Symmetry and new solutions of the equation for vibrations of an elastic beam

In this short paper it is studied the “not Lie” symmetry of the beam equation. All operators of symmetry linear di？erential until third order are constructed. It is noted that the resolution of this equation reduces to ？nding solutions of two Kolmogorov equations. Several class of new solutions of the beam equation are found, particularly those that verify the law of conservation of the initial areolar speed and other that verify the law of conservation of the initial elasticity. It is showed the equivalence between the solution of the problem Cauchy and the existence of a speci？c symmetry. It is found a parallelism between the beam equation and the wave equation. Using the Ansatz method a wide new family of exact solutions is built. This family includes particularly the solutions which describe the propagation of damped waves. All results of this paper are new