Calculus of Variations Part 2: Lines, Bubbles, and Lagrange

In the first part, we discussed the idea of a functional, what it means, and how to find its extrema using the calculus of variations. However, those equations don’t really capture how amazing and applicable calculus of variations really is so the following will be some examples of this. In fact, the drawn out results from the posts The Shape of a String and The Lagrangian are just two cases of the one equation.

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General Relativity

General Relativity is perhaps one of the most enlightening theories in all of physics that reconciles many fundamental ideas about gravity, mass, and spacetime. The understanding of such a theory requires, in some sense, a certain conceptual jump which in the past has been explained through a “stretchy fabric” where mass acts as wells and other masses fall into these wells acting as a conceptual picture of general relativity. If you don’t know what I’m talking about, watch this video. This is immensely flawed in many ways. Read More »

Quantum Teleportation

Quantum teleportation is the idea that entangling states can cause very fast information transfer. The name however is misleading as this information transfer is not instantaneous but simply travels at the speed of light. The basis of this idea is based on a very interesting trick that arises out of some of the math of quantum computing.Read More »

Speed of Light Derivation

The speed of light may seem like an arbitrary constant of nature but, in some sense, it is actually set by other properties of the world. These other properties are the strengths of the electric and magnetic fields which are defined by the constants that are used in determination of them, otherwise known as the permittivity constant () and permeability constant (). Because light is simply an electromagnetic wave, one can derive its speed using these constants.Read More »

Complex Impedance

There is a certain luxury of circuit calculations for systems contain direct current that alternating current systems really do not have. It is the idea that voltage and current are “synced.” An increase in voltage will create a corresponding increase in current seemingly instantaneously. However, an alternating current that experiences voltage oscillations experiences a delay. This can mean voltage is at the highest point in its fluctuations while current only reaches such a point a little bit later at which point voltage might already be at its lowest. The ratio of voltage to current is also unclear in these circuits. This makes it hard to describe the system easily.Read More »

On Conservation

Conservation of different properties in nature immensely simplifies calculations to the point where some are impossible without the consideration of them. In some cases, it seems completely intuitive and impossible not to consider. However, not only are there many conservations laws unknown to many but there are also, in some sense, “violations” to these laws.Read More »