A Lie point symmetry is a continuous transformation of independent and dependent variables that maps solutions of a differential equation to solutions.
A one-parameter point transformation has infinitesimal generatorIts flow recovers the finite transformations.
The prolongation of a point-symmetry generator is its induced vector field on derivatives of the dependent variable. A generator is a symmetry of a differential equation when its required prolongation sends to zero on the solution manifold.
A scaling symmetry acts by weighted dilations of the independent and dependent variables. Its infinitesimal generator is a weighted sum of Euler vector fields such as .
An invariant of a one-parameter symmetry group becomes a similarity variable that reduces a PDE to an ODE.
A group-invariant solution is constant along the prolonged symmetry orbits. For a scaling of two independent variables with fixed dependent variable, it depends only on a ratio such as .
The vector fieldgenerates the local transformationsIts second prolongation sends to times that expression, so it is a Lie point symmetry of the potential Burgers equation.
A similarity variable is constant along symmetry-group orbits and parametrizes group-invariant solutions.
Articles by others on the same topic
Lie point symmetry is a concept from the field of differential equations and mathematical physics, named after the mathematician Sophus Lie. It specifically refers to symmetries of differential equations that can be expressed in terms of point transformations of the independent and dependent variables. In simpler terms, if a differential equation remains invariant under a transformation that is generated by a continuous group of transformations, then it possesses a Lie point symmetry.