Abstract: It is well known that the theory for many dimensions non-linear mechanical systems is very difficult and it can be applied only to very particular cases. Generally speaking, the approximation of the solution is performed only for one or two dimension systems and, in addition, the equations of motions are assumed to be in a proper form. There is neither an approach for the systems with more than two degrees of freedom, nor indication about how we can obtain the desired form for the equations of neither motion, nor proofs if the theory applied in one or two degrees of freedom can be used in the general case. It is our goal to develop a general theory for these situations. In this paper we started with some backgrounds about the normal forms for vector fields. This theory leads us to a generalization for the multi dimensional systems that characterized the non-linear oscillations of the mechanical systems. Based on this, we obtain the simplest form for the differential equations of motion. We called this form the canonical form, which is met in the literature. The transformation to the canonical form is performed in the most general case and it also offers us the necessary and sufficient conditions for the stability of the motion. The theory developed in this paper is finally applied to a few mechanical systems which are completely solved.

Brief Biography of the Speaker:
Nicolae-Doru Stanescu (born 1965) graduated the Faculty of Machines Construction’s Technology at the “Politehnica” University of Bucharest in 1989, and the Faculty of Mathematics and Computer Science at the University of Pitesti in 1995. Since 2003 he is PhD in Mechanical Engineering at the University of Pitesti, and since 2008 he is PhD in Mathematics at the University of Bucharest. Now, he is Associate Professor at the Department of Applied Mechanics of the University of Pitesti, where he teaches Mechanics, Numerical Methods and Non-linear Vibrations. He wrote more than 100 articles and 6 books. He participated as researcher or was director at 8 grants. He is member of the International Institute of Accoustic and Vibration in USA, and of Societe des Ingineurs de l’Automobile, France, among other associations. He was invited professor at Instituto Superior Tecnico, Lisbon, Portugal. His fields of interests are: non-linear vibrations, dynamical systems, stability, chaos, and numerical analysis.