Symmetries and Conservation Laws

This chapter explores what symmetries have to do with the conservation laws. The Lagrangian is the difference between the kinetic energy and the potential energy, that is to be regarded as a function of the independent variables for the purposes of partial differentiation. The transition from classical to quantum mechanics is made more transparent by considering the Hamiltonian formulation. The idea, in the first place, is to find a change in variables which will replace the second order Euler-Lagrange equation by two linked first order equations. The chapter introduces an infinite number of oscillators each with minimum energy, so that the total energy of our "continuous chain of atoms" is infinite. The conventional view is that this is not a real problem, only the energy differences really matter.