Noether’s Theorem

Emmy Noether was a woman who made significant contributions to the field of mathematics, most notably abstract algebra. She played a big role in defining the axioms of ring theory that we use today. Noether also worked on non-commutative algebras and linear transformations. Today, however, I will discuss a contribution she made in the field of physics, namely Noether’s  Theorem. Noether’s Theorem applies to Lagrangian mechanics and states that if the Lagrangian of a system is symmetric under the translation q→q*=qQ, and q’→q*’=q’Q’then the quantity (∂L/∂q’)Q’ is conserved. Let’s explore what this means.

A bit of background on Lagrangian mechanics is necessary to begin understanding Noether’s Thoerem. Usually when solving a physics problem, one would need to find all the forces on the object of concern. They then use Newton’s second law, ∑ F=ma, which allows them to find the second derivative of the object’s position, from which they could (hopefully)  find a function for the object’s position. This works well enough for simple physics problems such as a block sliding down a ramp without friction. However the problem increases in difficulty if we allow the ramp to slide on the surface it rests on, of course without friction. This is where Lagrangian mechanics comes in. The idea behind it is to minimize what is called the action, or Lagrangian. The Lagrangian of a system is simply equal to the kinetic energy of the system minus the potential energy of the system, usually written L=T-U, where T is kinetic energy, and U is potential energy.

In order to truly understand Lagrangian mechanics, one needs a rudimentary understanding of the Euler-Lagrange equations, which is derived using calculus of variations. Simply put, the Euler-Lagrange equations allow one to minimize what’s called a functional, or a function of functions which are usually expressed as an integral over a set of functions.  It can be shown that in maximizing the functional whose integrand is the Lagrangian of the system will give the equations of motion of a system. Therefore the Euler-Lagrange equations are applied to the Lagrangian of a system to give the system’s equation of motion. Though this method is more complicated, and almost not worth it for simple problems, it is extremely useful with more complicated systems. The Euler-Lagrange equations are as follows:

Euler-Lagrange

Where q is the set of coordinates which describe the system. I think we may finally be ready to look at what Noether’s Theorem says.

Noether’s Theorem states that if the Lagrangian of a system is symmetric about some translation q→q*=qQ, and q’→q*’=q’Q’, then a quantity is conserved, namely (∂L/∂q’)Q’, is conserved. Often times this tells us that the conserved quantity is the momentum of the system. We can see this because U is almost always (if not always) independent of q’ so the partial derivative of the Legrangian with respect to q’ is equal to the partial derivative of the kinetic energy with respect to q’.

The kinetic energy of an object is defined to be (1/2)m(q’)², thus the partial derivative with respect to q’ is  mq’, which is the definition of momentum. Let’s now go through a quick and simple example to help solidify our understanding.

Consider a block sliding down a ramp with no friction (the ramp is fixed in our case). Let us choose our coordinate system as follows

Ramp

where the arrows on the right hand side represent the forces, so we can ignore them, and the z-axis is pointing out of the page. Since motion will be restricted to the x and z directions (the block will not bounce or jump off the surface of the ramp), there will be no kinetic energy in the y direction. Thus our Lagrangian is

L(x,z,x’,z’)=(m/2)(x’²+z’²)-mg(xsinθ),

where the potential energy was found using the idea of virtual gravity, or the portion of gravity which points down the ramp. If we make the following transformations

x→x*=x+0

x’→x*’=x’

z→z*=z+C

z’→z*’=z’

where C is a constant, we can see that L(x,z,x’,z’)=L(x*,z*,x*’,z*’), so our Lagrangian is symmetric about the above transformation. Thus, Noether’s Theorem tells us that

(∂L/∂x’)(0)+(∂L/∂z’)C=constant,

since the first term is zero, this simplifies to

(∂L/∂z’)=A

Where A is the above constant divided by C. As mentioned above, (∂L/∂z’) is an expression for the object’s momentum in the z direction. Therefore momentum is conserved in the z direction.

This beautiful and very useful theorem is just one of the contributions Emmy Noether has made to the fields of math and physics. Its foolish that we follow tradition in believing that men are better at math than women. I mean, sure there may be more men who do math than women, especially in Noether’s day. But men not letting women do math, and telling women that they are bad at math from a young age, so they don’t see their own potential, is worlds away from women being inherently bad at math. Emmy Noether is one of the best counter examples to disprove the notion that women cannot do math as well as men.

29 Hand in Cribbage: How likely is it?

I was working on an optional math problem given by my math professor John Golden at GVSU. The assignment was to analyze the probability of one of our favorite games. I chose to analyze the game Cribbage, as my grandpa taught me how to play and I’ve many fond memories of playing with him on warm summer days. For those of you who don’t know how to play cribbage, I’ll give you a quick run-down of the rules that will be relevant for our task here, which is to find the probability of obtaining a perfect hand in a two person game of cribbage.

In Cribbage each player is dealt 6 cards. They then choose two cards to discard into the “crib” which is essentially and extra hand for the dealer. After this is done, the remaining deck is cut and the top card flipped over. This card will be included in both players hands, as well as the crib, giving each player a hand of five cards (two hands for the dealer including the crib). One of the main ways points are scored is by grouping some or all of your cards to sum to 15, e.g. 6+9, 7+8, 2+5+8, 5+King. The last sum holds because in cribbage all face cards have a value of 10. Each 15 you make is worth two points. Another way to score is by having pairs of the same card, for example two fives. Each pair is worth two points, and sets of three cards the same counts as three pairs, thus is worth 6 points. Another seemingly random way to score points is by having the Jack of the same suite as the card which has been cut, this is called nobs and it is worth one point.

Now that we all know the ways to score the points in the perfect hand (there are a couple other ways to score which I have chosen to leave out here), we are ready to talk about what the perfect hand is. The perfect hand consists of three 5’s and a Jack which is a different suite than all three fives. Then the fourth 5 must be cut to be the fifth card in the hand. The perfect hand is worth 29 points but it will be helpful to see where all those points come from. Four 15’s can be made with the Jack and each of the 5’s, giving us 8 points. Four more 15’s can be made of three 5’s, giving us a total of 16 points. Because we have four 5’s, we have six pairs, each worth two points, giving us a new total of 28. Of course we cant forget nobs, giving us a grand total of 29 points.

I’ve known about this hand for some time now and have wondered how likely it is to occur. I decided to try to find out on my own. I did so by creating a table which mapped out the dealing of the cards. I had a column for who the card was being dealt to, O for opponent and D for dealer, one for the desired card, and a final column with the probability of dealing the desired card. Here’s a link to the table I created (forgive the awkward link title): Card goes to Card needed probability.

After thinking through each card dealt and its associated probability, I figured the hard work was done. I simply multiplied all the probabilities together to get the probability of all of the events occurring at once. The answer I got was 7.2×10^(-8) which is about 7.2 out of a hundred million. This answer seemed to be reasonable to me, knowing that obtaining the perfect hand is something that rarely occurs. However, I consulted wikipedia for confirmation and found a bit of a discrepancy (98%). The odds on the web were about 1 in 216580, or 4.617×10^(-6). I found similar numbers on other sites as well as one claiming to see such numbers “experimentally” in cribbage tournaments. How was I off by two orders of magnitude?

I did some reworking with a friend ignoring the dealers cards. We did this because in one of the calculations online (which we had a hard time following) involved multiplying by 1/46, which we deduced was associated with the probability of cutting the last 5. After going through the calculations as if there was only one person playing (such a sad game), we came up with the probability of 7.695×10^(-8), which was only slightly better than before. We then had a huge “ahaa!” moment when we realized that there were four perfect hands, i.e. one for each Jack. We multiplied our answer by a measly 4 and got 3.078×10^(-7).

We were getting much better, but an order of magnitude off is still quite a bit. After banging our heads against the wall for a half hour or so, we came to the realization that we were forcing the perfect hand to be dealt in the first four cards of the six. We did some thinking and came to the conclusion that there are in fact 15 different ways (6 choose 4) different ways to deal each perfect hand, each having the same probability. After multiplying our last probability by 15 we got 4.617×10^(-6)! We were able to match the probability given by wikepedia on our own, which excited us immensely.

We were both a little off put though that the accepted probability of obtaining the perfect hand was calculated in a situation where one person sits down and deals themselves 6 cards, cuts the deck and then flips the last card. After all, that is not how cribbage is really played. We took what we learned trying to match the probability posted on the internet and applied it to my original probability. After multiplying by a factor of 60, the probability we obtained was 4.32×10^(-6). Both my friend and I feel that this is a better approximation of the probability of obtaining a perfect hand in a two handed game of cribbage.

In doing this I learned that one should never accept what they see without skepticism. If one is indeed skeptical enough, they should try to work out the problem on their own. They may just find out that the accepted answer doesn’t fit their standards. Also, a lot is learned by working things out, opposed to taking the answers for granted.

Euler, Master of Us All

I recently finished reading Euler, Master of us All, by William Dunham. I was most interested in this book due to my fascination of the natural number. Before reading the book I had seen Euler’s identity, and of course Euler’s equation. I also remember reading somewhere that we denote the natural number with an e in honor of Euler himself. I’m not sure if this is true, but after reading Euler, Master of us All (EMA), I’ve been convinced that he deserves the honor regardless.

The book started off with a biography of Euler to give the reader an idea of what he did when, mostly noting where he was working at the time. I liked this a lot because I honestly hadn’t a clue when Euler lived at all (he lived at about the same time as Ben Franklin). The rest of the book didn’t follow any chronological order, but was organized by different types of mathematics.

I felt that the style of the book was a bit repetitive and predictable at times. The author used the same formatting for each chapter, but this was only minutely annoying. It didn’t detract from my amazement at some of the things Euler did. He truly was a master of mathematics. There were a few trends that I noticed throughout the text. Euler loved logarithms. The second chapter of the book is dedicated to his work on logarithms and points out that Euler was the first to recognize the relationship between logarithms and exponentials. He discovered the “golden rule of logarithms” which is used to change base of a logarithm with a given base. Though the second chapter is the only one with logarithms in its title, by no means is it the only place they show up. Euler used logarithms in many of his “proofs” even when they didn’t seem applicable at all. One of my favorites was to show the sum of reciprocals of all primes diverges. In this particular proof Euler not only used the natural logarithm, but also expressed an infinite series of natural logs as an infinite sum of infinite series.

This brings me to another theme I picked up on, namely that Euler loved infinite series. They were another tool on his proof tool belt. As I was reading it felt like every other proof involved an infinite series. One of my favorite excerpts from the book was how Euler calculated e to 20 digits. His method seemed to me to be very loosely related to taking derivatives. It involved exponentiation by an infinitesimal number. He realized that this would be very near to one, and wrote it as a^w=1+kw where w was the infinitesimal, a was some number, and k was a constant he found to be dependent on a. After some change of variables and a series expansion, he set k to be one, and found that a must be e, the natural number! I thought this was brilliant, as the set up reminded me vaguely of Newton’s method.

I would recommend this book to a fellow math fanatic. There is so much great math history and so many cool math tricks, though “hand wavy” they might be. I would warn a potential reader of the amount proofs included in the text. It’s totally possible to read the text without carefully following the proofs (as I did for some sections), however I found it much more interesting to try to at least see the logic behind the proofs. This slowed me down a bit while reading, but I was ok with it. I’ve always been one of those who oddly enjoys math proofs. In short, I would definitely recommend this book to any math major with the cautioning of light to mild proofs involved. It was a good read and I learned a lot about a great mathematician.

Relationship Between Math and Physics

I’ve always thought that math physics have had a very intimate relationship. After all, it seems as if one could never get very far in studying physics without a firm grasp on a few mathematical concepts. What I would argue is most basic is an understanding algebra. Many people view physics as that hard course that they needed to take in college for some reason unbeknownst to them. To most, this course consists of a large set of random equations. The task of those taking the course then becomes figuring out what equations to use for which problem. To do this, one definitely needs to be comfortable manipulating and rearranging equations.

As one progresses in their physics career, math becomes more of a necessity. It seems like math is the language of physics. A prime example can be seen when studying electrodynamics. One can have a large set of equations which apply only to specific situations. But with a little knowledge of vector calculus one can greatly reduce the number of equations needed to four, namely Maxwell’s equations. Though he discovered none of them and only “fixed” one, his name is associated with these equations because he was one of the first people to recognize how closely they were related. For those of you who aren’t familiar, Maxwell’s equations are as follows:  

 Maxwells equations

Where E denotes electric vector field, B denotes magnetic vector field, ρ denotes the electric charge density, J denotes electric current density, and μ (naught) is the permeability of free space, and ε (naught) is the permittivity of free space. Let’s just examine the first equation here. First let’s make sure everyone knows what the del operator (the “upside down” triangle). It can be thought of a differential vector. In three dimensions it is

Del operator

When this operator is multiplied by a vector field via scalar product, as it is in the first of Maxwell’s equations, it is called divergence. In loose terms, it gives a measure of how much the vector field is expanding from that point. If we are to integrate both sides of the first of Maxwell’s equations we are able to apply the divergence theorem on the left hand side. The divergence theorem states that the integral of the divergence of a vector field over some volume is equal to the surface integral of the flux of the vector field over the surface enclosing the volume. More clearly stated:

Divergence theoremAfter integrating, the left hand side of the equation becomes the integral of the flux over a surface integral. The right hand side becomes the charge enclosed by the surface, divided by the permittivity constant. But what does this mean? It means that the amount of electric flux through a surface is directly related to the amount of charge enclosed in that surface. What if we extend our surface without enclosing any more charge? Assuming there’s some good symmetry (usually a sphere), we can deduce that the strength of the electric field decreases as we move away from the source (the charge enclosed). This may seem like a very round about way to say what could be said in a sentence or two. This is true to an extent, but there is no way one could conclude anything quantitative from a couple of sentences. Thus the mathematical representation is much more robust than saying a few words about how electric fields behave.

I’d say that’s enough of that, so lets get back to the question at hand. How exactly are math and physics related? Is it true that math is merely a tool on the belt of a physicist? If this were true, I think the relationship could be classified as being communal, meaning that physics is the only beneficiary. One needs only to come up with a counter example to disprove this claim. The perfect counter example can be found in one of my favorite mathematical stories, and perhaps one of the best known. Newton’s work on the development of differential calculus is an obvious way in which physics has helped to advance mathematics (I know Leibniz played his role as well, but let’s not get into that). Newton made leaps and bounds towards the calculus we know today, but was no mathematician. He didn’t prove what he know worked, as is the case for many physicists. He did however lay down the ground work.

I’m sure there are numerous other examples where physics has lent ideas to mathematics and vice versa, but I’m running out time to write this post. What we can take away though is that math and physics are intimately related in a mutual relationship. Both help to advance the other. I feel that this should be kept in mind when teaching the two subjects. This allows for students to see how close the two subjects are related. It also will prepare them to make the same types of advancements that have been made throughout history. If we teach math and physics as separate entities, in the future we might treat them as so, which could hinder progress in both fields.

Construction of Specific Nets

The other day in math class my professor showed us a paper cube he brought from home, however this was no ordinary cube. It was held together by magnets and was able to split into three different parts. The cube was split in half into two equal right triangular prisms. One of the prisms was then split once more, into a square pyramid with its vertex directly above one of its corners, and its height equal to the height of the original cube, and a third piece I can only think to describe as an irregular tetrahedron. We were split into groups to reconstruct the three polyhedra starting with their nets. I thought this was a fun exercise, and thought I would take it further outside of class.

I began with the task of constructing a specific cone, i.e. one with given dimensions. I decided I would specify the radius of the base, r, and the angle, θ, to be measured between one end of a diameter, the vertex, and the opposite end of the diameter. I first drew a qualitative net of a cone, and labeled the parts I would need to find. In my case, I needed to find the radius, R, of the pie shaped portion of the net (in the picture below, its one mighty piece of pie), as well as an angle, Φ, which must be spanned by the piece so the arc length equals the circumference of the base. I began by drawing a cross section of the cone through a diameter and the vertex. This was an isosceles triangle which I split into two congruent right triangles. I used simple trigonometry to find in terms of r  and phi. I found that R = r/sin(θ/2). I then found Φ in terms of given quantities. The key to this was simply to realize that the arc spanned by Φ must equal the circumference of the base. The formal representation of this is ΦR = 2πr, which after substituting in our value for  and solving for Φ, I concluded that Φ = 2πsin(θ/2). To test this I chose θ = 90° and a non specific radius (it ended up being 3.04 “Geogebra cms”), and used Geogebra to create my net. After folding, sure enough the angle was (about) 90°!

IMG_1925IMG_1927

Feeling that the quantitative aspect cones and pyramids would reduce to trig problems, I decided to do some qualitative constructions too. I started with the problem of constructing a square pyramid whose vertex was directly above the midpoint of one of its edges, for the sake of explanation let it be the right edge. I came to the conclusion that such a net consists of two different isosceles triangles, two congruent right triangles, and one square. The right side (where the vertex lies) is an isosceles triangle with its height equal to the height of the pyramid, and its base equal to the edge of the square base. The front and back side are two congruent triangles with one leg equal to the edge of the square base, and another equal to the equal legs of the isosceles triangle on the right. The left side is the second isosceles triangle, and has two equal legs that are equal to the hypotenuse of the right triangles previously described. I again used Geogebra to make the net and construct the solid.

IMG_1915IMG_1918

Wanting yet a little more, I turned my attention to constructing a net for a cylinder with a whole in the middle. I could easily construct a net for a cylinder with holes in its bases, but figuring out how to connect to that the inner surface necessary to make our cylinder a solid was a bit trickier. I started by drawing two cylinders, one large (with holes in its bases), and one small. For the smaller cylinder, I used dotted lines to represent the base, as the “base” of the small cylinder is the hole in the base of the large cylinder. I tried to superimpose the two images (matching the “base” of the small cylinder to the hole in the big one), but could not. This was due to the overlap of the base of the large cylinder with the side of the small cylinder. I was ready to claim that the net was impossible to construct when the idea of splitting one of the large bases in half hit me. This would allow me to attach the side of the small cylinder to its “base” (the hole in the big base) without forcing any overlap of the big base and the small side. I found out the hard way that in order for the base splitting to work, one must be careful to attach the halves at opposite sides of the side. This is not opposite ends on the net, as the two ends come together at the same point. After a failed attempt, and one minor change, I was able to construct the net for a cylinder with a hole in it.

IMG_1919 IMG_1921 IMG_1924

Throughout the process I learned that when working with nets it is very helpful to first do a qualitative analysis before a quantitative one. I also found that a good method was to start where things are simple, i.e. a base or known side, and stem from there. I would be interested to work out the quantitative description of the cylinder with a hole removed from its center. I also think it could be easily generalized in a way that the hole and cylinder need not be coaxial.

Construction

I have personally always been fascinated with the idea of construction in the context of geometry. For those of you who aren’t familiar with it, the method of construction allows one only to use a straight edge and a compass, both without any markings. With these two tools, and quite a bit of smarts, ancient mathematicians were able to do such things as construct the perpendicular bisector of a line segment or the angle bisector of an arbitrary angle. Using the method of construction, ancient mathematicians figured out how to create regular polygons with 3, 4, and 5 sides. Because they were also able to construct the midpoint of a line segment, they were able to double the number of sides in their polygons, giving them 3x, 4x, and 5x sided polygons where x=2^k for some positive integer k. However, they were never able to generalize their results to construct a polygon with n sides for any integer n. In 1796, when he was 19 years old, Carl Friedrich Gauss came up with a proof that a heptadecagon, or a 17-sided regular polygon, could be created using construction. A method of construction for a heptadecagon wasn’t discovered until 1825 by Johannes Erchinger. Though Gauss himself never constructed any 17-gons, he was able to generalize his proof a bit, giving the sufficient conditions for a n-sided polygon to have the possibility of begin constructed. That condition is that n is a distinct Format prime multiplied by 2^k, where k is any non-negative integer. Gauss was unable to prove that this condition was also necessary, though a hand was lent by Pierre Wantzel. Hence the name Gauss-Wantzel Theorem.

What is Math?

What is mathematics? A fair question, though one I haven’t considered too much, even though I’m a student whose done math for quite some time (15 years and counting). I would  say that mathematics is a language used to study and discuss patterns found in nature. The fact that math is a language makes it a very important tool in many sciences. It has allowed humans to communicate in order to understand the way the world and universe around us works. In math, as in any language, the more one knows about the language and the structure, the more adept one becomes at being able to convey and understand complex thoughts and ideas.

As one might guess, there are numerous (perhaps countably infinite?) milestones in the history of mathematics. I’ve never been too good with history, so I’m not at all sure about order, and I’ll just start with my favorite, which is Isaac Newton’s discovery of calculus. I am still astonished to this day, when I imagine someone my age coming up with a completely new type of mathematics in order to solve a problem they were working on. Another one of my favorites is the acceptance of zero as a number, an idea I’ve been raised to accept, but which took mathematicians years of debate to conclude. A milestone that I can personally relate to is the acceptance of i as a numberl. My older brother taunted me for a couple weeks when I was in 4th grade with the simple question: “What is the square root of -16?” (of which my answers were -4!… no 4! … no…). Also high up on my list of favorite math milestones would have to be the acceptance of irrationals as numbers. I feel like my list would not be complete without the two completely different milestones that I will lump into one underrated milestone: the discovery of e and pi. What kind of a subject would math be without the most beautiful expression, e^(i(pi))+1=0?