Introduction to Lagrangian Mechanics

Abstract If modern mechanics began with Isaac Newton, modern analytical mechanics can be said to have begun with the work of the 18th-century mathematicians who elaborated his ideas. Without changing Newton’s fundamental principles, Leonhard Euler, Pierre-Simon Laplace, and Joseph Louis Lagrange developed elegant computational methods for the increasingly complex problems to which Newtonian mechanics was being applied. The Lagrangian formulation of mechanics is, at first glance, merely an abstract way of writing Newton’s second law: the law of angular momentum. This chapter discusses Lagrangian mechanics as well as configuration space, Newton’s second law in Lagrangian form, arbitrary generalised coordinates, generalised velocities in the q-system, generalised forces in the q-system, expression of Lagrangian mechanics in the q-system, invariance of the Lagrange equations, generalised momenta in the q-system, ignorable coordinates, generalised energy function, generalized energy and total energy, and velocity dependent potentials.