Lie symmetry analysis and exact solutions to the quintic nonlinear beam equation

In this paper,the exact solutions to the equation of motion of the nonlinear vibration of euler-bernoulli beam which is governed by the quintic nonlinear equation are investigated by using lie symmetry analysis. the leading tools for transforming the equation of motion which is in the form of partial differential equation into an ordinary differential equation are the infinitesimal generators. these generators are calculated by using technique of group transformation. the lie algebra of the infinitesimal generator is spanned by four linearly independent generators. an optimal system of subalgebra is constructed. invariants are calculated by solving the characteristic system and then designate one of invariant as a function of the others. then the partial differential equation can be transformed to the ordinary differential equation. based on an optimal system,in some cases the ordinary differential equation can be solved and exact solutions are obtained.