Lie point symmetry

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 (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. 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.
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. For Lie point symmetries, the coefficients of the infinitesimal generators depend only on coordinates, denoted by formula_1.
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, and the moving frame-based reduction process. 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 formula_2 be the set of coordinates on which a system is defined where formula_3 is the cardinal of formula_1. An infinitesimal generator formula_5 in the field formula_6 is a linear operator formula_7 that has formula_8 in its kernel and that satisfies the Leibniz rule:
In the canonical basis of elementary derivations formula_10, it is written as:
where formula_12 is in formula_6 for all formula_14 in formula_15.
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 formula_16 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 formula_17 such a dynamical system, more precisely, a (left-)action of a group formula_18 on a manifold formula_19:
such that for all point formula_1 in formula_19:
A continuous dynamical system is defined on a group formula_25 that can be identified to formula_8 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 formula_25 acting on a manifold formula_19. For the sake of clarity, we restrict ourselves to n-dimensional real manifolds formula_33 where formula_3 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 formula_35 be a finite set of rational functions over the field formula_8 where formula_37 and formula_38 are polynomials in formula_39 i.e. in variables formula_2 with coefficients in formula_8. An "algebraic system" associated to formula_42 is defined by the following equalities and inequalities:
An algebraic system defined by formula_44 is "regular" (a.k.a. smooth) if the system formula_42 is of maximal rank formula_46, meaning that the Jacobian matrix formula_47 is of rank formula_46 at every solution formula_1 of the associated semi-algebraic variety.
Definition of Lie point symmetries.
The following theorem (see th. 2.8 in ch.2 of ) gives necessary and sufficient conditions so that a local Lie group formula_25 is a symmetry group of an algebraic system.
Theorem. Let formula_25 be a connected local Lie group of a continuous dynamical system acting in the n-dimensional space formula_52. Let formula_53 with formula_54 define a regular system of algebraic equations:
Then formula_25 is a symmetry group of this algebraic system if, and only if,
for every infinitesimal generator formula_5 in the Lie algebra formula_16 of formula_25.
Example.
Consider the algebraic system defined on a space of 6 variables, namely formula_61 with:
The infinitesimal generator
is associated to one of the one-parameter symmetry groups. It acts on 4 variables, namely formula_64 and formula_65. One can easily verify that formula_66 and formula_67. Thus the relations formula_68 are satisfied for any formula_1 in formula_70 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 formula_71 be a derivation w.r.t. the continuous independent variable formula_72. We consider two sets formula_73 and formula_74. The associated coordinate set is defined by formula_75 and its cardinal is formula_76. With these notations, a "system of first-order ODEs" is a system where:
and the set formula_44 specifies the evolution of state variables of ODEs w.r.t. the independent variable. The elements of the set formula_79 are called "state variables", these of formula_80 "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. The infinitesimal generator formula_5 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 formula_17 be a continuous dynamical system and formula_84 its infinitesimal generator. A continuous dynamical system formula_85 is a Lie point symmetry of formula_17 if, and only if, formula_85 sends every orbit of formula_17 to an orbit. Hence, the infinitesimal generator formula_89 satisfies the following relation based on Lie bracket:
where formula_91 is any constant of formula_84 and formula_89 i.e. formula_94. These generators are linearly independent.
One does not need the explicit formulas of formula_17 in order to compute the infinitesimal generators of its symmetries.
Example.
Consider Pierre François Verhulst's logistic growth model with linear predation, where the state variable formula_96 represents a population. The parameter formula_97 is the difference between the growth and predation rate and the parameter formula_98 corresponds to the receptive capacity of the environment:
The continuous dynamical system associated to this system of ODEs is:
The independent variable formula_101 varies continuously; thus the associated group can be identified with formula_8.
The infinitesimal generator associated to this system of ODEs is:
The following infinitesimal generators belong to the 2-dimensional symmetry group of formula_17:
Software.
There exist many software packages in this area. For example, the package liesymm of Maple provides some Lie symmetry methods for PDEs. 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.