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Akano's Blog

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Truly a landmark day

Posted by Akano Toa of Electricity , in Life Apr 23 2013 · 77 views

I have finished my last Jackson E&M homework ever. Of all time.

*insert angelic choir here*

Also, NEW ZELDA 3DS GAME HOLY MUKAU I CAN'T WAIT! [/slowpoke]

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Equation of the Day #6: The Lagrangian

Posted by Akano Toa of Electricity , in Math/Physics Apr 22 2013 · 57 views

Classical, Mechanics, Physics and 1 more...

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 t₁ and t₂ 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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Science Journals

Posted by Akano Toa of Electricity , in Life, Math/Physics Mar 19 2013 · 35 views

Neutrons, Winter Wrap-Up

So, in an earlier blog entry I talked about a journal article one of our professors presented at our department's journal club discussing neutrons in a purely gravitational potential well. Well, I decided to read it and am going to present it to the math and science grad students on Friday because I think it's pretty dang awesome.

Related: Airy functions are weird. And cool. Perhaps I'll discuss them later...

Also, tomorrow spring is here! (If it weren't for ponies, I would not say that with so much excitement.)

`

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Goodbye, Spring Break

Posted by Akano Toa of Electricity , in Life Mar 17 2013 · 43 views

Alas, my Spring Break has come to a close. I did have fun, though. I got Sonic & Knuckles for the SEGA Genesis and enjoyed time with friends and family.

This week I get a midterm for my grad level E&M class. We have over two weeks to do it, though, which is quite nice.

Also, Happy St. Patrick's Day, all! If you are old enough to drink, please do so responsibly.

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Special Relativity

Posted by Akano Toa of Electricity , in Math/Physics Mar 01 2013 · 75 views

A particle moving through space at the speed of light (i.e. a "massless" particle) does not experience time, and particles that are at rest travel through time at the speed of light.

So the next time you feel lazy loafing on your couch or computer chair, just remember that you are traveling at light speed, no matter how fast or slowly you move.

This entry brought to you by SCIENCE!

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Set Review: 79003 An Unexpected Gathering

Posted by Akano Toa of Electricity , in Set Review, LEGO Feb 23 2013 · 53 views

Bag End, Bilbo, Gandalf, Dwarves and 1 more...

I've posted another set review! This time it's Bag End, so head on over here and check out some thoughts on and pictures of this wonderful LEGO set.

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Anyone else having Brickshelf troubles?

Posted by Akano Toa of Electricity , Feb 23 2013 · 50 views

I am currently unable to upload photos to Brickshelf, and I'm not sure why. I'm trying to upload a .zip file of pics I want to use for a review, but after a long time of "uploading," the folder I attempt to upload to is empty. I tried doing one photo at a time, but that's failing as well.

Anyone else experiencing this?

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A True, True Friend

Posted by Akano Toa of Electricity , in My Little Pony Feb 21 2013 · 79 views

Ponies, Season 3 Finale

Confound these ponies!

They drive me to sing.

I liked the finale, but I kinda wish it were a two-parter. It felt slightly rushed as one episode.

Also, SONG OVERLOAD.

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Tekulo's birthday entry

Posted by Akano Toa of Electricity , in Life Feb 16 2013 · 31 views

Tekulo, birthday, Vamlumtimes Day and 1 more...

I was going to make this a topic when it happened, but unfortunately the forums were offline, so I'll make an entry for it instead. My younger brother, Tekulo, turned 21 this past Thursday, Valentine's Day, so go ahead and wish him a belated happy birthday!

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The Hobbit Soundtrack

Posted by Akano Toa of Electricity , in The Hobbit/LOTR, Music, Life Feb 12 2013 · 46 views

Hobbit, LOTR, Music, Howard Shore

Oh, Howard Shore, your music is wonderful. My favorite tracks, in no particular order, are

Blunt the Knives
Misty Mountains
The Adventure Begins
An Ancient Enemy
Radagast the Brown
Erebor
The Edge of the Wild

I'm looking forward to the next couple movies and the delightful music they will bring.

Also, The Wizard Battle set is totally getting added to my collection when it comes out.

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About Me

Akano Toa of Electricity
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Name: Akano
Real Name: Forever Shrouded in Mystery
Age: 23
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