The History of Gamecube

January 8, 2014

I may write mostly about science, and from the look of this blog, you might think that is my only interest. But I also like videogames. I mainly play Nintendo games and occasional PC games like Civ V. Anyways, I have found this fascinating article about the creation and lifetime of Nintendo Gamecube. There are a lot of information of the decisions the company made and the reason behind them. Basically, tons of stuff I didn’t know.

If you don’t know, Gamecube was Nintendo’s least successful console, and was mainly a niche platform for gaming. It had some great games, but like all post SNES Nintendo consoles, it suffered long stretches of time without software releases. In the case of the Gamecube, well, nothing could save it. Despite that, it managed to make a nice profit for Nintendo, even if the profit generated wasn’t exactly mind blowing. A lot of people who owned it look back at it fondly thanks to some excellent exclusive games.


Feynman Lectures Are Online!

January 7, 2014

Well, looks like they put all three volumes of Feynman’s Lecture on Physics online! Well, volume 1 and 3 have been around for a while, but volume 2 has finally been posted.

What are they about? You might ask if you don’t know about this. Well, they are transcribed version of Feynman’s Caltech undergraduate physics lectures decades ago. Volume one is mechanics and thermodynamics, volume two is electricity/magnetism and matter, and volume three is quantum mechanics. If you want to read them, I suggest a background in algebra, calculus, vectors, differential equations, and linear algebra. You don’t need to know all of those at the same time, it depends on the area of physics you are covering, but at the very least if you don’t have algebra, you are dead if you read these books.

Overall, it covers a large portion of what physics is about nicely. Just remember that just because you read this book doesn’t mean you mastered the material. It takes doing actual problem sets in order to know how to apply these stuff.


The Huge Scale of Andromeda

January 5, 2014

Have you wondered what Andromeda would look like if it were a lot brighter? Well, most of you probably haven’t, but it certainly would look incredible (credit for the image goes to this person):

Notice how it would look so much larger than the moon? Now think about this. The Andromeda galaxy is around 2.5 million light years away. Imagine how large it has to be in order to look like that even from that unimaginable distance! In a way, this image gives you a sense of how large 100,000 light years (the visible part, there are invisible parts that stretches Andromeda to 220,000 light years) is, well not completely since such sizes are unfathomable, but this will do.


Minimizing Quantity Part 5: Principle of Stationary Action

December 15, 2013

Last post, I hinted that finding the path that minimizes quantities is very fundamental to physics. You can formulate an alternative to Newton’s laws using it. With Newton’s second law, you use F=ma in order to build a differential equation that models an object’s motion. In something called Lagrangian mechanics, you do it by using the Euler Lagrange equation on a functional called the Lagrangian. The Lagrangian is the kinetic energy minus the potential energy. In a Lagrangian, kinetic energy is T and the potential is V, so L=T-V. By using the E-L equation on the Lagrangian, you find the path in which a quantity S called action is stationary. It looks like this:

1) S=\int^{t2}_{t1} L(t, q, \dot{q}) dt

2) 0= \frac{\partial L}{\partial q} - \frac{d}{dt}(\frac{\partial L}{\partial \dot{q}})

This is called the principle of stationary action, and it is another way to formulate the laws of mechanics. And the above statement is equivalent to Newton’s second law. First of all, kinetic energy is:

3) T=\frac{1}{2} m \dot{q}^2

Since kinetic energy is represented by the velocity, the partial derivative of it over position is zero, so:

4) \frac{\partial L}{\partial q}=\frac{dV}{dq}

Since the potential of the force is defined as the integral of the force over position, and energy is conserved, the negative of its reverse is true:

5) -\frac{dV}{dq}=F

As for the right component of the right and side of equation 2, since the potential does not depend on the velocity, that is zero. So:

6) \frac{d}{dt} (\frac{\partial}{\partial \dot{q}} \frac{1}{2}m \dot{q}^2) = \frac{d}{dt} m \dot{q}

7) \frac{d}{dt} \dot{q} = m \ddot{q}

Inserting 7 and 5 into 2 gets us Newton’s second law as expected:

8) F=m \ddot{q}

What an amazing result! This is telling us that every object moves in such a way that it minimizes the quantity called action. At the same time, it does seems suspicious, doesn’t it? What is this Lagrangian and action and how did anyone think of doing things this way? And we have Newton’s second law, why would we ever want to use this Lagrangian mechanics? It is, after all, going to get us the same results.

Well, all of these scientific principles involving variational calculus were done by people like Maupertuis, D’Alember, Lagrange, Euler, Fermat, Hamilton, etc who believed that nature did things in ways that was the most efficient. The fact that Fermat’s principle worked to explain the path of light stoked their ambition. The Lagrangian itself comes out of something called D’Alembert principle. D’Alembert principle basically puts all forces in a standing still reference frame. Objectss that are being accelerated are to be made seem standing still by adding a pretend force that opposes such acceleration, creating a non inertial frame of reference. Here is a material I found which explains more about the Lagrangian and how it comes out of D’Alembert.

As for why we would ever use Lagrangian mechanics, well, it allows us to use other kinds of coordinate systems instead of just the cartesian one. Some problems are easier in alternate coordinate systems. You also don’t have to deal with a bunch of equations built from the various forces. As an example, I will show you how we can do the pendulum problem using this method, and we shall do it in polar coordinates, which consists of angle and radius from the center of the coordinate. I will label the radius l and the angle theta. First of all, here is the setup, with height being zero at the pendulum’s lowest point:

pendulum setup

 

Now, instead of drawing the forces, we have to determine the kinetic and potential energy:

 

1) T=\frac{1}{2}mv^2

In order to transform it for polar coordinates, you have to remember that in a rotating motion:

2) v=\dot{\theta} l

Plugging equation 2 back in the first one:

3) T=\frac{1}{2}ml^2 \dot{\theta}^2

Potential energy is much simpler:

4) V=mgh=mgl(1-cos(\theta))

So the Lagrangian is:

5) L=\frac{1}{2} ml^2 \dot{\theta}^2 - mgl(1-cos(\theta))

Use the following Euler Lagrange equation:

6) \frac{\partial L}{\partial \theta} - \frac{d}{dt}( \frac{\partial L}{\partial \dot{\theta}})=0

Doing the partial derivatives gets us:

7) -mgl sin(\theta) - ml^2 \ddot{\theta} = 0

Dividing both sides by -ml^2 gets us:

8) \ddot{\theta} + \frac{g}{l} sin(\theta) = 0

That is indeed the equation of motion for the pendulum. Further simplification is possible by having the angle close to zero. After all, I don’t feel like dealing with the elliptical integral of  the first kind, which I don’t even know how to use. The linear expansion of sin(x) is x, so:

9)  \ddot{\theta} + \frac{g}{l} \theta = 0

See? It gets us the same results, but instead by using concepts of energy and variational calculus.

The five parts posts on minimizing, or as you have eventually learned, finding the stationary points, is a taste of what variational calculus can bring into physics. As you can tell, a lot of problems involving extremizing behavior that are hard to grasp becomes possible with these new set of tools, and there is the added bonus of having another framework of analyzing classical mechanics problem via the Lagrangian. Now, I know that for many of you, these series of posts are pretty useless for your personal life. For most of you it is probably true, and these posts were mostly done in order to learn about variational calculus physics myself, and crystallize what I have learned. Despite all that, hopefully you got something out of this, whether your interest in physics has been piked, or your mind got blown away by the principle of stationary action (it did with mine!) or you learned new ways to do certain physics problems.


No More Comet

December 2, 2013

Wellp, looks like Comet Ison is likely dead. Disappointing since I was looking forward to watching it myself after sunset. It was only visible before sunrise previously. As of right now, astronomers haven’t completely ruled out that it is intact, but I am not getting my hopes up.


If You Missed Comet Ison

November 23, 2013

If you tried catching the recent comet Ison in the night sky, it is too late. I is now in the sun’s glare. But, there will be a second chance after it swings around the sun in November 28. You will be able to see it in a more convenient time, after sunset, I think. Before you had to get up really early to see it before the sun rises. Anyways, keep checking astronomy news and the charts. Also, check if the comet survives the trip around the sun, as it goes really close to the sun, and there is a possibility it might break up. If you do see it eventually, though, enjoy!


Minimizing Quantity Part 4: Euler-Lagrange Equation

November 23, 2013

In the last three posts, I talked about paths that minimize time. In all those cases, it involves objects going through a path and minimizing certain quantities. But is there a single equation that covers all physical situations that involve finding the path that minimizes quantity? The answer is yes, and it is called the Euler Lagrange equation. Read the rest of this entry »


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