| 𝑈𝜇𝑈𝜗 𝐎1𝑟′ 𝜇𝜗𝛜1⋯𝛜𝑟′ + ￜ̃ￜ21 𝜋 |ￜ̃ￜ𝜋| 𝑈𝜇 𝐎𝐞 𝜇𝛜1⋯𝛜𝑟′] + + 2 𝑘𝐵 𝐎𝑟1 𝛌𝛌1⋯𝛌𝑟𝜈 [ 𝐎11 0 𝑚2 ￜ̃ￜ𝑞 ℎ𝜈𝜇 𝑈𝜗 𝐎1𝑟′ 𝜈𝜗𝛜1⋯𝛜𝑟′ − 𝐵2 𝑐 2 3 𝑚 ￜ̃ￜ𝑞 ℎ𝜈𝜇 𝐎𝐞 𝜇𝛜1⋯𝛜𝑟′] + HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 199 + 1 𝑘𝐵 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 𝐎1𝑟′ 𝜗𝛜𝛜1⋯𝛜𝑟′ [ 15 2 𝑚 𝐵1 ℎ𝜗<𝛟ℎ𝛿>3𝛜 − 2 𝑝 𝑚 𝑐2ￜ̃ￜ𝑞 𝑈𝛿 ℎ𝛟𝜗 𝑈𝛜 − ￜ̃ￜ33 𝜋 3 𝑚 𝑐2|ￜ̃ￜ𝜋| (𝑈𝛟𝑈𝛿 + 1 3 ℎ𝛟𝛿) ℎ𝜗𝛜 + ￜ̃ￜ23 𝜋 𝑚 𝑐2|ￜ̃ￜ𝜋| (𝑈𝛟𝑈𝛿 + 1 3 ℎ𝛟𝛿)𝑈𝜗𝑈𝛜] + + 1 𝑘𝐵 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 𝐎𝐞 𝜗𝛜1⋯𝛜𝑟′ [− 2 𝐵4 3 𝑚 ￜ̃ￜ𝑞 𝑈𝛿 ℎ𝛟𝜗 + ￜ̃ￜ13 𝜋 𝑐2|ￜ̃ￜ𝜋| (𝑈𝛟𝑈𝛿 + 1 3 ℎ𝛟𝛿)𝑈𝜗] , 𝒜𝜇 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑠′ = − 1 𝑘𝐵 𝐵𝑟𝑠′ 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑠′ + + 1 𝑘𝐵 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟 [ ￜ̃ￜ31 𝜋 3 |ￜ̃ￜ𝜋| ℎ𝜗𝛜 𝐵1𝑠′ 𝜗𝛜𝛜1⋯𝛜𝑠′ + ￜ̃ￜ21 𝜋 |ￜ̃ￜ𝜋| 𝑈𝜗𝑈𝛜 𝐵1𝑠′ 𝜗𝛜𝛜1⋯𝛜𝑠′ − ￜ̃ￜ11 𝜋 |ￜ̃ￜ𝜋| 𝑈𝜗 𝐎𝐞 𝜗𝛜1⋯𝛜𝑠′] + + 𝐎𝑟1 𝛌𝛌1⋯𝛌𝑟𝜈 𝑘𝐵 𝑈𝜈 𝑐2 [ ￜ̃ￜ33 𝜋 3 |ￜ̃ￜ𝜋| ℎ𝜗𝛜 𝐵1𝑠′ 𝜗𝛜𝛜1⋯𝛜𝑠′ 𝑚 + ￜ̃ￜ23 𝜋 |ￜ̃ￜ𝜋| 𝑈𝛜𝑈𝜗 𝐵1𝑠′ 𝜗𝛜𝛜1⋯𝛜𝑠′ 𝑚 + ￜ̃ￜ13 𝜋 |ￜ̃ￜ𝜋| 𝑈𝜗 𝐎𝑉𝐞 𝜗𝛜1⋯𝛜𝑠′] + + 2 𝑘𝐵 𝐎𝑟1 𝛌𝛌1⋯𝛌𝑟𝜈 𝑚 [ 𝐎11 0 𝑚 ￜ̃ￜ𝑞 ℎ𝜈𝜗 𝑈𝛜 𝐵1𝑠′ 𝜗𝛜𝛜1⋯𝛜𝑠′ − 𝐵2 𝑐 2 3 ￜ̃ￜ𝑞 ℎ𝜈𝜗 𝐎𝑉𝐞 𝜗𝛜1⋯𝛜𝑠′] + + 1 𝑘𝐵 15 2 𝑚 𝐵1 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 𝐵1𝑠′ 𝜗𝛜𝛜1⋯𝛜𝑠′ ℎ𝛟<𝜗ℎ𝛜>3𝛿 + + 1 𝑘𝐵 𝑐 2 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 (𝑈𝛟𝑈𝛿 + 1 3 ℎ𝛟𝛿) [ ￜ̃ￜ33 𝜋 3 𝑚 |ￜ̃ￜ𝜋| ℎ𝜗𝛜 𝐵1𝑠′ 𝜗𝛜𝛜1⋯𝛜𝑠′ + ￜ̃ￜ23 𝜋 𝑚 |ￜ̃ￜ𝜋| 𝐵1𝑠′ 𝜗𝛜𝛜1⋯𝛜𝑠′ 𝑈𝜗𝑈𝛜 + + ￜ̃ￜ13 𝜋 |ￜ̃ￜ𝜋| 𝑈𝜗 𝐎𝑉𝐞 𝜗𝛜1⋯𝛜𝑠′] + 4 𝑘𝐵 𝑐 2 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 𝑈𝛟 [ 𝐎11 0 𝑚2 ￜ̃ￜ𝑞 𝑈𝛜 ℎ𝛿𝜗 𝐵1𝑠′ 𝜗𝛜𝛜1⋯𝛜𝑠′ − 𝐵2 𝑐 2 3 𝑚 ￜ̃ￜ𝑞 ℎ𝛿𝜗 𝐎𝑉𝐞 𝜗𝛜1⋯𝛜𝑠′] . The expressions of 𝒜𝑉𝜋 𝛌𝛌1⋯𝛌𝑠 , 𝒜𝑉𝜇 𝛌𝛌1⋯𝛌𝑠 , 𝒜𝑉𝑞 𝛌𝛌1⋯𝛌𝑠𝛿 , 𝒜𝑉𝑡 𝛌𝛌1⋯𝛌𝑠𝛟𝛿 , 𝒜𝑉𝜆 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑟′ 𝒜𝑉𝜇 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑠′ can be obtained from the above ones by substituting 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟 with 𝐎𝑉𝐞 𝛌𝛌1⋯𝛌𝑠, 𝐎𝑟1 𝛌𝛌1⋯𝛌𝑟𝜈 with 𝐵𝑠1 𝛌𝛌1⋯𝛌𝑠𝜈, 𝐎𝑟2 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 with 𝐵𝑠2 𝛌𝛌1⋯𝛌𝑠𝛟𝛿 , 𝐎𝑟𝑟′ 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑟′ with 𝐵𝑠𝑟′ 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑟′ , 𝐵𝑟𝑠′ 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑠′ with 𝐶𝑠𝑠′ 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑠′. In fact, this is what comes out from the comparison between (26) 1 and (26) 2; obviously, the contribute of the Lagrange multipliers is the same so that nothing else must be changed. So we avoid to report such expressions for the sake of brevity. The Equations (33) jointly with (23) give the requested closure. Obviously, in Equations (33) the Lagrange multipliers 𝜇, 𝜆𝛜1⋯𝛜𝑟′ , 𝜇𝛜1⋯𝛜𝑠′ still appear between the independent variables. If we want to express them too in terms of physical variables, we have firstly to clarify what these physical variables are besides those already introduced. In my opinion they are those whose non relativistic limit gives the variables which are derivated respect to time, or still better, the deviations from their equilibrium value. In other words, we have to consider the equations: Δ = 1 𝑐4 𝑈𝛌𝑈𝛌1𝑈𝛌2 (𝐎 𝛌𝛌1𝛌2 − 𝐎𝐞 𝛌𝛌1𝛌2) , (34) Δ𝛌1⋯𝛌𝑟 = 𝑈𝛌(𝐎 𝛌𝛌1⋯𝛌𝑟 − 𝐎𝐞 𝛌𝛌1⋯𝛌𝑟) = 𝑈𝛌 𝒜𝜋 𝛌𝛌1⋯𝛌𝑟 𝜋 + 𝑈𝛌 𝒜𝜇 𝛌𝛌1⋯𝛌𝑟 𝜇 + 𝑈𝛌 𝒜𝑞 𝛌𝛌1⋯𝛌𝑟𝛿 𝑞𝛿 + (35) +𝑈𝛌 𝒜𝑡 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 𝑡<𝛟𝛿>3 + ∑ 𝑆+2 𝑟′=3 𝑈𝛌 𝒜𝜆 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑟′ 𝜆𝛜1⋯𝛜𝑟′ + ∑ 𝑆 𝑠′=0 𝑈𝛌 𝒜𝜇 𝛌𝛌1⋯𝛌𝑟𝛜1⋯𝛜𝑠′ 𝜇𝛜1⋯𝛜𝑠′ , Δ𝑉 𝛌1⋯𝛌𝑠 = 𝑈𝛌(𝐎𝑉 𝛌𝛌1⋯𝛌𝑠 − 𝐎𝑉𝐞 𝛌𝛌1⋯𝛌𝑠) = 𝑈𝛌 𝒜𝑉𝜋 𝛌𝛌1⋯𝛌𝑠 𝜋 + 𝑈𝛌 𝒜𝑉𝜇 𝛌𝛌1⋯𝛌𝑠 𝜇 + 𝑈𝛌 𝒜𝑉𝑞 𝛌𝛌1⋯𝛌𝑠𝛿 𝑞𝛿 + +𝑈𝛌 𝒜𝑉𝑡 𝛌𝛌1⋯𝛌𝑟𝛟𝛿 𝑡<𝛟𝛿>3 + ∑ 𝑆+2 𝑟′=3 𝑈𝛌 𝒜𝑉𝜆 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑟′ 𝜆𝛜1⋯𝛜𝑟′ + ∑ 𝑆 𝑠′=0 𝑈𝛌 𝒜𝑉𝜇 𝛌𝛌1⋯𝛌𝑠𝛜1⋯𝛜𝑠′ 𝜇𝛜1⋯𝛜𝑠′ . HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 200 The left hand sides of these equations are the additional physical variables; Equations (34) for 𝑟 = 3,⋯ 𝑆 + 2 and 𝑠 = 0,⋯ 𝑆 have to be used to determine 𝜇, 𝜆𝛜1⋯𝛜𝑟′, 𝜇𝛜1⋯𝛜𝑠′ in terms of the physical variables. The result has to be substituted in (33) so obtaining the closure all in terms of physical variables. Can we do this? Yes, we can. But the equations present in this article are complicated enough to want to burden them further. Therefore we refrain from doing it. In any case, when we want to make a practical application of the model, we must first choose in harmony with the experimental results the number 𝑆 to stop at. In this case, since 𝑆 is a given number, these further steps can be carried out easily. So the last step in the first part of the flowchart present in the Introduction has been obtained, i.e., the closure of the present general relativistic model (6). 6. Conclusions In this article, it was found that the relativistic counterpart of the classical model for polyatomic gases takes into account both the vibrational and rotational modes. As is common in Extended Thermodynamics, in the balance equations not only independent variables appear but also other additional tensors; the closure is obtained when the expressions of these tensors are found as functions of the independent variables. This end is here reached by imposing universal principles such as the Entropy Principle, the Maximum Entropy Principle, and, obviously, the covariance of all the equations and the variables involved. As a bonus, the field equations assume the symmetric form and are hyperbolic; this is important because assures the respect of the cause and effect principle and the fact that the wave velocities don’t exceed the speed of light. Another nice mathematical property in this way is the continuous dependence on the initial data. The validity of the present model has already been tested because in the simplest case of 16 moments, it coincides with that already known in the literature. Certainly, the field equations have become somewhat complicated due to the fact that independent variables more appealing to the common reader have been chosen. If we use the Lagrange multipliers as independent variables, everything becomes simpler. The results here obtained give indications on how to structure the non-relativistic model. In fact, the classical model with an arbitrary number of moments known in literature proposes only 3 hierarchies with infinite equations. It is not shown how to interrupt these 3 blocks in order to obtain a finite system, except for the 2 simplest cases. This aspect is clarified here and, in particular, in section 2. The optimal choice of moments here presented in the subsystem with only one mode becomes the same as that already known in literature. 6. Declarations 6.1. Data Availability Statement Data sharing is not applicable to this article. 6.2. Funding This work has been partially supported by GNFM/INdAM and by the Italian MIUR through the PRIN2017 project Multi-scale phenomena in Continuum Mechanics: singular limits, off-equilibrium and transitions (Project Number: 2017YBKNCE). 6.3. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 7. References [1] Arima, T., Ruggeri, T., & Sugiyama, M. (2018). Extended Thermodynamics of Rarefied Polyatomic Gases: 15-Field Theory Incorporating Relaxation Processes of Molecular Rotation and Vibration. Entropy, 20(4), 301. doi:10.3390/e20040301. [2] Arima, T., Ruggeri, T., & Sugiyama, M. (2020). Rational extended thermodynamics of dense polyatomic gases incorporating molecular rotation and vibration. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 378(2170), 20190176. doi:10.1098/rsta.2019.0176. [3] Liu, I.-S., & MÃŒller, I. (1983). Extended thermodynamics of classical and degenerate ideal gases. Archive for Rational Mechanics and Analysis, 83(4), 285–332. doi:10.1007/bf00963838 [4] MÃŒller, I., & Ruggeri, T. (1998). Rational Extended Thermodynamics. Springer Tracts in Natural Philosophy. doi:10.1007/978- 1-4612-2210-1. [5] Arima, T., Taniguchi, S., Ruggeri, T., & Sugiyama, M. (2011). Extended thermodynamics of dense gases. Continuum Mechanics and Thermodynamics, 24(4-6), 271–292. doi:10.1007/s00161-011-0213-x. HighTech and Innovation Journal Vol. 2, No. 3, September, 2021 201 [6] Ruggeri, T., & Sugiyama, M. (2015). Rational Extended Thermodynamics beyond the Monatomic Gas. doi:10.1007/978-3-319- 13341-6. [7] Pennisi, S., & Ruggeri, T. (2020). Classical Limit of Relativistic Moments Associated with Boltzmann–Chernikov Equation: Optimal Choice of Moments in Classical Theory. Journal of Statistical Physics, 179(1), 231–246. doi:10.1007/s10955-020- 02530-2. [8] Boillat, G., & Ruggeri, T. (1997). Hyperbolic Principal Subsystems: Entropy Convexity and Subcharacteristic Conditions. Archive for Rational Mechanics and Analysis, 137(4), 305–320. doi:10.1007/s002050050030. [9] Carrisi, M. C., & Pennisi, S. (2019). A 16 moments model in relativistic extended thermodynamics of rarefied polyatomic gas. Ricerche Di Matematica. doi:10.1007/s11587-019-00468-6. [10] Borghero, F., Demontis, F., & Pennisi, S. (2006). The Non-Relativistic Limit Of Relativistic Extended Thermodynamics With Many Moments- Part I: The Balance Equations. Waves and Stability in Continuous Media. Proceedings Wascom (2005), 47– 52. doi:10.1142/9789812773616_0008. [11] Pennisi, S., & Ruggeri, T. (2017). Relativistic extended thermodynamics of rarefied polyatomic gas. Annals of Physics, 377, 414–445. doi:10.1016/j.aop.2016.12.012. [12] Mentrelli, A., & Ruggeri, T. (2020). Shock structure in extended thermodynamics with second-order maximum entropy principle closure. Continuum Mechanics and Thermodynamics, 33(1), 125–150. doi:10.1007/s00161-020-00892-2. [13] Brini, F. (2001). Hyperbolicity region in extended thermodynamics with 14 moments. Continuum Mechanics and Thermodynamics, 13(1), 1–8. doi:10.1007/s001610100036. [14] Ruggeri, T., & Trovato, M. (2004). Hyperbolicity in Extended Thermodynamics of Fermi and Bose gases. Continuum Mechanics and Thermodynamics, 16(6), 551–576. doi:10.1007/s00161-004-0180-6. [15] Brini, F., & Ruggeri, T. (2019). Second-order approximation of extended thermodynamics of a monatomic gas and hyperbolicity region. Continuum Mechanics and Thermodynamics, 32(1), 23–39. doi:10.1007/s00161-019-00778-y. [16] Brini, F., & Ruggeri, T. (2020). Hyperbolicity of first and second order extended thermodynamics theory of polyatomic rarefied gases. International Journal of Non-Linear Mechanics, 124, 103517. doi:10.1016/j.ijnonlinmec.2020.103517. [17] Carrisi, M. C., & Pennisi, S. (2013). Extended thermodynamics of charged gases with many moments. Journal of Mathematical Physics, 54(2), 023101. doi:10.1063/1.4789544.