1. The Lagrangian

We will in this chapter use calculus of variations to get the equations of motion for a mechanical system. I next part we will see that the motion of the system is such that a specific integral is minimized. Consider a mechanical system


that is




are the so called generalized coordinates and


are called generalized velocities. If aij are known functions of the coordinates y1,y2,…,yn, we denote 


the generalized kinetic energy and the generalized potential energy as


We can now define the Lagrangian as 


If we consider the general coordinates y1,y2,…,yn, as coordinates in Rn the equations


can be considered parametric equationes for a curve C connecting two states S0 and S1 in space.