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.

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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.


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 »


Dark Matter Still Dark

October 30, 2013

Aw, the first run of the new dark matter detector LUX was a disappointment. Alas, that is how science goes. Sometimes, you find nothing. It still has over two years to go, though, so my fingers are crossed for something to be found. And no, even if no WIMP (the dark matter candidate) is found, MOND is not it. The Bullet Cluster shot that theory dead and it is now a corpse in the graveyard of scientific ideas.


Minimizing Quantity Part 3: The Brachistochrone

October 30, 2013

Last post, I finished the post cryptically, mentioning that there is a certain path that an object falling will take less time to travel than any other. Now, one of the solutions, as I mentioned, is just a straight fall. That is only one of the solutions, but only if the endpoint is directly underneath the starting point. What if the endpoint is to the side, what curve would the fastest path follow?

brachsetup Read the rest of this entry »


Minimizing Quantity Part 2: Snell’s Law

July 1, 2013

Last post, I talked about the fastest route between two mediums, and how light seems to follow that route when travelling from one medium to another. From what I just summarized, one would think light did that just because it is in a rush to get somewhere or something, but it is instead due to the wave nature of light. In classical physics, one can model how light reflects and refracts by applying something called Huygens Principle. Basically, a light wave at its edge forms a spherical wave at every point. Those spherical waves combine to form a single smooth wave. In the case of refraction, it looks like this:

snell image

Interestingly, the following model gives you the same math: Instead of the waves making more waves, let’s imagine instead that there are a chain of people holding hands and walking on the beach at constant speed. They suddenly reach the ocean, but at an angle. The person who reaches it first then bend the path, and then go forward. Otherwise, if the person wants to remain on the same straight path along with the others, the person must also move sideways. If you can’t picture it, it looks like this:

snell2

Now, let’s set the geometry up so that it shows the position of the line of people on the sand right before they go in the water and the line of people on the water right after they left land. Again, the math is the same, but I thought I could make it more intuitive. Notice that the length of the shore is the same for the triangle above and below it. That will be the key to creating the ratio for Snell’s law:

snell math

Before we go on, I have to mention that angle 1 in triangle BFC and angle 2 in triangle ADE are the angles given in Snell’s law. Then how do we know that angle 1 in ABC and angle 2 in AEF are the same respectively? ABF and BFC are similar triangles. So is AEF and ADE. The proof of that is not the subject for this post, so I might as well leave it as your homework assignment 😉 .

Now, since we want to find the ratio, we want to find which line is the same for the triangle ABF and AEF. Line l is the same. So we want to find something that looks like this:

l=\frac{stuff \hfill a}{stuff \hfill b}=\frac{stuff \hfill c}{stuff \hfill d}

Well, we can use trigonometry for that. For triangle ABF:

1a) sin(\theta_1)=\frac{\lambda_1}{l}

1b) \frac{sin(\theta_1)}{\lambda_1}=\frac{1}{l}

Doing the same for triangle AEF gets us:

2) \frac{sin(\theta_2)}{\lambda_2}=\frac{1}{l}

As you can see equation 1=equation 2:

3) \frac{sin(\theta_1)}{\lambda_1}=\frac{sin(\theta_2)}{\lambda_2}

Since distance=velocity*time:

4) \lambda_1=v_1*T and \lambda_2=v_2*T

Plugging it back in equation 3 gets us:

5) \frac{sin(\theta_1)}{v_1}=\frac{sin(\theta_2)}{v_2}

Multiplying both sides by the speed of light c gets us the generic Snell’s law equation with n standing for index of refraction, which is equal c/v:

6) n_1*sin(\theta_1)=n_2*sin(\theta_2)

Cool, the light bending its path due to the wave slowing down results in a path that is naturally the fastest possible route!

Interestingly, if you use the math of Snell’s law and imagine stacking infinitely thin layers of mediums, the ones next to each other being only infinitesimally different in their travel velocity, you can solve another interesting problem that involves the fastest route from top to bottom by gravity as we shall see in the next part of this series! And no, just letting go off the object so it falls straight down is not the answer (although it is A answer).


Minimizing Quantity Part 1: The Fastest Route

May 22, 2013

This is a classical type of problem in physics. Imagine you are on the beach, and you throw a ball somewhere in the ocean. You the decide to have a race with someone who can get to the ball faster. Both of you run at the same speed, and when you enter the ocean, you slow down due to the water’s resistance. Therefore, the only thing that will affect who will win the race is what path each one of you takes. So the question is, which is the fastest path?

shortest route

Read the rest of this entry »