Classical mechanics embraces an unusually broad range of topics, far more than can be discussed during this term. Classical mechanics addresses basic problems ranging from motion of astronomical bodies to nuclear and particle physics, from one-body to many-body motion, statistical and quantum mechanics. The foundations of classical mechanics were first established during the seventeenth century by Newton and Galileo; this approach is called Newtonian mechanics. The essential physics contained in the Newtonian formulation of classical mechanics is contained in Newton's three laws of motion. The Newtonian formulation is the most intuitive approach to classical mechanics in that it is based on vector quantities like force, momentum, acceleration, etc, which are easy to visualize, while the theory has embedded into it both cause and effect.
Lagrange, Euler, Hamilton and Jacobi developed powerful algebraic formulations of classical mechanics during the eighteenth and nineteenth centuries. These approaches are based on work done by von Liebnitz contemporaneously with Newton. The Lagrangian formulation is cast in terms of kinetic and potential energies, that involve only scalar functions, and the philosophical belief is that the physical universe follows paths through time and space that are extrema. The Hamiltonian and Hamilton-Jacobi formulations of classical mechanics are closely related to the Lagrange formulation.
From the late seventeenth century, until the dawn of modern physics at the start of the twentieth century, classical mechanics was a primary driving force in the development of physics. These advances contributed to, and benefitted from, parallel developments in mathematics. Introduction of the Theory of Relativity in 1905 necessitated modifications to classical mechanics for relativistic velocities and can be considered to be an extended theory of classical mechanics. Since the 1930's quantal physics has superseded classical mechanics in the microscopic domain. In spite of this, Hamiltonian and Lagrangian dynamics provided the foundation upon which quantum mechanics, statistical mechanics and quantum field theory were built. For example, you will be familiar with the use of the Hamiltonian in the Schrödinger equation while both Dirac and Feynman used Lagrangian mechanics in quantum field theory. Although quantum physics has played the leading role in the development of physics during much of the past century, classical mechanics still is a vibrant field of physics that recently has led to exciting developments associated with chaos theory that has spawned new branches of physics and mathematics as well as changing our notion of causality.
A thorough knowledge of classical mechanics is essential to a full understanding of physics as well as providing an introduction to powerful new mathematical methods that underlie many branches of physics. Some of these techniques were first developed for quantum mechanics and then applied back to classical physics. It will be assumed that you already have taken an introductory course in Newtonian mechanics. Therefore P235W will start with a brief review of the elements of the Newtonian formulation of classical mechanics together with the necessary mathematics of scalar and vector fields. Applications of Newtonian mechanics will be discussed followed by a discussion of oscillatory systems, non-linear mechanics and chaos. The Lagrangian and Hamiltonian formulations of classical mechanics then will be introduced and will form the backbone of the remainder of this course. It will be necessary to introduce variational calculus that underlies the Lagrangian approach. Applications to two-body motion, motion in non-inertial frames, the dynamics of rigid-body motion, and coupled oscillatory systems will be discussed. Finally the modern approach to the Hamiltonian formulation of classical mechanics will be introduced as well as its relation to quantum physics. This will be followed by a discussion of the implications to classical mechanics of the Theory of Relativity.