2019. Vol. 14. Issue 4, Pp. 268–273

URL: http://mfs.uimech.org/mfs2019.4.034,en

DOI: 10.21662/mfs2019.4.034

URL: http://mfs.uimech.org/mfs2019.4.034,en

DOI: 10.21662/mfs2019.4.034

The chain of embedded invariant submodels for conic motions

Mukminov T.F.

Mavlyutov Institute of Mechanics UFRC RAS, Ufa , Russia

The equations of continuum mechanics are invariant in relation to the Galilean group generalized by extention. Its 11-dimensional Lie algebra has many subalgebras, which form the optimal system of dissimilar subalgebras. Subalgebras from the optimal system form the graph of embedded subalgebras. There are many chains of subalgebras in the graph. We consider the chain of embedded subalgebras containing operators of space and time translation, the rotation and uniform extension of all independent variables for the models of the continuous medium mechanics. We choose concordant invariants for each subalgebra from the chain. The chain of invariant submodels is constructed in a cylindrical coordinates based on chosen invariants. It is proved that solutions of a submodel constructed on a subalgebra of higher dimension will be part of solutions of submodels constructed on subalgebra of smaller dimensions for the considered chain. Thus, the chain of embedded invariant submodels is constructed by the example of equations of ideal gas dynamics.

Galilean group,

Lie algebra,

chain of embedded subalgebras,

concordant invariants,

invariant submodels

The equations of continuum mechanics are invariant in relation to the Galilean group generalized by extention. Its 11-dimensional Lie algebra has many subalgebras, which form the optimal system of dissimilar subalgebras. Subalgebras from the optimal system form the graph of embedded subalgebras. There are many chains of subalgebras in the graph. We consider the chain of embedded subalgebras containing operators of space and time translation, the rotation and uniform extension of all independent variables.
**Purpose:** To construct the chain of embedded invariant submodels of gas dynamics using the chain of embedded subalgebras of 11-dimensional Lie algebra.

**Methodology:** For the selected chain, concordant invariants were found in the cylindrical coordinate system. Representations of the invariant solution for each submodel were derived from it. By substitution of them into the system of equations of gas dynamics, embedded invariant submodels were obtained.

**Findings:** The chain of embedded invariant submodels of ranks from 0 to 3 for the models of the continuous medium mechanics.

**Value:** It is demonstrated that solutions of a submodel constructed from a subalgebra of higher dimension are part of solutions of submodels constructed on subalgebra of smaller dimensions from the considered chain.

- Mukminov T.F., Khabirov S.V.
[Graph of embedded subalgebras of 11-dimensional symmetry algebra for continuous medium].
Sibirskie e’lektronnye matematicheskie izvestiya [Siberian Electronic Mathematical Reports]. 2019. V. 16. Pp. 121–143. (In Russian)

DOI: 10.33048/semi.2019.16.006 - Khabirov S.V. [Analytical methods in gas dynamics] Analiticheskie metody v gazovoj mexanike. Ufa: BSU, 2013. P. 224 (in Russian)

https://elibrary.ru/item.asp?id=19444103 - Khabirov S.V. A hierarchy of submodels of differential equations. Siberian Mathematical Journal. 2013. V. 54, No. 6. Pp. 1110–1119.

DOI: 10.1134/S0037446613060189 - Chirkunov Yu.A., Khabirov S.V. Elements of symmetry analysis of differential equations ofcontinuum mechanics. Novosibirsk: NSTU, 2012. P. 659.

https://elibrary.ru/item.asp?id=21714062