I’ve been going through Russ Tedrake’s notes on robotics, which got me thinking about their so-called monogram notation. The result of this deliberation was a new notation for spatial velocities that bridges the gap between abstract Lie groups and their applications to robotics and computer graphics.

A principal bundle is a fiber bundle, that additionally has the right-action of a group that preserves fibers. Beginning with coset spaces, we look at some interesting examples of principal bundles and the things we can do with them.

Some results about the differentials (i.e., pushforwards) of the exponential and logarithm maps of a Lie group. I rely extensively on the interpretation of tangent vectors as equivalence classes of curves.

Poincaré’s 1901 paper introduces (in just a humble 3 pages) the Euler-Poincaré equations, which are the specialization of the Euler-Lagrange equations to the case where a Lie group acts on the configuration manifold. I work through Poincaré’s paper without making too many identifications.

The Fourier transform maps a complex-valued function to a function on its Pontryagin dual space. Generalizations of this concept to non-Abelian compact and locally compact Lie groups are reviewed.

Despite having encountered the Lagrangian and Hamiltonian formalisms of mechanics several times in a variety of engineering and physics settings, I had never been able to retain it in my memory. I had maintained a similar dissatisfaction with the many formulae of multivariable calculus, which only really ✨clicked✨ for me once I learned about the exterior derivative and the generalized Stokes’ theorem. In this post, I would like to collect my thoughts on the differential geometric treatment of Lagrangian and Hamiltonian mechanics, which assume a very simple and memorable form once we introduce the language of symplectic geometry . Of course, to have made our mathematical journey to this point, where we are able to say anything at all about symplectic geometry, was a not-so-simple task. ...

Here, I solve Problem 5-5 from John Lee’s book on Riemannian Manifolds, which demonstrates the non-flatness of the 2-sphere. This problem is particularly interesting because it serves as the motivating example for a later chapter in the book on curvature.

A Lie group is a group that is also a (continuous, differentiable) topological space. To measure lengths and volumes (and relatedly, to define and integrate probability densities) we need to endow the group with additional structure so that it is not merely a manifold, but a Riemannian manifold.

There are multiple ways to construct new groups from old ones. I provide an intuition for how these constructions work, and also go over some of the additional structures that can be imposed on Lie groups, paving the path towards doing differential geometry and calculus on Lie groups.

A topological group is a set of elements that has both a group operation and a topology. The group operation satisfies the usual axioms (same as those of finite groups), and the presence of a topology lets us say things like ’the group is connected’ and ’the group operation is continuous'.