Ray tracing, ABCD matrices

First the review on ray-tracing is discussed. Then specific examples of calculation of ABCD matrix for lens, mirrors, dielectric interface, prism, cavity is presented.
It’s a lecture by Profs. Debdeep Jena and Clif Pollock at Cornell University.

Laser modes, maxwell equations

More discussion on gain, multi-level lasing, multi-mode systems, Maxwell’s equation for light waves is presented.
It’s a lecture by Profs. Debdeep Jena and Clif Pollock at Cornell University.

Laser basics

Basic properties of Lasers are discussed. Mathematical expression of light wave is introduced. It’s a lecture by Profs. Debdeep Jena and Clif Pollock at Cornell University.

Chaotic waterwheel

Schematic diagram of the waterwheel. Video of waterwheel in action. Derivation of governing equations for the waterwheel. Contuinuity equation and torque balance. Amplitude equations. Using orthogonality of the Fourier modes. A miracle: Exact decoupling of a three-dimensional subsystem from the rest of the modes. It’s a lecture at Cornell University.

Conservative systems

Mechanical systems with one degree of freedom. Particle in a double well. Symmetry. Homoclinic orbits. Energy surface. Theorem about nonlinear centers. Pendulum. Cylindrical phase space. It’s a lecture at Cornell University.

Two dimensional nonlinear systems fixed points

Linearization. Jacobian matrix. Borderline cases. Example: Centers are delicate. Polar coordinates. Example of phase plane analysis: rabbits versus sheep (Lotka-Volterra model of competition in population biology). Stable manifold of a saddle point. It’s a lecture at Cornell University.

Two dimensional linear systems

Phase plane analysis. Eigenvectors and eigenvalues. Classification of 2-D linear systems. Saddle points. Stable and unstable nodes and spirals. Centers. Non-isolated fixed points. It’s a lecture at Cornell University.

Model of an insect outbreak

Model of spruce budworm outbreaks in the forests of northeastern Canada and United States. Nondimensionalization. Saddle-node bifurcations. Jump phenomena. Hysteresis. Cusp catastrophe. It’s a lecture at Cornell University.

One dimensional systems

Linearization for 1-D systems. Existence and uniqueness of solutions. Bifurcations. Saddle-node bifurcation. Bifurcation diagrams. It’s a lecture at Cornell University.

Mohammed and the Arab Conquests

In this lecture, Professor Freedman at Yale University introduces Islam. He begins with a discussion of its geographical context: the dry desert lands of the Arabian peninsula. The Bedouins, or nomadic Arabs of the region, lived in a tribal society somewhat similar to the Germanic tribes discussed earlier in the course. Their raids against the Byzantine and the Persian Empire, for lack of strong opposition, would lead to the Arab conquests. The second half of the lecture focuses on the life of Mohammed (570/580 — 632) and the early years of Islam. Mohammed’s revelation was one of the unity of God and a progressive interpretation of God’s prophets, with Mohammed as the last of these. Early Islam was slow to differentiate itself for Christianity and Judaism, though this process accelerated after Mohammed’s flight to Medina in 622. Professor Freedman ends with a discussion of the tenets of Islam and anticipates the discussion of the Arab conquests in the next lecture.