These are the lecture notes of a talk I gave in the Graduate Mathematical Physics Seminar at the University of Arizona on Wednesday September 3, 2008.

You have probably seen Newton's second law F=ma, but have you ever wondered why does the world actually evolve (to some approximation) according to this equation?! What is so special about it?

'Why' questions are always difficult. But the answer to this (vaguely stated...) question was given by the 'A-team': Lagrange, Euler and Hamilton about 200 years ago. Surprisingly enough the answer is a BEAUTIFUL geometric interpretation of the laws of nature. Fortunately, it also involves a lot of beautiful mathematics.

We start start by describing Newtonian mechanics in a nutshell, deriving the beautiful Euler-Lagrange equation and using it to prove that the shortest path between two points is a straight-line. Then we follow Hamilton's principle to formulate the Lagrangian mechanics, and discuss how naturally this theory 'fits' into the theory of differentiable manifolds. The last part discusses Lagrangian formulation of the more subtle case, in which energy is not conserved, and the importance of the Lagrangian formulation as a uniting principle in physics.