A Brief Introduction to Classical Mechanics with Illustrative Problems

The following sections are included:

• Preface
• Contents

Isaac Newton (1642–1727) defined his famous three laws of motion in the Principia in the following form:

• Law I: Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.
• Law II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
• Law III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts

Newton’s law of gravity specifies that two masses, m₁ and m₂, attract each other with a force

Ernest Rutherford (1871–1937) deduced the structure of the atom with a classical experiment performed in 1909 by his collaborators Hans Geiger (1882–1945) and Ernest Marsden (1889–1970). They fired alpha particles (nuclei of helium, as we now know) to a thin sheet of gold and noted that most of them were not deflected from their initial trajectories, but a few suffered a strong scattering. This effect can only be explained if the alpha particles are scattered by an almost point-like charge, in this case, the nuclei of gold atoms…

Until now, we considered fixed reference frames without inertial forces. In this chapter, we study the motion in a rotating frame in which there are inertial forces that must be taken into account. Consider then a two-dimensional Cartesian coordinate system, (x′, y′) rotating around the origin of a fixed system (x, y) with angular velocity Ω, as shown in Fig. 4.1. The relation between the two systems of coordinates is given by the formula

Up to this point, we have considered the motion of bodies that can be approximated as point particles. That is the case, for instance, of a planet orbiting the Sun: its size is much smaller than the orbit and can therefore be approximated by a point. However, there are other problems in which it is necessary to take into account the finite size of the body under study. This chapter is devoted to the motion of solid bodies of finite size.

This chapter is devoted to the formalism of Lagrange, which, together with the Hamiltonian formalism (see Chapter 7), is the core of analytical mechanics. The basic idea of the Lagrangian formalism is the following. Until now, we have used mainly Cartesian or, occasionally, spherical coordinates, adapted to the geometry of each particular problem. However, a mechanical system may be constrained to move with some restrictions, and a particular coordinate system may be the most appropriate to deal with the problem. For instance, a spherical pendulum moves in a three-dimensional space, but its motion is restricted to a sphere that can be described by two coordinates, and the force acting along the string is compensated by its tension (Newton’s third law!). Another simple example is a particle moving on a fixed surface (Fig. 6.1); the force of gravity acting on the particle has two components, one parallel to the surface and another perpendicular to it: it is only the first force that produces motion, while the latter is compensated by the reaction of the surface and does not contribute to the motion.

Given a mechanical system with F degrees of freedom, the Lagrangian formalism permits one to deduce F equations of motion for the F generalized coordinates, which are second-order differential equations. An alternative approach is to redefine the first temporal derivatives of each generalized coordinate and thus obtain 2F differential equations of first-order. This is the approach taken by Hamilton in his formulation of mechanics.

Having learned how to minimize a functional such as the action, we can use the same procedure to solve a variety of problems. Let us work out three illustrative examples in the following sections.

The following sections are included:

• Appendix A: Derivation of the Lagrange Equations
• Appendix B: Prolate and Oblate Coordinates
• Appendix C: Laplace Equation

The following sections are included:

• Selected Bibliography
• Index