The Squishy Pendulum

Over my two-week vacation before I returned to Berkeley for the summer, I read Leonard Susskind and George Hrabovsky’s The Theoretical Minimum: What You Need to Know to Start Doing Physics. This book was based on the first course in a series of courses called the Theoretical Minimum, taught by Leonard Susskind at Stanford targeted at curious older students who had, in their lives, fallen through the cracks of physics education but wanted to learn.

This book was focused on mechanics, but dived pretty rapidly into more advanced formulations of mechanics that I had never really learned in a class. Coincidentally, I’m scheduled to take that class next semester, so I wanted to dive in to get a brief taste of Lagrangian and Hamiltonian mechanics.

So, to start, I set out to simulate the squishy pendulum.

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Linear Algebra and Quantum Mechanics

When I was younger, I would occasionally hear about higher math classes that one was able to take. To me, then a naïve high schooler, AP Calculus represented an attainable pinnacle of mathematical knowledge beyond which lie a plethora of weird maths to explore. I had heard of multivariable calculus, which sounded like more of the same with more letters, differential equations, which was just calculus with more tricky problems, and so on and so forth.

However one, linear algebra, seemed like a mystery. It’s name evoked, to me, the kinds of problems done in middle school where I was painstakingly asked to grind through systems of three equations to find xy, and zSure, I thought, maybe there’s a use for solving ever bigger systems of equations of ever increasing complexity with bigger and better techniques. But, if multivariable and diff-E.Q. were “more of the same,” jumping back to middle school lines and planes was definitely going be a bore.

Spoiler alert: It wasn’t, and, while related, linear algebra really isn’t about that stuff. It’s actually about a lot of other, cooler stuff, including really cool stuff like quantum mechanics.

Note: This is just a primer on linear algebra. I introduce the axioms, and then paint over the subject with a broad brush that isn’t meant to be comprehensive. Quantum mechanics is inseparable from linear algebra, so I try to get to the meat of linear algebra while not glossing over too much. At the same time, this obviously shouldn’t be taken as a substitute for a more rigorous treatment of linear algebra.

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Vector Calculus and Electromagnetism

Vector calculus is an extremely interesting and important branch of math with very relevant applications in physics. Specifically, vector calculus is the language in which (classical) electromagnetism is written. It is fascinating to me that Maxwell’s equations can so succinctly and elegantly express so many phenomena, from electric and magnetic interactions to light (electromagnetic waves). One popular formulation of Maxwell’s equations is the following:

maxwell-equations

So what does this all mean? Let’s break it down, arduously and tediously, from the nitty-gritty of the vector calculus requisite for understanding the elegance of these equations.

Note: I presume basic knowledge of calculus in this post. Obviously, this is not meant to be a substitute for a more rigorous foundation with more computational practice.

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