Lie point symmetry
Lie groups and Lie algebras |
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Lie point symmetry is a concept in advanced
Roughly speaking, a Lie point symmetry of a system is a local group of transformations that maps every solution of the system to another solution of the same system. In other words, it maps the solution set of the system to itself. Elementary examples of Lie groups are
The Lie symmetry theory is a well-known subject. In it are discussed continuous symmetries opposed to, for example, discrete symmetries. The literature for this theory can be found, among other places, in these notes.[5][6][7][8][9]
Overview
Types of symmetries
Lie groups and hence their infinitesimal generators can be naturally "extended" to act on the space of independent variables,
Applications
Lie symmetries were introduced by Lie in order to solve ordinary differential equations. Another application of symmetry methods is to reduce systems of differential equations, finding equivalent systems of differential equations of simpler form. This is called reduction. In the literature, one can find the classical reduction process,[4] and the moving frame-based reduction process.[11][12][13] Also symmetry groups can be used for classifying different symmetry classes of solutions.
Geometrical framework
Infinitesimal approach
Lie's fundamental theorems underline that Lie groups can be characterized by elements known as infinitesimal generators. These mathematical objects form a Lie algebra of infinitesimal generators. Deduced "infinitesimal symmetry conditions" (defining equations of the symmetry group) can be explicitly solved in order to find the closed form of symmetry groups, and thus the associated infinitesimal generators.
Let be the set of coordinates on which a system is defined where is the cardinality of . An infinitesimal generator in the field is a linear operator that has in its kernel and that satisfies the Leibniz rule:
- .
In the canonical basis of elementary derivations , it is written as:
where is in for all in .
Lie groups and Lie algebras of infinitesimal generators
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Lie algebras can be generated by a generating set of infinitesimal generators as defined above. To every Lie group, one can associate a Lie algebra. Roughly, a Lie algebra is an algebra constituted by a vector space equipped with Lie bracket as additional operation. The base field of a Lie algebra depends on the concept of invariant. Here only finite-dimensional Lie algebras are considered.
Continuous dynamical systems
A
such that for all point in :
- where is the neutral element of ;
- for all in , .
A continuous dynamical system is defined on a group that can be identified to i.e. the group elements are continuous.
Invariants
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An invariant, roughly speaking, is an element that does not change under a transformation.
Definition of Lie point symmetries
In this paragraph, we consider precisely expanded Lie point symmetries i.e. we work in an expanded space meaning that the distinction between independent variable, state variables and parameters are avoided as much as possible.
A symmetry group of a system is a continuous dynamical system defined on a local Lie group acting on a manifold . For the sake of clarity, we restrict ourselves to n-dimensional real manifolds where is the number of system coordinates.
Lie point symmetries of algebraic systems
Let us define
Algebraic systems
Let be a finite set of rational functions over the field where and are polynomials in i.e. in variables with coefficients in . An algebraic system associated to is defined by the following equalities and inequalities:
An algebraic system defined by is regular (a.k.a.
Definition of Lie point symmetries
The following theorem (see th. 2.8 in ch.2 of [5]) gives necessary and sufficient conditions so that a local Lie group is a symmetry group of an algebraic system.
Theorem. Let be a connected local Lie group of a continuous dynamical system acting in the n-dimensional space . Let with define a regular system of algebraic equations:
Then is a symmetry group of this algebraic system if, and only if,
for every infinitesimal generator in the Lie algebra of .
Example
Consider the algebraic system defined on a space of 6 variables, namely with:
The infinitesimal generator
is associated to one of the one-parameter symmetry groups. It acts on 4 variables, namely and . One can easily verify that and . Thus the relations are satisfied for any in that vanishes the algebraic system.
Lie point symmetries of dynamical systems
Let us define systems of first-order
Systems of ODEs and associated infinitesimal generators
Let be a derivation w.r.t. the continuous independent variable . We consider two sets and . The associated coordinate set is defined by and its cardinal is . With these notations, a system of first-order ODEs is a system where:
and the set specifies the evolution of state variables of ODEs w.r.t. the independent variable. The elements of the set are called state variables, these of parameters.
One can associate also a continuous dynamical system to a system of ODEs by resolving its equations.
An infinitesimal generator is a derivation that is closely related to systems of ODEs (more precisely to continuous dynamical systems). For the link between a system of ODEs, the associated vector field and the infinitesimal generator, see section 1.3 of.[4] The infinitesimal generator associated to a system of ODEs, described as above, is defined with the same notations as follows:
Definition of Lie point symmetries
Here is a geometrical definition of such symmetries. Let be a continuous dynamical system and its infinitesimal generator. A continuous dynamical system is a Lie point symmetry of if, and only if, sends every orbit of to an orbit. Hence, the infinitesimal generator satisfies the following relation
where is any constant of and i.e. . These generators are linearly independent.
One does not need the explicit formulas of in order to compute the infinitesimal generators of its symmetries.
Example
Consider
The continuous dynamical system associated to this system of ODEs is:
The independent variable varies continuously; thus the associated group can be identified with .
The infinitesimal generator associated to this system of ODEs is:
The following infinitesimal generators belong to the 2-dimensional symmetry group of :
Software
There exist many software packages in this area.[15][16][17] For example, the package liesymm of Maple provides some Lie symmetry methods for PDEs.[18] It manipulates integration of determining systems and also differential forms. Despite its success on small systems, its integration capabilities for solving determining systems automatically are limited by complexity issues. The DETools package uses the prolongation of vector fields for searching Lie symmetries of ODEs. Finding Lie symmetries for ODEs, in the general case, may be as complicated as solving the original system.
References
- ^ Lie, Sophus (1881). "Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen". Archiv for Mathematik og Naturvidenskab (in German). 6: 328–368.
- ^ Lie, Sophus (1890). Theorie der Transformationsgruppen (in German). Vol. 2. Teubner, Leipzig.
- ^ Lie, Sophus (1893). Theorie der Transformationsgruppen (in German). Vol. 3. Teubner, Leipzig.
- ^ a b c Olver, Peter J. (1993). Applications of Lie Groups to Differential Equations (Second ed.). Springer-Verlag.
- ^ a b Olver, Peter J. (1995). Equivalence, Invariance and Symmetry. Cambridge University Press.
- ^ Olver, Peter J. (1999). Classical Invariant Theory (First ed.). Cambridge University Press.
- ^ Bluman, G.; Kumei, S. (1989). Symmetries and Differential Equations. Applied Mathematical Sciences Series. Vol. 81 (Second ed.). New York: Springer-Verlag.
- ^ a b Stephani, H. (1989). Differential Equations (First ed.). Cambridge University Press.
- S2CID 17161506.
- ^ Noether, E. (1918). "Invariante Variationsprobleme. Nachr. König. Gesell. Wissen". Math.-Phys. Kl. (in German). Göttingen: 235–257.
- ^ Cartan, Elie (1935). "La méthode du repère mobile, la théorie des groupes continus et les espaces généralisés". Exposés de géométrie - 5 Hermann (in French). Paris.
- S2CID 6681218.
- S2CID 826629.
- ^ Murray, J. D. (2002). Mathematical Biology. Interdisciplinary Applied Mathematics. Vol. 17. Springer.
- ^ Heck, A. (2003). Introduction to Maple (Third ed.). Springer-Verlag.
- doi:10.1137/1030094.
- ^ Dimas, S.; Tsoubelis, T. (2005). "SYM: A new symmetry-finding package for Mathematica" (PDF). The 10th International Conference in MOdern GRoup ANalysis. University of Cyprus, Nicosia, Cyprus: 64–70. Archived from the original (PDF) on 2006-10-01.
- .