analytical mechanics

Courses and textbooks, at least in undergrad, often gloss over the details of the Legendre transformation, which converts convex functions of one variable into another convex function of the “conjugate” variable. Used ubiquitously in physics, from thermodynamics to quantum field theory, this mathematical method plays a central role in connecting some of the most fundamental concepts, and yet, at least to me for a while, was a mysterious black-box procedure. I am now going to attempt to illuminate this technique....

Pseudovectors, or axial vectors as they are sometimes referred to as, are commonly encountered, but mysterious mathematical objects. In physics they arise in many different areas, particularly when cross products are involved - e.g., magnetic fields, angular momentum, and torque. The goal of this post is to address two seemingly different definitions of a pseudovector, in 3 dimensions, and in the process unravel this mystery. Definition 1 A pseudovector is a tensor on $\mathbb{R}^3$ that transforms like a vector under proper rotations, but picks up a sign under an improper rotation, like a reflection....