SUBJECT MATTER

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, for velocities ranging from zero to the velocity of light, from one-body to statistical many-body motion, and extensions to quantum mechanics. The roadmap given below illustates that 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 alternative algebraic approaches are based on ideas of least action and variational calculus as suggested by 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 powerful Hamiltonian and Hamilton-Jacobi formulations of classical mechanics are closely related to the Lagrange formulation. The algebraic formulations provide the only viable approach to solving many-body systems. For over two centuries the Newtonian formulation reigned supreme in classical mechanics. It was only in 1905 that the full significance and superiority of the algebraic variational formulations of classical mechanics became widely accepted. The Theory of Relativity requires that the laws of nature are invariant to the reference frame. This is not satisfied by the Newtonian formulation of mechanics which assumes one absolute frame of reference and a separation between space and time. In contrast, the Lagrangian and Hamiltonian formulations of the principle of least action remain valid in the Theory of Relativity once the Lagrangian is written in an invariant form. The complete invariance of the variational approach to coordinate frames is precisely the formalism necessary for relativistic mechanics. The Poisson bracket and Hamilton-Jacobi formulations of Hamiltonian mechanics naturally incorporate the underlying physics of quantum mechanics and quantum field theory. As a consequence, the philosophical opposition to the variational approach, which some considered to be speculative and "metaphysical", no longer exists and now it has become the supreme and most useful formulation of classical mechanics.

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.

The primary goal of this course is to introduce you to these powerful variational methods that play such a pivotal role in many branches of science and engineering, and, in particular, their use in classical dynamics. A secondary goal is to stimulate, interest, and challenge you at the crucial period when you first encounter the sophistication and challenge of junior/senior courses as well as for developing your problem solving skills. The connections and applications of classical mechanics to modern physics will be emphasized throughout the course. This course will follow the general features of the roadmap shown in the figure and in the syllabus. 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. 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 and Hamiltonian formulations. Applications to two-body motion, motion in non-inertial frames, the dynamics of rigid-body motion, and coupled oscillatory systems will be discussed. This will be followed by a discussion of non-linear dynamics plus chaos and the Special theory of Relativity. The course will end with a brief introduction to advanced topics in Hamiltonian physics and the connection between classical mechanics and quantum plus statistical physics.

Roadmap of the development of classical mechanics that will be discussed in P235W