1. Overview
     Types of symmetries    Applications  

2. Geometrical framework
     Infinitesimal approach    Lie groups and Lie algebras of infinitesimal generators    Continuous dynamical systems    Invariants  

3. Definition of Lie point symmetries
     Lie point symmetries of algebraic systems    Algebraic systems    Definition of Lie point symmetries    Example    Lie point symmetries of dynamical systems    Systems of ODEs and associated infinitesimal generators    Definition of Lie point symmetries    Example  

4. Software

5. References

Towards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group in order to study the solutions of ordinary differential equations^[1][2][3] (ODEs). He showed the following main property: the order of an ordinary differential equation can be reduced by one if it is invariant under one-parameter Lie group of point transformations.^[4] This observation unified and extended the available integration techniques. Lie devoted the remainder of his mathematical career to developing these continuous groups that have now an impact on many areas of mathematically based sciences. The applications of Lie groups to differential systems were mainly established by Lie and Emmy Noether, and then advocated by Élie Cartan.

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 translations, rotations and scalings.

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][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, state variables (dependent variables) and derivatives of the state variables up to any finite order. There are many other kinds of symmetries. For example, contact transformations let coefficients of the transformations infinitesimal generator depend also on first derivatives of the coordinates. Lie-Bäcklund transformations let them involve derivatives up to an arbitrary order. The possibility of the existence of such symmetries was recognized by Noether.^[10] For Lie point symmetries, the coefficients of the infinitesimal generators depend only on coordinates, denoted by .

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.^[12][13][14] 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 cardinal 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

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 dynamical system (or flow) is a one-parameter group action. Let us denote by such a dynamical system, more precisely, a (left-)action of a group on a manifold :

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

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 used in the forthcoming symmetry definition.

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. smooth) if the system is of maximal rank , meaning that the Jacobian matrix is of rank at every solution of the associated semi-algebraic variety.

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

Let us 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 ODEs used in the forthcoming symmetry definition.

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^[8] based on Lie bracket:

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 Pierre François Verhulst's logistic growth model with linear predation,^[18] where the state variable represents a population. The parameter is the difference between the growth and predation rate and the parameter corresponds to the receptive capacity of the environment:

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.^[19][20][21] For example, the package liesymm of Maple provides some Lie symmetry methods for PDEs.^[22] 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

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