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This is one that I didn't really know much about until recently, so I thought I'd share it. Today's equation is known as the Virial theorem,

or, in component form,

The word "virial" comes from the Latin vis, which means "force" or "energy," and looking at the equation, it makes sense why it's called that. Here the big Σ means sum, the "k" index denotes the k^{th} particle of a system of N particles, V is the potential energy function affecting the k^{th} particle, T is the potential energy of all the particles in the system, and r_{k} is the position of the k^{th} particle. This essentially relates the kinetic energy of all the particles to the positions and forces exerted on each particle (since -grad V is the force when energy is conserved, which is an assumption we are making). The brackets 〈 〉 denote that we're taking an average, so 〈T〉 is the average kinetic energy, etc.

Now, you may be thinking, "okay, that's a cute equation, I guess, but I don't see how it's particularly useful." Okay, here's where the usefulness comes in. Let's say I want to know the mass of some distant galaxy, but I don't have a good galaxy-weighing device on hand. We know that the gravitational potential energy of an object is given by

where m is the mass of the star, M is the mass of the center of the galaxy, and r is the distance from the center of the galaxy. Taking the distance r and multiplying by the gradient of the potential yields...the potential again, with a negative sign out front. So, for gravity,

Plugging this into the Virial theorem above and noting that 2T = mv^2 (where v is speed), we get that, for an object in the gravitational pull of an object of mass M,

Thus, we have at our disposal a way of measuring the mass of something like a galaxy by measuring only the speeds of stars and their distance away from the center. That's pretty incredible.

This actually is one of the ways scientists support the idea that there is dark matter in the universe; the Virial theorem gives an average of what speeds the stars in our galaxy should have based on their distance away from the center of the Milky Way, but what we actually observe is startlingly different. Thus, we can conclude that something is wrong with our knowledge of how gravity within a galaxy works. Based on this and other observations, the idea that there's extra stuff that can't be seen that adds to the gravitational force of a galaxy seems to be a reasonable idea.

In my research on diatomic hydrogen (H_{2}), the Virial theorem is used in a different capacity. When figuring out the potential energy of an electron (or two) around the two positively charged protons, the virial has the Coulomb force term (which is just -V, just like gravity) and an additional term that pops up from assuming that the electrons are keeping the protons at equilibrium. I won't go too much into the physics, but the final product is

where E, T, and V are the total energy, kinetic energy, and potential energy of the electron(s), respectively, and R is the distance between the nuclei. This tells us something useful about the energy of the electrons; more specifically, it tells us about how the energy changes as you move the nuclei farther apart or closer together. In other words, since E = T + V,

which is very useful when constructing potential energy curves for hydrogen.

On a slightly related note, our lab's paper got published! Akano is now a for reals, published scientist! 8D

I have absolutely no life to the point that I just spent the better part of my afternoon going back through my blog posts and recording the view count, reply count, and word count of all 185 entries I've made prior to this one. These are the results of my labor:

All time averages: Views: 57.4 ± 45.8 Replies: 2.2 ± 2.6 (LOL) Word Count: 176 ± 242 (BIGGER LOL)

Largest stats: Views: 266 (courtesy of "Ask Akano" blog entry; Fort Legoredo review got 265, a close second) Replies: 24 (again, "Ask Akano") Word Count: 1711 (Vampyre Castle review)

I've bloggedbefore about the physics of bouncing a particle like a neutron on a table or other similar surface. Well, someone at the University of Arkansas has made an animation of how the spread of said particle over space evolves in time if it starts out in a "Gaussian wavepacket," which is a fancy way of saying that the neutron starts out looking more like a particle than a wave by being localized in space. The animation is here, while the full page containing the full Quicktime movie is here.

The red dot is a classical bouncing ball evolving over time (it's pretty boring comparatively). On the left is a plot of the probability of finding the particle at a certain height (the vertical axis) at any given time of the movie. The quantum particle does sort of bounce, become kind of wavy and messy, and then bounces again, but out of phase with the classical ball.

TL;DR: Cool animation of doing mundane physics with a quantum system with results that are anything but mundane. Click the links to have your mind blown.

So, in the last few weeks, my family came to visit (including Tekulo and KK) and we spent the latter half of the week seeing sights and enjoying each other's company. My mom won a game of Trivial Pursuit on a category that was supposed to stump her. Story of my Trivial Pursuit life.

When they left, they abandoned KK with me, which has led to me stepping into the nerd realm of playing Dungeons & Dragons. We're doing a campaign in the land of Hyrule with the races of Hyrule being used as analogs of D&D races. We're currently in the Forest Temple seeking an herb to cure the Great Deku Tree's muteness.

I'm also working in a new physics lab where I'm studying the energy states of the hydrogen molecule (H_{2}). I'm thoroughly enjoying it, since I'm learning computational stuffs and learning my way around Linux. (Emacs rules the school.) The program I'm working with is in Fortran, which is my native programming language but was written by someone else with a lot more skill than I possess.

You may have learned once that classical mechanics all stems from Newton's laws of motion, and while that is true, it is not necessarily the best way to solve a given physical problem. Often when we look at a physical system, we take note of certain physical parameters: energy, momentum, and position. However, these can be more generalized to fit the physical situation in question better. This is where Lagrange comes in; he thought of a new way to formulate mechanics. Instead of looking at the total energy of a system, which is the potential energy plus the kinetic energy, he instead investigated the difference in those two quantities,

where T is the kinetic energy and V is the potential energy. Since the kinetic and potential energy, in general, depend on the coordinate position and velocity of the particle in question, as well as time, so too does the Lagrangian. You're probably thinking, "okay, what makes that so great?" Well, if we were to plot the Lagrangian and calculate the area under the curve with respect to time, we get a quantity known as the action of the particle.

where t_{1} and t_{2} are the starting and ending times of interest. Usually if the motion is periodic, the difference between these times is one period. Now, it turns out that for classical motion, the action is minimized with respect to a change in the path along which the particle moves for the physical path along which the particle actually moves. This sounds bizarre, but what it means is that there is only one path along which the particle can move while keeping the action minimized. Physicists call this the Principle of Least Action; I like to call it "the universe is inherently lazy" rule. When you do the math out, you can calculate an equation related to the Lagrangian for which the action is minimized. We call these the Euler-Lagrange Equations.

These are the equations of motion a particle with Lagrangian L in generalized coordinates q_{i} with velocity components denoted by q_{i} with a dot above the q (the dot denotes taking a time derivative, and the time derivative of a coordinate is the velocity in that coordinate's direction). This is one of the advantages of the Lagrangian formulation of mechanics; you can pick any coordinate system that is best-suited for the physical situation. If you have a spherically symmetric problem, you can use spherical coordinates (altitude, longitude, colatitude). If your problem works best on a rectangular grid, use Cartesian coordinates. You don't have to worry about sticking only with Cartesian (rectilinear) coordinates and then converting to something that makes more sense; you can just start out in the right coordinate system from the get go! Now, there are a couple of special attributes to point out here. First, the quantity within the time derivative is a familiar physical quantity, known as the conjugate momenta.

Note that these do not have to have units of linear momentum of [Force × time]. For instance, in spherical coordinates, the conjugate momentum of longitude is the angular momentum in the vertical direction, which has units of action, [Energy × time]. The Euler-Lagrange equations tell us to take the total time derivative of these momenta, i.e. figure out how they change in time. This gives us a sort of conjugate force, since Newton's second law reads that the change in momentum over time is force. The other quantity gives special significance when it equals zero,

This is just fancy math language for saying that if one of our generalized coordinates, q_{i}, doesn't appear at all in our Lagrangian, then that quantity's conjugate momentum is conserved, and the coordinate is called "cyclic." In calculating the Kepler problem – the physical situation of two particles orbiting each other (like the Earth around the Sun) – the Lagrangian is

Note that the only coordinate that doesn't appear in the Lagrangian is ϕ, the longitude in spherical coordinates. Thus, the conjugate momentum of ϕ, which is the angular momentum pointing from the North pole vertically upwards, is a conserved quantity. This reveals a symmetry in the problem that would not be seen if we used the Lagrangian for the same problem in Cartesian coordinates:

That just looks ugly. Note that all three coordinates are present, so there are no cyclic coordinates in this system. In spherical coordinates, however, we see that there is a symmetry to the problem; the symmetry is that the situation is rotationally invariant under rotations about an axis perpendicular to the plane of orbit. No matter what angle you rotate the physical situation by about that axis, the physical situation remains unchanged.

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Name: Akano Real Name: Forever Shrouded in Mystery Age: 25 Gender: Male Likes: Science, Math, LEGO, Bionicle, Ponies, Comics, Yellow, Voice Acting Notable Facts: One of the few Comic Veterans still around Has been a LEGO fan since ~1996 Bionicle fan from the beginning Misses the 90's. A lot. Twitter: @akanotoe