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. 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. 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 Z {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, and the moving frame-based reduction process. 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 Z = ( z 1 , … , z n ) {displaystyle Z=(z_{1},dots ,z_{n})} be the set of coordinates on which a system is defined where n {displaystyle n} is the cardinal of Z {displaystyle Z} . An infinitesimal generator δ {displaystyle delta } in the field R ( Z ) {displaystyle mathbb {R} (Z)} is a linear operator δ : R ( Z ) → R ( Z ) {displaystyle delta :mathbb {R} (Z) ightarrow mathbb {R} (Z)} that has R {displaystyle mathbb {R} } in its kernel and that satisfies the Leibniz rule: In the canonical basis of elementary derivations { ∂ ∂ z 1 , … , ∂ ∂ z n } {displaystyle left{{frac {partial }{partial z_{1}}},dots ,{frac {partial }{partial z_{n}}} ight}} , it is written as: where ξ z i {displaystyle xi _{z_{i}}} is in R ( Z ) {displaystyle mathbb {R} (Z)} for all i {displaystyle i} in { 1 , … , n } {displaystyle left{1,dots ,n ight}} .