To quote from Lanczos [LAN 70], "by founding the analytical mechanics, Lagrange added nothing fundamentally different to Newton's laws, but provided an immensely powerful weapon to solve any mechanical problem on the basis of pure calculation" and his book "M canique Analytique", first published in 1788 "may be considered rightly as an extraordinary achievement, which opened an entirely new world to mathematical modelling in mechanics". Analytical mechanics is entirely formulated in terms of generalized coordinates, providing thus mathematical expressions which hold independently of the specific coordinate system chosen. It introduces, as prime concepts, scalar quantities such as kinetic and potential energies, instead of force. Then, a variational principle concerning energy is used to derive the equations of motion in a purely analytical way. Variational principles and methods are very appealing in various branches of theoretical physics, as they bring out the symmetrical nature of the fundamental laws of physics. Following the approach of many textbooks devoted to mechanical engineering, the formalism of Lagrange's equations will be introduced here starting from the principle of virtual work. Nevertheless, it will be restated in the next chapter, starting from the principle of least action, as formulated by Hamilton, which offers richer perspectives both for a deeper understanding of physics and for the development of mathematical methods to solve practical problems.