Lie point symmetry

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 ${\displaystyle Z}$.

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.

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 ${\displaystyle Z=(z_{1},\dots ,z_{n})}$ be the set of coordinates on which a system is defined where ${\displaystyle n}$ is the cardinality of ${\displaystyle Z}$. An infinitesimal generator ${\displaystyle \delta }$ in the field ${\displaystyle \mathbb {R} (Z)}$ is a linear operator ${\displaystyle \delta$ :\mathbb {R} (Z)\rightarrow \mathbb {R} (Z)} that has ${\displaystyle \mathbb {R} }$ in its kernel and that satisfies the Leibniz rule:

${\displaystyle \forall (f_{1},f_{2})\in \mathbb {R} (Z)^{2},\delta f_{1}f_{2}=f_{1}\delta f_{2}+f_{2}\delta f_{1}}$.

In the canonical basis of elementary derivations ${\displaystyle \left\{{\frac {\partial }{\partial z_{1}}},\dots ,{\frac {\partial }{\partial z_{n}}}\right\}}$, it is written as:

${\displaystyle \delta =\sum _{i=1}^{n}\xi _{z_{i}}(Z){\frac {\partial }{\partial z_{i}}}}$

where ${\displaystyle \xi _{z_{i}}}$ is in ${\displaystyle \mathbb {R} (Z)}$ for all ${\displaystyle i}$ in ${\displaystyle \left\{1,\dots ,n\right\}}$.

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 ${\displaystyle {\mathfrak {g}}}$ 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.

A dynamical system (or flow) is a one-parameter group action. Let us denote by ${\displaystyle {\mathcal {D}}}$ such a dynamical system, more precisely, a (left-)action of a group ${\displaystyle (G,+)}$ on a manifold ${\displaystyle M}$:

${\displaystyle {\begin{array}{rccc}{\mathcal {D}}:&G\times M&\rightarrow &M\\&\nu \times Z&\rightarrow &{\mathcal {D}}(\nu ,Z)\end{array}}}$

such that for all point ${\displaystyle Z}$ in ${\displaystyle M}$:

• ${\displaystyle {\mathcal {D}}(e,Z)=Z}$ where ${\displaystyle e}$ is the neutral element of ${\displaystyle G}$;
• for all ${\displaystyle (\nu ,{\hat {\nu }})}$ in ${\displaystyle G^{2}}$, ${\displaystyle {\mathcal {D}}(\nu ,{\mathcal {D}}({\hat {\nu }},Z))={\mathcal {D}}(\nu +{\hat {\nu }},Z)}$.

A continuous dynamical system is defined on a group ${\displaystyle G}$ that can be identified to ${\displaystyle \mathbb {R} }$ i.e. the group elements are continuous.

An invariant, roughly speaking, is an element that does not change under a transformation.

Let us define algebraic systems used in the forthcoming symmetry definition.

Let ${\displaystyle F=(f_{1},\dots ,f_{k})=(p_{1}/q_{1},\dots ,p_{k}/q_{k})}$ be a finite set of rational functions over the field ${\displaystyle \mathbb {R} }$ where ${\displaystyle p_{i}}$ and ${\displaystyle q_{i}}$ are polynomials in ${\displaystyle \mathbb {R} [Z]}$ i.e. in variables ${\displaystyle Z=(z_{1},\dots ,z_{n})}$ with coefficients in ${\displaystyle \mathbb {R} }$. An algebraic system associated to ${\displaystyle F}$ is defined by the following equalities and inequalities:

${\displaystyle {\begin{array}{ccc}\left\{{\begin{array}{l}p_{1}(Z)=0,\\\vdots \\p_{k}(Z)=0\end{array}}\right.&{\mbox{and}}&\left\{{\begin{array}{l}q_{1}(Z)\neq 0,\\\vdots \\q_{k}(Z)\neq 0.\end{array}}\right.\end{array}}}$

An algebraic system defined by ${\displaystyle F=(f_{1},\dots ,f_{k})}$ is regular (a.k.a. smooth) if the system ${\displaystyle F}$ is of maximal rank ${\displaystyle k}$, meaning that the Jacobian matrix ${\displaystyle (\partial f_{i}/\partial z_{j})}$ is of rank ${\displaystyle k}$ at every solution ${\displaystyle Z}$ of the associated semi-algebraic variety.

The following theorem (see th. 2.8 in ch.2 of [5]) gives necessary and sufficient conditions so that a local Lie group ${\displaystyle G}$ is a symmetry group of an algebraic system.

Theorem. Let ${\displaystyle G}$ be a connected local Lie group of a continuous dynamical system acting in the n-dimensional space ${\displaystyle \mathbb {R} ^{n}}$. Let ${\displaystyle F:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{k}}$ with ${\displaystyle k\leq n}$ define a regular system of algebraic equations:

${\displaystyle f_{i}(Z)=0\quad \forall i\in \{1,\dots ,k\}.}$

Then ${\displaystyle G}$ is a symmetry group of this algebraic system if, and only if,

${\displaystyle \delta f_{i}(Z)=0\quad \forall i\in \{1,\dots ,k\}{\mbox{ whenever }}f_{1}(Z)=\dots =f_{k}(Z)=0}$

for every infinitesimal generator ${\displaystyle \delta }$ in the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of ${\displaystyle G}$.

Consider the algebraic system defined on a space of 6 variables, namely ${\displaystyle Z=(P,Q,a,b,c,l)}$ with:

${\displaystyle \left\{{\begin{array}{l}f_{1}(Z)=(1-cP)+bQ+1,\\f_{2}(Z)=aP-lQ+1.\end{array}}\right.}$

The infinitesimal generator

${\displaystyle \delta =a(a-1){\dfrac {\partial }{\partial a}}+(l+b){\dfrac {\partial }{\partial b}}+(2ac-c){\dfrac {\partial }{\partial c}}+(-aP+P){\dfrac {\partial }{\partial P}}}$

is associated to one of the one-parameter symmetry groups. It acts on 4 variables, namely ${\displaystyle a,b,c}$ and ${\displaystyle P}$. One can easily verify that ${\displaystyle \delta f_{1}=f_{1}-f_{2}}$ and ${\displaystyle \delta f_{2}=0}$. Thus the relations ${\displaystyle \delta f_{1}=\delta f_{2}=0}$ are satisfied for any ${\displaystyle Z}$ in ${\displaystyle \mathbb {R} ^{6}}$ that vanishes the algebraic system.

Let us define systems of first-order ODEs used in the forthcoming symmetry definition.

Let ${\displaystyle d\cdot /dt}$ be a derivation w.r.t. the continuous independent variable ${\displaystyle t}$. We consider two sets ${\displaystyle X=(x_{1},\dots ,x_{k})}$ and ${\displaystyle \Theta =(\theta _{1},\dots ,\theta _{l})}$. The associated coordinate set is defined by ${\displaystyle Z=(z_{1},\dots ,z_{n})=(t,x_{1},\dots ,x_{k},\theta _{1},\dots ,\theta _{l})}$ and its cardinal is ${\displaystyle n=1+k+l}$. With these notations, a system of first-order ODEs is a system where:

${\displaystyle \left\{{\begin{array}{l}{\dfrac {dx_{i}}{dt}}=f_{i}(Z){\mbox{ with }}f_{i}\in \mathbb {R} (Z)\quad \forall i\in \{1,\dots ,k\},\\{\dfrac {d\theta _{j}}{dt}}=0\quad \forall j\in \{1,\dots ,l\}\end{array}}\right.}$

and the set ${\displaystyle F=(f_{1},\dots ,f_{k})}$ specifies the evolution of state variables of ODEs w.r.t. the independent variable. The elements of the set ${\displaystyle X}$ are called state variables, these of ${\displaystyle \Theta }$ 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 ${\displaystyle \delta }$ associated to a system of ODEs, described as above, is defined with the same notations as follows:

${\displaystyle \delta ={\dfrac {\partial }{\partial t}}+\sum _{i=1}^{k}f_{i}(Z){\dfrac {\partial }{\partial x_{i}}}\cdot }$

Here is a geometrical definition of such symmetries. Let ${\displaystyle {\mathcal {D}}}$ be a continuous dynamical system and ${\displaystyle \delta _{\mathcal {D}}}$ its infinitesimal generator. A continuous dynamical system ${\displaystyle {\mathcal {S}}}$ is a Lie point symmetry of ${\displaystyle {\mathcal {D}}}$ if, and only if, ${\displaystyle {\mathcal {S}}}$ sends every orbit of ${\displaystyle {\mathcal {D}}}$ to an orbit. Hence, the infinitesimal generator ${\displaystyle \delta _{\mathcal {S}}}$ satisfies the following relation[8] based on Lie bracket:

${\displaystyle [\delta _{\mathcal {D}},\delta _{\mathcal {S}}]=\lambda \delta _{\mathcal {D}}}$

where ${\displaystyle \lambda }$ is any constant of ${\displaystyle \delta _{\mathcal {D}}}$ and ${\displaystyle \delta _{\mathcal {S}}}$ i.e. ${\displaystyle \delta _{\mathcal {D}}\lambda =\delta _{\mathcal {S}}\lambda =0}$. These generators are linearly independent.

One does not need the explicit formulas of ${\displaystyle {\mathcal {D}}}$ in order to compute the infinitesimal generators of its symmetries.

Consider Pierre François Verhulst's logistic growth model with linear predation,[14] where the state variable ${\displaystyle x}$ represents a population. The parameter ${\displaystyle a}$ is the difference between the growth and predation rate and the parameter ${\displaystyle b}$ corresponds to the receptive capacity of the environment:

${\displaystyle {\dfrac {dx}{dt}}=(a-bx)x,{\dfrac {da}{dt}}={\dfrac {db}{dt}}=0.}$

The continuous dynamical system associated to this system of ODEs is:

${\displaystyle {\begin{array}{rccc}{\mathcal {D}}:&(\mathbb {R} ,+)\times \mathbb {R} ^{4}&\rightarrow &\mathbb {R} ^{4}\\&({\hat {t}},(t,x,a,b))&\rightarrow &\left(t+{\hat {t}},{\frac {axe^{a{\hat {t}}}}{a-(1-e^{a{\hat {t}}})bx}},a,b\right).\end{array}}}$

The independent variable ${\displaystyle {\hat {t}}}$ varies continuously; thus the associated group can be identified with ${\displaystyle \mathbb {R} }$.

The infinitesimal generator associated to this system of ODEs is:

${\displaystyle \delta _{\mathcal {D}}={\dfrac {\partial }{\partial t}}+((a-bx)x){\dfrac {\partial }{\partial x}}\cdot }$

The following infinitesimal generators belong to the 2-dimensional symmetry group of ${\displaystyle {\mathcal {D}}}$:

${\displaystyle \delta _{{\mathcal {S}}_{1}}=-x{\dfrac {\partial }{\partial x}}+b{\dfrac {\partial }{\partial b}},\quad \delta _{{\mathcal {S}}_{2}}=t{\dfrac {\partial }{\partial t}}-x{\dfrac {\partial }{\partial x}}-a{\dfrac {\partial }{\partial a}}\cdot }$