The Brachistochrone problem is a very famous problem in the history of physics and is as follows: find the path in which a particle under only the action of gravity will move from one point to another fastest and in the shortest amount of time possible.

In this lesson, we'll give a brief explanation of what generalized coordinates are. Up till this point, we got away with a very vague and imprecise definition. But for some of the problems that we'll look at later (such as the double-pendulum problem), the more technical definition we develop in this lesson will be very useful.

In this section, we'll find the solution (which is a curve x(y)) to the Euler-Lagrange equation where the quantity being minimized is the arc length between any two points on a cylinder. That might sound like a mouthful, but we'll see that hopefully it isn't too bad.

Guass's law: for any charge distribution, whether a system of a finite number of point charges or a continuous distribution of charges, if I draw any arbitrary closed surface (which can be of any shape, size, and location), then the only contributions to the electric flux through that closed surface will be due to only the charges \(q_{enc}\) enclosed within that surface and that electric flux is given by the amount \(\frac{q_{enc}}{\epsilon_0}\). It is important to appreciate the word *any* to fully grasp the applicability of Guass's law: it applies to *any* closed surface and to *any *distribution of charge.

Everything in this section will summarize some—a fairly good portion—of what we'll be studying in much more detail in the subsequent sections.

Column's law describes the electric force between any distribution and number of charged particles. It is derived empirically and falls off inversely with the square of the distance, analogous to Newton's law of gravity. The practical usefulness of this law is that it allows one to determine the effect that any distribution of charge has on any other distribution of charge; this is analogous to how Newton's law of gravity allows one to determine the effects of one mass distribution on another. One very gradually learns this by solving many problems. But in this section, we'll start off with Column's law and then proceed to do some algebra and calculus in order to derive equations which describe the electric force that systems and continuous charge distributions exert on an individual particle.

In this section, we'll derive the Euler-Lagrange equation. The Euler-Lagrange equation is a differential equation whose solution minimizes some quantity which is a functional. There are many applications of this equation (such as the two in the subsequent sections) but perhaps the most fruitful one was generalizing Newton's second law. This equation, together with Hamilton's principle, allows us to generalize Newtonian mechanics and find the motion of systems using generalized coordinates—all of a sudden, double-pendulums don't seem so scary.

In this lesson, we'll see that Guass' law makes the same prediction about the electric field produced by a point charge as Column's law.

Unlike the flux of the velocity field of a fluid, the notion of electric flux is much more abstract. Like concepts such as energy, it is very abstract but nonetheless very useful in analyzing certain situations. In this section, we derive the electric flux of any electric field through any arbitrary surface. Only later one, when we discuss Guass's Law, will we learn about the usefulness of this idea.

Noether's Theorem is one of the most profound theorems in all of science, let alone physics. It implies that the very fabric of space and time give rise to the laws of nature and lead the mathematician David Hilbert to independently discover the general theory of relativity, alongside Einstein. Noether's theorem, in simplest terms, is the statement that for every symmetry (what a symmetry is is something we'll get to in this lecture) there is an associated conservation law. Our goal in this lecture will be to prove that and show that various different conservation laws follow from an associated symmetry.