Fall, 2007
Our exploration of the theoretical underpinnings of modern physics begins with classical mechanics, the mathematical physics worked out by Isaac Newton (16421727) and later by Joseph Lagrange (17361813) and William Rowan Hamilton (18051865). We will start with a discussion of the allowable laws of physics and then delve into Newtonian mechanics. We then study three formulations of classical mechanics respectively by Lagrange, Hamiltonian and Poisson. Throughout the lectures we will focus on the relation between symmetries and conservation laws. (Image credit: The International Astronomical Union/Martin Kornmesser)
Lectures in this Course

State diagrams and the nature of physical laws
This first lecture is a general discussion of the nature of the laws of physics and in particular classical mechanics. The notions of configuration, reversibility, determinism, and conservation law are introduced for simple systems with a finite... [more] 
Newton's laws, principle of least action
Consider Newton's law relating force and acceleration. Define kinetic energy and potential energy. Derive conservation of energy for a system of particles. Define the Lagrangian and derive the Principle of Least Action starting from Newton's Law. ... [more] 
EulerLagrange equations, symmetry and conservation laws
Define the EulerLagrange equations as a result of the Principle of Least Action. Define the canonical momentum. Differential conservation laws result from a continuous symmetry. 
Symmetry and conservation Laws
This lecture focuses on he relation between continuous symmetries of the Lagrangian and conserved quantities. Generalized coordinates and canonical conjugate momentum are introduced. 
Lagrangians and Hamiltonians
Example simple pendulum: derive the EulerLagrange equations, the kinetic energy, potential energy, Lagrangian and angular momentum. Calculate the Hamiltonian. Review the trajectory of the simple pendulum in phase space. Derive conservation of area... [more] 
Hamilton's equations
The Hamiltonian function determines the evolution of a system Closed cycles in phase space represents a conservation law Hamilton's equations are first order equations of motion. Newton's equations are second order equations of motion. Canonical... [more] 
Liouville’s theorem
Prove Liouville’s theorem from Hamilton’s equations. Liouville’s theorem can be thought of as information conservation. The laws of mechanics are equivalent to the rules governing state transition. Ignored degrees of freedom, such as friction, can... [more] 
Motion in an electromagnetic field
Recapitulate symmetries and conservation laws. According to Liouville's theorem, a system evolving according to Hamilton's equations conserves the volume of phase space. Information is conserved. The Principle of Least Action underlies Lagrangian... [more] 
Poisson brackets formulation
Poisson brackets formulation describes flow in phase space. Define the axiomatic structure of Poisson brackets. Canonical rransformation of Poisson brackets preserves the Poisson bracket axioms. Define the generator of an infinitesimal canonical... [more]