CHAPTER 1 - THE EQUATIONS OF MOTION

Abstract

The relations among accelerations, velocities, and coordinates are called the equations of motion. They are second-order differential equations for the functions and their integration makes possible, in principle, the determination of these functions and so of the path of the system. This chapter reviews generalized coordinates and the principle of least action. The most general formulation of the law governing the motion of mechanical systems is the principle of least action or Hamilton's principle. The chapter also discusses Galileo's relativity principle. Galileo's relativity principle states that the laws of nature are invariant under transformation. The chapter further discusses the determination of the form of the Lagrangian and considers all the simplest case that of the free motion of a particle relative to an inertial frame of reference. As the homogeneity of space and time, the Lagrangian of a free particle cannot depend explicitly on either the position vector r or the time t , that is, L is a function of the velocity ν only; because of the isotropy of space, the Lagrangian must also be independent of the direction of the vector v and is, therefore, a function only of its magnitude, that is, v 2 = ν 2 : L = L( ν 2 ). This form of this function is uniquely determined by Galileo's relativity principle, which shows that L( ν 2 ) must have the same form in every inertial frame of reference. The chapter also explains a system of particles that interact with one another but with no other bodies. This is called a closed system.