APPLYING SYMMETRY PROPERTIES OF FINITE SYSTEM TO ITS FIELD THEORY DESCRIPTION

2004 ◽  
Vol 18 (26) ◽  
pp. 3443-3450
Author(s):  
B. A. SEREDYUK ◽  
K. D. TOVSTYUK ◽  
N. K. TOVSTYUK
Keyword(s):  
Symmetry Group ◽  
Particle System ◽  
Finite System ◽  
Wave Functions ◽  
Point Symmetry ◽  
Relaxation Processes ◽  
Point Symmetry Group ◽  
Field Techniques ◽  
Self Energy ◽  
Symmetry Properties

The self-energy is received using the theory group and field techniques, based on the wave functions of the classes of point symmetry group and accounting for the two-particle system. It contains components responsible not only for relaxation processes but also for the oscillating ones, caused by a different degree of occupation of the class of point symmetry group by particles of the structure. Analyzing the character of energy oscillations depending on the degree of the class occupation, the mechanisms of diffusion, catalysis, chemical reactions and Le-Shatelje principle are described.

A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems

1993 ◽  
Vol 48 (4) ◽  
pp. 535-550 ◽  
Author(s):  
H. Kötz
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Classification Problem ◽  
Similarity Solutions ◽  
Point Symmetry ◽  
Point Symmetry Group ◽  
Relativistic Case ◽  
Lie Point Symmetry

"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group Gare an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra Gof the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of Gvary in case of 3 ≤ s ≤ r, depending on the solvability of G. If the r-dimensional Lie algebra Gof the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of Gone has to determine the optimal subsystems of semisimple subalgebras of Gin order to construct the full optimal systems of s-dimensional subalgebras of Gwith 3 ≤ s ≤ r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra Gof the eight-dimensional Lie point symmetry group Gadmitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case.

Determination of Irreducible Rotational Operators on the Rotation SO(3) Group and Td Point Symmetry Group

2018 ◽  
Vol 61 (3) ◽  
pp. 516-520
Author(s):  
S. Chan ◽  
O. V. Gromova ◽  
E. S. Bekhtereva ◽  
C. Leroy ◽  
O. N. Ulenikov
Keyword(s):  
Symmetry Group ◽  
Point Symmetry ◽  
Point Symmetry Group

scholarly journals A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. I. Optimal Systems of Solvable Lie Subalgebras

1992 ◽  
Vol 47 (11) ◽  
pp. 1161-1174 ◽  
Author(s):  
H. Kötz
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Lie Group ◽  
Similarity Solutions ◽  
Point Symmetry ◽  
Invariant Solutions ◽  
Point Symmetry Group ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Lie Point Symmetry

Abstract Lie group analysis is a powerful tool for obtaining exact similarity solutions of nonlinear (integro-) differential equations. In order to calculate the group-invariant solutions one first has to find the full Lie point symmetry group admitted by the given (integro-)differential equations and to determine all the subgroups of this Lie group. An effective, systematic means to classify the similarity solutions afterwards is an "optimal system", i.e. a list of group-invariant solutions from which every other such solution can be derived. The problem to find optimal systems of similarity solutions leads to that to "construct" the optimal systems of subalgebras for the Lie algebra of the known Lie point symmetry group. Our aim is to demonstrate a practicable technique for determining these optimal subalgebraic systems using the invariants relative to the group of the inner automorphisms of the Lie algebra in case of a finite-dimensional Lie point symmetry group. Here, we restrict our attention to optimal subsystems of solvable Lie subalgebras. This technique is applied to the nine-dimensional real Lie point symmetry group admitted by the two-dimensional non-stationary ideal magnetohydrodynamic equations

Point symmetry group of the Lagrangian

1991 ◽  
Vol 30 (11) ◽  
pp. 1461-1472 ◽  
Author(s):  
M. Aguirre ◽  
J. Krause
Keyword(s):  
Symmetry Group ◽  
Point Symmetry ◽  
Point Symmetry Group
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