Chapter Six: Energy Conservation

OVERVIEW

KEY FEATURES

The key features of this chapter are the energy principle for a particle, conservative fields of force, potential energies and energy conservation.

In this Chapter, we introduce the notion of mechanical energy and its conservation. Although energy methods are never indispensible ^[*] for the solution of problems, they do give a greater insight and allow many problems to be solved in a quick and elegant manner. Energy has a fundamental r ole in the Lagrangian and Hamiltonian formulations of mechanics. More generally, the notion of energy has been so widely extended that energy conservation has become the most pervasive and important principle in the whole of physics.

6.1 THE ENERGY PRINCIPLE

Suppose a particle P of mass m moves under the influence of a force F. Then its equation of motion is

where ? is the velocity of P at time t. At this stage we place no restrictions on the force F. It may depend on the position of P, the velocity of P, the time, or anything else; if more than one force is acting on P, then F means the vector resultant of these forces. On taking the scalar product of both sides of equation (6.1) with ?, we obtain the scalar equation

and, since

this can be written in the form

where T