Lie point symmetry is a concept in advanced mathematics.
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.
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.
Elementary examples of Lie groups are translations, rotations and scalings.
Lie-Bäcklund transformations let them involve derivatives up to an arbitrary order.
[10] For Lie point symmetries, the coefficients of the infinitesimal generators depend only on coordinates, denoted by
Another application of symmetry methods is to reduce systems of differential equations, finding equivalent systems of differential equations of simpler form.
These mathematical objects form a Lie algebra of infinitesimal generators.
in its kernel and that satisfies the Leibniz rule: In the canonical basis of elementary derivations
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.
A dynamical system (or flow) is a one-parameter group action.
such a dynamical system, more precisely, a (left-)action of a group
An invariant, roughly speaking, is an element that does not change under a transformation.
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.
For the sake of clarity, we restrict ourselves to n-dimensional real manifolds
Let us define algebraic systems used in the forthcoming symmetry definition.
be a connected local Lie group of a continuous dynamical system acting in the n-dimensional space
is a symmetry group of this algebraic system if, and only if, for every infinitesimal generator
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.
Let us define systems of first-order ODEs used in the forthcoming symmetry definition.
For the link between a system of ODEs, the associated vector field and the infinitesimal generator, see section 1.3 of.
associated to a system of ODEs, described as above, is defined with the same notations as follows: Here is a geometrical definition of such symmetries.
Consider Pierre François Verhulst's logistic growth model with linear predation,[14] where the state variable
[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.
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.