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| {{Lie groups |Algebras}}
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| 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]]<ref name="Lie1881"/><ref name="Lie1890"/><ref name="Lie1893"/> (ODEs). He showed the following main property: the order of an ordinary differential equation can be reduced by one if it is [[invariant (mathematics)|invariant]] under one-parameter Lie group of [[point transformation]]s.<ref name="Olver1993"/> This observation unified and extended the available integration techniques. Lie devoted the remainder of his mathematical career to developing these [[continuous group]]s that have now an impact on many areas of mathematically-based sciences. The applications of Lie groups to [[differential system]]s were mainly established by Lie and [[Emmy Noether]], and then advocated by [[Élie Cartan]].
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| 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 [[Scaling (geometry)|scalings]].
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| The Lie symmetry theory is a well-known subject. In it are discussed [[continuous symmetry|continuous symmetries]] opposed to, for example, [[discrete symmetry|discrete symmetries]]. The literature for this theory can be found, among other places, in these notes.<ref name="Olver1995"/><ref name="Olver1999"/><ref name="BlumanKumei1989"/><ref name="Stephani1989"/><ref name="LeviWinternitz2006"/>
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| == Overview ==
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| === Types of symmetries ===
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| Lie groups and hence their infinitesimal generators can be naturally "extended" to act on the space of independent variables, [[state variable]]s (dependent variables) and derivatives of the state variables up to any finite order. There are many other kinds of symmetries. For example, [[contact geometry|contact transformations]] let coefficients of the transformations infinitesimal generator depend also on first derivatives of the coordinates. [[Lie-Bäcklund transformation]]s let them involve derivatives up to an arbitrary order. The possibility of the existence of such symmetries was recognized by Noether.<ref name="Noether1918"/> For Lie point symmetries, the coefficients of the infinitesimal generators depend only on coordinates, denoted by <math>Z</math>.
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| === Applications ===
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| 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 (mathematics)|reduction]]. In the literature, one can find the classical reduction process,<ref name="Olver1993"/> and the [[moving frame]]-based reduction process.<ref name="Cartan1935"/><ref name="FelsOlver1998"/><ref name="FelsOlver1999"/> Also symmetry groups can be used for classifying different symmetry classes of solutions.
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| == Geometrical framework ==
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| === Infinitesimal approach ===
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| Lie's fundamental theorems underline that Lie groups can be characterized by their [[infinitesimal generator]]s. 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.
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| Let <math>Z=(z_1,\dots,z_n)</math> be the set of coordinates on which a system is defined where <math>n</math> is the cardinal of <math>Z</math>. An infinitesimal generator <math>\delta</math> in the field <math>\mathbb{R}(Z)</math> is a linear operator <math>\delta : \mathbb{R}(Z)\rightarrow \mathbb{R}(Z)</math> that has <math>\mathbb{R}</math> in its kernel and that satisfies the [[Product rule|Leibniz rule]]:
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| :<math>\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</math>.
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| In the canonical basis of elementary derivations <math>\left\{\frac{\partial}{\partial z_1},\dots,\frac{\partial}{\partial z_n}\right \}</math>, it is written as:
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| :<math>\delta = \sum_{i=1}^{n} \xi_{z_i}(Z) \frac{\partial}{\partial z_i}</math> | |
| where <math>\xi_{z_i}</math> is in <math>\mathbb{R}(Z)</math> for all <math>i</math> in <math>\left\{1,\dots,n\right\}</math>.
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| === Lie groups and Lie algebras of infinitesimal generators ===
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| {{Expand section|date=May 2010}}
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| [[Lie algebra]]s can be generated by a generating set of [[infinitesimal generator]]s. To every Lie group, one can associate a Lie algebra. Roughly, a Lie algebra <math>\mathfrak{g}</math> is an [[algebra]] constituted by a vector space equipped with [[Lie bracket of vector fields|Lie bracket]] as additional operation. The base field of a Lie algebra depends on the concept of [[invariant (mathematics)|invariant]]. Here only finite-dimensional Lie algebras are considered.
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| === Continuous dynamical systems === | |
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| A [[dynamical system]] (or [[Flow (mathematics)|flow]]) is a one-parameter [[group action]]. Let us denote by <math>\mathcal{D}</math> such a dynamical system, more precisely, a (left-)action of a group <math>(G,+)</math> on a [[manifold]] <math>M</math>:
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| :<math>
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| \begin{array}{rccc}
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| \mathcal{D} : & G\times M & \rightarrow & M \\
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| & \nu \times Z & \rightarrow & \mathcal{D}(\nu,Z)
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| \end{array}
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| </math>
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| such that for all point <math>Z</math> in <math>M</math>:
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| * <math>\mathcal{D}(e,Z)=Z</math> where <math>e</math> is the neutral element of <math>G</math>;
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| * for all <math>(\nu, \hat{\nu})</math> in <math>G^2</math>, <math>\mathcal{D}(\nu,\mathcal{D}(\hat{\nu},Z))=\mathcal{D}(\nu+\hat{\nu},Z)</math>.
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| A continuous dynamical system is defined on a group <math>G</math> that can be identified to <math>\mathbb{R}</math> i.e. the group elements are continuous.
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| === Invariants ===
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| {{Expand section|date=May 2010}}
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| An [[invariant (mathematics)|invariant]], roughly speaking, is an element that does not change under a transformation.
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| == Definition of Lie point symmetries ==
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| 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.
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| A symmetry group of a system is a continuous dynamical system defined on a local Lie group <math>G</math> acting on a manifold <math>M</math>. For the sake of clarity, we restrict ourselves to n-dimensional real manifolds <math>M=\mathbb{R}^n</math> where <math>n</math> is the number of system coordinates.
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| === Lie point symmetries of algebraic systems ===
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| Let us define [[algebraic system]]s used in the forthcoming symmetry definition.
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| ==== Algebraic systems ====
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| Let <math>F=(f_1,\dots,f_k)=(p_1/q_1,\dots,p_k/q_k)</math> be a finite set of rational functions over the field <math>\mathbb{R}</math> where <math>p_i</math> and <math>q_i</math> are polynomials in <math>\mathbb{R}[Z]</math> i.e. in variables <math>Z=(z_1,\dots,z_n)</math> with coefficients in <math>\mathbb{R}</math>. An ''algebraic system'' associated to <math>F</math> is defined by the following equalities and inequalities:
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| :<math>
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| \begin{array}{ccc}
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| \left\{
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| \begin{array}{l}
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| p_1(Z)= 0, \\
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| \vdots \\
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| p_k(Z)=0
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| \end{array}
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| \right.& \mbox{and} &
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| \left\{
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| \begin{array}{l}
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| q_1(Z) \neq 0, \\
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| \vdots \\
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| q_k(Z) \neq 0.
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| \end{array}
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| \right.
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| \end{array}
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| </math>
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| An algebraic system defined by <math>F=(f_1,\dots,f_k)</math> is ''regular'' (a.k.a. [[smooth]]) if the system <math>F</math> is of maximal rank <math>k</math>, meaning that the [[Jacobian matrix]] <math>(\partial f_i / \partial z_j)</math> is of rank <math>k</math> at every solution <math>Z</math> of the associated semi-algebraic [[algebraic variety|variety]].
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| ==== Definition of Lie point symmetries ====
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| The following theorem (see th. 2.8 in ch.2 of <ref name="Olver1995"/>) gives necessary and sufficient conditions so that a local Lie group <math>G</math> is a symmetry group of an algebraic system.
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| '''Theorem'''. Let <math>G</math> be a connected local Lie group of a continuous dynamical system acting in the n-dimensional space <math>\mathbb{R}^n</math>. Let <math>F: \mathbb{R}^n \rightarrow \mathbb{R}^k</math> with <math>k \leq n</math> define a regular system of algebraic equations:
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| :<math>
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| f_i(Z)=0 \quad \forall i \in \{1,\dots,k\}.
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| </math>
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| Then <math>G</math> is a symmetry group of this algebraic system if, and only if,
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| :<math>
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| \delta f_i(Z)=0 \quad \forall i\in \{1,\dots,k\} \mbox{ whenever } f_1(Z)=\dots=f_k(Z)=0
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| </math>
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| for every infinitesimal generator <math>\delta</math> in the Lie algebra <math>\mathfrak{g}</math> of <math>G</math>.
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| ==== Example ====
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| Let us consider the algebraic system defined on a space of 6 variables, namely <math>Z=(P,Q,a,b,c,l)</math> with:
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| :<math>
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| \left \{
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| \begin{array}{l}
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| f_1(Z)=(1-cP)+bQ + 1, \\
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| f_2(Z)=aP - lQ +1.
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| \end{array}
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| \right .
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| </math>
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| The infinitesimal generator
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| :<math>
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| \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}
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| </math>
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| is associated to one of the one-parameter symmetry groups. It acts on 4 variables, namely <math>a,b,c</math> and <math>P</math>. One can easily verify that <math>\delta f_1 = f_1 -f_2</math> and <math>\delta f_2 = 0</math>. Thus the relations <math>\delta f_1 = \delta f_2 = 0</math> are satisfied for any <math>Z</math> in <math>\mathbb{R}^6</math> that vanishes the algebraic system.
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| === Lie point symmetries of dynamical systems ===
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| Let us define systems of first-order [[ordinary differential equations|ODEs]] used in the forthcoming symmetry definition.
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| ==== Systems of ODEs and associated infinitesimal generators ====
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| Let <math>d\cdot / dt</math> be a derivation w.r.t. the continuous independent variable <math>t</math>. We consider two sets <math>X=(x_1,\dots,x_k)</math> and <math>\Theta = (\theta_1,\dots,\theta_l)</math>. The associated coordinate set is defined by <math>Z=(z_1,\dots,z_n)=(t,x_1,\dots,x_k,\theta_1,\dots,\theta_l)</math> and its cardinal is <math>n=1+k+l</math>. With these notations, a ''system of first-order ODEs'' is a system where:
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| :<math>
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| \left \{
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| \begin{array}{l}
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| \dfrac{d x_i}{dt} = f_i(Z) \mbox{ with } f_i \in \mathbb{R}(Z) \quad \forall i \in \{1,\dots,k\}, \\
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| \dfrac{d \theta_j}{dt} = 0 \quad \forall j \in \{1,\dots,l\}
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| \end{array}
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| \right .
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| </math>
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| and the set <math>F=(f_1,\dots,f_k)</math> specifies the evolution of state variables of ODEs w.r.t. the independent variable. The elements of the set <math>X</math> are called ''state variables'', these of <math>\Theta</math> ''parameters''.
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| One can associate also a continuous dynamical system to a system of ODEs by resolving its equations.
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| 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.<ref name="Olver1993"/> The infinitesimal generator <math>\delta</math> associated to a system of ODEs, described as above, is defined with the same notations as follows:
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| :<math>
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| \delta = \dfrac{\partial}{\partial t} + \sum_{i=1}^{k} f_i(Z) \dfrac{\partial}{\partial x_i}\cdot
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| </math>
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| ==== Definition of Lie point symmetries ====
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| Here is a geometrical definition of such symmetries. Let <math>\mathcal{D}</math> be a continuous dynamical system and <math>\delta_\mathcal{D}</math> its infinitesimal generator. A continuous dynamical system <math>\mathcal{S}</math> is a Lie point symmetry of <math>\mathcal{D}</math> if, and only if, <math>\mathcal{S}</math> sends every orbit of <math>\mathcal{D}</math> to an orbit. Hence, the infinitesimal generator <math>\delta_\mathcal{S}</math> satisfies the following relation<ref name="Stephani1989"/> based on [[Lie bracket]]:
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| :<math>
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| [\delta_\mathcal{D}, \delta_\mathcal{S}] = \lambda \delta_\mathcal{D}
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| </math>
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| where <math>\lambda</math> is any constant of <math>\delta_\mathcal{D}</math> and <math>\delta_\mathcal{S}</math> i.e. <math>\delta_\mathcal{D}\lambda = \delta_\mathcal{S} \lambda = 0</math>. These generators are linearly independent.
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| One does not need the explicit formulas of <math>\mathcal{D}</math> in order to compute the infinitesimal generators of its symmetries.
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| === Example ===
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| Consider [[Pierre François Verhulst]]'s [[logistic growth]] model with linear predation,<ref name="Murray2002"/> where the state variable <math>x</math> represents a population. The parameter <math>a</math> is the difference between the growth and predation rate and the parameter <math>b</math> corresponds to the receptive capacity of the environment:
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| :<math>
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| \dfrac{dx}{dt}= (a-bx)x, \dfrac{da}{dt}=\dfrac{db}{dt}=0.
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| </math>
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| The continuous dynamical system associated to this system of ODEs is:
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| :<math>
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| \begin{array}{rccc}
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| \mathcal{D}: & (\mathbb{R},+) \times \mathbb{R}^4 & \rightarrow & \mathbb{R}^4 \\
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| & (\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).
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| \end{array}
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| </math>
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| The independent variable <math>\hat{t}</math> varies continuously; thus the associated group can be identified with <math>\mathbb{R}</math>.
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| The infinitesimal generator associated to this system of ODEs is:
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| :<math>
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| \delta_\mathcal{D} = \dfrac{\partial}{\partial t} + ((a-bx)x)\dfrac{\partial}{\partial x}\cdot
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| </math>
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| The following infinitesimal generators belong to the 2-dimensional symmetry group of <math>\mathcal{D}</math>:
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| :<math>
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| \delta_{\mathcal{S}_1} = -x \dfrac{\partial}{\partial x}+b\dfrac{\partial}{\partial b},
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| \quad
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| \delta_{\mathcal{S}_2} = t\dfrac{\partial}{\partial t}-x\dfrac{\partial}{\partial x}-a\dfrac{\partial}{\partial a}
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| \cdot
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| </math>
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| == Software ==
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| There exist many software packages in this area.<ref name="Heck2003"/><ref name="Schwarz1988"/><ref name="Dimas2005"/> For example, the package liesymm of [[Maple (software)|Maple]] provides some Lie symmetry methods for [[partial differential equation|PDEs]].<ref name="CarminatiDevittFee1992"/> It manipulates integration of determining systems and also [[differential form]]s. 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 field]]s for searching Lie symmetries of ODEs. Finding Lie symmetries for ODEs, in the general case, may be as complicated as solving the original system.
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| == References ==
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| {{Reflist|refs=
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| <ref name="BlumanKumei1989">{{cite book |title= Symmetries and Differential Equations|last1= Bluman |first1 = G. |last2=Kumei |first2= S.|year=1989 |publisher= Springer-Verlag|location= New York |edition= Second |series= Applied Mathematical Sciences Series|volume= 81}}</ref>
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| <ref name="Murray2002">{{cite book |title= Mathematical Biology |last= Murray |first= J. D. |year= 2002 |publisher= Springer |series= Interdisciplinary Applied Mathematics|volume= 17}}</ref>
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| <ref name="Heck2003">{{cite book |title= Introduction to Maple|last= Heck|first= A.|year= 2003|publisher= Springer-Verlag |edition=Third}}</ref>
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| <ref name="Lie1890">{{cite book |title= Theorie der Transformationsgruppen|last= Lie |first= Sophus |year= 1890| publisher= Teubner, Leipzig |volume =2| language =german}}</ref>
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| <ref name="Lie1893">{{cite book |title= Theorie der Transformationsgruppen|last= Lie |first= Sophus |year= 1893| publisher= Teubner, Leipzig |volume =3| language =german}}</ref>
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| <ref name="Olver1993">{{cite book |title= Applications of Lie Groups to Differential Equations|last= Olver |first= Peter J. |year= 1993 |publisher= Springer-Verlag | edition= Second }}</ref>
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| <ref name="Olver1995">{{cite book |title= Equivalence, Invariance and Symmetry|last= Olver |first= Peter J. |year= 1995 |publisher= Cambridge University Press}}</ref>
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| <ref name="Olver1999">{{cite book |title= Classical Invariant Theory|last= Olver |first= Peter J. |year= 1999 |publisher= Cambridge University Press|edition=First}}</ref>
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| <ref name="Stephani1989">{{cite book |title= Differential Equations |last= Stephani|first= H. |year= 1989|publisher= Cambridge University Press |edition=First}}</ref>
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| <ref name="FelsOlver1998">{{cite journal |last1= Fels|first1= M.|last2= Olver |first2= Peter J.|date=April 1998|title= Moving Coframes: I. A Practical Algorithm|journal= Acta Applicandae Mathematicae|volume=51 |issue= 2|pages= 161–213}}</ref>
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| <ref name="FelsOlver1999">{{cite journal |last1= Fels|first1= M.|last2= Olver |first2= Peter J.|date=January 1999|title= Moving Coframes: II. Regularization and theoretical foundations|journal= Acta Applicandae Mathematicae|volume=55 |issue= 2|pages= 127–208 |doi=10.1023/A:1006195823000}}</ref>
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| <ref name="LeviWinternitz2006">{{cite journal |last1= Levi|first1= D.|last2= Winternitz|first2= P.|year= 2006|title= Continuous symmetries of difference equations|journal=Journal of Physics: Mathematical and General |volume= 39 |pages= R1-R63}}</ref>
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| <ref name="Schwarz1988">{{cite journal |last= Schwarz|first= F.|year= 1988|title= Symmetries of differential equations: from Sophus Lie to computer algebra|journal= [[SIAM Review]]|volume= 30 |pages= 450–481}}</ref>
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| <ref name=Dimas2005>{{cite journal|last=Dimas|first=S.|coauthors=Tsoubelis, T.|title=SYM: A new symmetry-finding package for Mathematica|journal=The 10th International Conference in MOdern GRoup ANalysis|year=2005|pages=64–70|location=University of Cyprus, Nicosia, Cyprus|url=http://www2.ucy.ac.cy/~mogran10/Proof/Dimas.pdf}}</ref>
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| <ref name="CarminatiDevittFee1992">{{cite journal |last1= Carminati|first1= J. |last2= Devitt|first2= J. S. |last3= Fee |first3=G. J. |year= 1992 |title= Isogroups of differential equations using algebraic computing|journal= Journal of Symbolic Computation|volume= 14|issue= 1 |pages= 103–120}}</ref>
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| <ref name="Cartan1935">{{cite journal |last= Cartan|first= Elie |year= 1935|title= La méthode du repère mobile, la théorie des groups continus et les espaces généralisés.|journal= Exposés de géométrie - 5 Hermann |location= Paris|language= french}}</ref>
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| <ref name="Lie1881">{{cite journal |last= Lie|first= Sophus |year= 1881 |title= Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen|journal= Arch. for Math|volume= 6 |pages= 328–368 |language= german}}</ref>
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| <ref name="Noether1918">{{cite journal |last= Noether |first= E. |year= 1918 |title= Invariante Variationsprobleme. Nachr. König. Gesell. Wissen.|journal= Math.-Phys. Kl. |pages= 235–257|location=Göttingen |language= german}}</ref>
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| }}
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| {{DEFAULTSORT:Lie Point Symmetry}}
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| [[Category:Lie groups]]
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| [[Category:Symmetry]]
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