Talk 1. Jon Herman - 1:00pm
Pure Mathematics Department, University of Waterloo
“Hamiltonian and Lagrangian Mechanics”
I will first introduce some of the basic concepts in Lagrangian mechanics. In particular, I will show how the Euler-Lagrange equations arise as a solution to a specific variational problem, the principle of least action. In contrast to Newton’s 2nd law, which only holds in an inertial reference frame, the Euler-Lagrange equations can be used to describe a motion in any coordinate system. Next I will discuss Hamilton’s formulation. In particular, I will show how the Legendre transform produces an isomorphism between the tangent and cotangent bundle, which in turn gives a relationship between the Hamilton and Euler-Lagrange equations.
Talk 2. Justin Shaw - 2:30pm
Pure Mathematics Department, University of Waterloo
“Invariant Vector Calculus 2”
We continue our study of vector calculus from the differential geometry viewpoint. In this talk we will cover the first half of a course in continuum mechanics as we begin to develop the Navier-Stokes equations. We will discuss the reasoning behind fluid modelling, give a brief review of the flows of vector fields, define the material derivative, and prove a theorem about the movement of a volume through a flow. We will also consider the physical interpretations of each topic covered. This presentation does not depend directly on the results proven in Vector Calculus 1 and so is suitable for anyone who has taken some differential geometry or vector calculus.