Summer Reading Program on the Mathematics of Classical Mechanics

Schedule

Below is an approximate schedule, subject to change. Primary reading material will always refer to sections of [Arnold1].

Part I - Reading and project selection.

• May 10 (F): Introduction, phase diagrams
  We started by addressing two pitfalls of this branch of math: using multivariable calculus without confusing oneself and recognizing that differential equations are hard. We reviewed the definition of the derivative in multiple variables, as well as the chain rule, and we discussed the notion of phase diagrams.
  Primary reading material: §4 of [Arnold1]. Supplementary reading material: [Halmos, §8], [Spivak, §2], [Arnold2, §1.1.1-1.1.4, §1.2.1]
• May 15 (W): Higher dimensional phase space and examples of manifolds
  We continued the analysis of differential equations using phase diagrams by looking at a higher dimensional example. This is all in the service of obtaining a qualitative understanding of differential equations. This led us naturally to talk about dimensions, the implicit function theorem and see our first examples of manifolds.
  Primary reading material: §5. Supplementary reading material: the section on the implicit function theorem in [Spivak, §2]
• May 17 (F): More on the geometry of phase space and an extended example
  We discussed a problem that requires "differentiation under the integral" and we thought extensively about the spherical pendulum and how to get our hands on the phase curves.
  Primary reading material: §4-5.
• May 22 (W): Calculus of Variations and the Euler-Lagrange equations
  We will begin in earnest the study of Lagrangian mechanics.
  Primary reading material: §12-13.
• May 24 (F):
• May 29 (W): Coordinate invariance
  We proved that the Euler-Lagrange equations are invariant under change of coordinates and we talked about duality in the context of the Legendre transformation.
  Primary reading material: §12.D, §13-14.
• May 31 (F): Differentiable manifolds
  After a few comments on the Legendre transformation and Hamilton's formulation, we defined differentiable manifolds and considered the example of stereographic projection on the unit 2-sphere. §15 on Hamilton's equations was asigned reading.
  Primary reading material: §14 and §18.
• June 5 (W): SO(3)
  We thought carefully about the manifold SO(3) by working through example 8 of §14: the configuration space of a triangle with one vertex fixed at the origin of 3-dimensional Euclidean space.
  Primary reading: §14.B. Seconday: In §2 of [Spivak], the Implicit Function Theorem.
• June 7 (V): Embedded manifolds and the tangent space
  We kept reading §14 and gave a definition of tangent vectors and the tangent space, in order to finall make sense of the definition of Lagrangian mechanics when the configuration space is a non-trivial manifold: a Lagrangian function is a smooth real-velued function on the tangent bundle of a smooth manifold.
  Primary reading: §14 and §15.

Part II - Project refinement and Presentations

• July ??
• August ??