Equation of the Day

Equation of the Day #6: Newton’s Second Law

May 12, 2017 Alex1 Comment

Born on Christmas Day, 1642, Isaac Newton was one of the most brilliant minds the world has ever seen. He invented calculus, discovered that white light is composed of all the colors of the rainbow, formulated the law of universal gravitation (which we now use to send people and robots into orbit around Earth and other objects in the Solar System), and his famous laws of motion (among a host of other achievements). Today I’m focusing on his Second Law of Motion, which you’ve probably seen written as

${\bf F}=m {\bf a},$

where F is force, m is mass, and a is acceleration. The boldface formatting indicates a vector, which is a mathematical entity that has both magnitude (or “strength”) and direction (up/down, left/right, forward/backward). What this means is that an object whose speed (the magnitude of velocity) or direction of travel is changing must be subject to an unbalanced force. This ties in to Newton’s First Law of Motion, which states that the natural state of an object is to move in a straight line at constant speed. However, the above equation is actually not fully general – it only applies when the accelerating object has constant mass. This works fine for projectile motion of a pitcher hurling a baseball or a cannonball launched at an enemy fortress, but it fails for, say, a rocket delivering a payload to the International Space Station, since fuel is being burned and ejected from the rocket. In general, Newton’s Second Law is given by

${\bf F}=\dfrac{d{\bf p}}{dt},$

where p is the total momentum of the system. Momentum is a measurement of the “oomf” that an object has due to its motion. Explicitly, it is the product of mass and velocity, or ${\bf p} = m{\bf v}$ . The expression d/dt indicates the rate of change of a quantity, in this case momentum, over time. So Newton’s Second Law states that a change in an object’s motion is due to an unbalanced force, which sounds like what I said for the equation ${\bf F}=m {\bf a},$ but this takes into account a change in mass as well.

Conceptually, this was a big breakthrough at the time and is something that students in introductory physics classes struggle with today. (Note: Newton figured all this stuff out and more by the time he was 26. Let that sink in.) On a quantitative level, this law allows us to predict the motions of objects that subjected to certain kinds of forces. If we know the nature of the force an object is subjected to, we can make predictions of what path the object will travel along. Similarly, if we know the trajectory of an object, we can predict the behavior of the force acting upon it. This was how Newton figured out the nature of gravity. Observations of his time showed that, from a sun-centered view of the Solar System, the planets orbit in ellipses, not circles. Newton surmised that this meant that the force of gravity must diminish with the square of the separation between two objects.

But how? The presence of acceleration in Newton’s Second Law indicates a differential equation. Acceleration is the rate of change of velocity, which itself is the rate of change of position (which I will denote as r). So, taking that into account, we get the equation

$\dfrac{d^2{\bf r}}{dt^2} = \dfrac{{\bf F}}{m}$

This can be solved for an expression of position as a function of time when we know the form that F takes. If the force is constant in space (like gravity near Earth’s surface), r takes on the form of parabolic motion. If the force obeys Hooke’s Law, $F = -k \Delta r$ , then the solution is simple harmonic oscillation. And if the force obeys an inverse square law, the position takes the functional form of conic sections, which are ellipses for the planets bound to the Sun in our solar system.

Conic sections: 1. Parabola, 2. Circle (bottom) and ellipse (top), 3. Hyperbola. (Image: Wikimedia Commons)

This is the model we use when we send objects into orbit or when we want to model a baseball being hit out of the park (among many other things), and it has been extended to explain phenomena that Newton himself could not have imagined. When scientists or mathematicians describe equations as beautiful, this pops into my head. It has such a simple form, but explains a wealth of real phenomena.

Equation of the Day #5: The Wave Equation

April 17, 2017May 11, 2017 Alex1 Comment

Waves are ubiquitous throughout nature, from water waves to sound waves to light and even matter itself. For this reason, one of my favorite equations is the wave equation,

$\Large\dfrac{1}{c^2} \dfrac{\partial^2 \psi}{\partial t^2} = \nabla^2 \psi.$

This is a differential equation, where ∂/∂t notation indicates the derivative operator with respect to the variable t, which I’ll get back to in a moment. In this equation, $\psi = \psi(x,y,z,t)$ represents the deformation the wave is making at a given point in space (x,y,z) at a given time t, and c is the speed of the wave. The right hand side is known as the Laplacian of $\psi$ . Physically, this is a measure of how much $\psi(x,y,z,t)$ deviates from its average local value. If $\psi$ is at a minimum, $\nabla^2\psi$ is positive; it is negative if $\psi$ is at a maximum.

A derivative is the rate of change of a function, like the deformation $\psi$ . So, $\partial\psi/\partial t$ is the rate at which the deformation changes over time. The second derivative, which is in the wave equation, is the rate of change of this rate of change, like how acceleration is the rate of change of velocity, which is itself the rate of change of the position of an object. Thus, the left hand side indicates how the variation in $\psi$ with respect to time changes in time at a given point in space. These spatial averages and temporal changes feed each other in a constant mutual feedback, resulting in the wave taking on a certain shape. All wave phenomena are governed by this equation (or a slight modification of it). In one dimension, the wave equation simplifies to

$\dfrac{1}{c^2} \dfrac{\partial^2 \psi}{\partial t^2} = \dfrac{\partial^2 \psi}{\partial x^2},$

which, for pure frequencies/wavelengths has the solutions

$\psi(x,t) = A \sin(kx-\omega t) + B \cos(kx - \omega t),$

where A and B are determined by appropriate boundary conditions (like if a string is fixed at both ends, or free at one end and fixed at the other), $\omega$ is the angular temporal frequency of the wave, and $k$ is the angular spatial frequency of the wave.

(Image: Wikimedia Commons)

This equation can be rewritten in terms of the wavelength $\lambda$ (shown above) and period T of the wave,

$\psi(x,t) = A \sin\left[\tau \left(\dfrac{x}{\lambda}-\dfrac{t}{T}\right)\right] + B \cos(kx - \omega t),$

where $\tau$ is the circle constant. These quantities are related via the speed of the wave, $c = \lambda/T = \omega/k$ . These solutions govern things like vibrations of a string, sound made by an air column in a pipe (like that of an organ, trumpet, or didgeridoo), or even waves created by playing with a slinky. They also govern the resonances of certain optical cavities, such as lasers or etalons. Adding up a bunch of waves with pure tones can create waves of almost any imaginable shape, such as this:

(Image: Wikimedia Commons)

Or this:

(Image: Wikimedia Commons)

Waves do not have to be one dimensional, of course, but I’ll save the two- and three-dimensional cases for another entry.

-A

Equation of the Day #4: The Pentagram of Venus

March 3, 2017October 3, 2021 Alex1 Comment

The above image is known as the Pentagram of Venus; it is the shape of Venus’ orbit as viewed from a geocentric perspective. This animation shows the orbit unfold, while this one shows the same process from a heliocentric perspective. There are five places in Venus’ orbit where it comes closest to the Earth (known as perigee), and this is due to the coincidence that

$\dfrac{1\text{ Venusian Year}}{1\text{ Earth Year}} = \dfrac{224.701\ \text{days}}{365.256\ \text{days}} =0.6151877 \approx \dfrac{8}{13} =0.6153846$

When two orbital periods can be expressed as a ratio of integers it is known as an orbital resonance (similar to how a string has resonances equal to integer multiples of its fundamental frequency). The reason that there are five lobes in Venus’ geocentric orbit is that $13-8=5$ . Coincidentally, these numbers are all part of the Fibonacci sequence, and as a result many people associate the Earth-Venus resonance with the golden ratio. (Indeed, pentagrams themselves harbor the golden ratio in spades.) However, Venus and Earth do not exhibit a true resonance, as the ratio of their orbital periods is about $0.032\%$ off of the nice fraction $8/13$ . This causes the above pattern to precess, or drift in alignment. Using the slightly more accurate fraction of orbital periods, $243/395$ , we can see this precession.

This is the precession after five cycles (40 Earth years). As you can see, the pattern slowly slides around without the curve closing itself, but the original $13:8$ resonance pattern is still visible. If we assume that $243/395$ is indeed the perfect relationship between Venus and Earth’s orbital periods (it’s not; it precesses $0.8^\circ$ per cycle), the resulting pattern after one full cycle (1944 years) is

Which is beautiful. The parametric formulas I used to plot these beauties are

$\begin{aligned} x(t) &= \sin t + r^{2/3} \sin\left(\frac{t}{r} \right)\\ y(t) &= \cos t + r^{2/3} \cos\left(\frac{t}{r} \right) \end{aligned}$

Where $t$ is time in years and $r$ is the ratio of orbital periods (less than one).

-A

Equation of the Day #3: Triangular Numbers

February 27, 2017March 4, 2017 AlexLeave a comment

I like triangles. I like numbers. So what could possibly be better than having BOTH AT THE SAME TIME?! The answer is nothing!

The triangular numbers are the numbers of objects one can use to form a triangle.

Anyone up for billiards? Or bowling? (Image: Wikimedia Commons)

Pretty straightforward, right? To get the number, we just add up the total number of things, which is equal to adding up the number of objects in each row. For a triangle with $n$ rows, this is equivalent to

$\displaystyle T_n = 1+2+3+ \ldots + n = \sum_{k=0}^n k$

This means that the triangular numbers are just sums from 1 to some number $n$ . This gives us a good definition, but is rather impractical for a quick calculation. How do we get a nice, shorthand formula? Well, let’s first add sequential triangular numbers together. If we add the first two triangular numbers together, we get $1 + 3 = 4$ . The next two triangular numbers are $3 + 6 = 9$ . The next pair is $6 + 10 = 16$ . Do you see the pattern? These sums are all square numbers. We can see this visually using our triangles of objects.

(Image: Wikimedia Commons)

You can do this for any two sequential triangular numbers. This gives us the formula

$T_n + T_{n-1} = n^2$

We also know that two sequential triangular numbers differ by a new row, or $n$ . Using this information, we get that

$\begin{aligned} n^2 &= T_n + (T_n - n) \\ 2 T_n & = n^2 + n = n(n+1) \\ T_n &= \frac{n(n+1)}{2} = \begin{pmatrix} n+1 \\ 2 \end{pmatrix} \end{aligned}$

Now we finally have an equation to quickly calculate any triangular number. The far right of the final line is known as a binomial coefficient, read “ $n$ plus one choose two.” It is defined as the number of ways to pick two objects out of a group of $n + 1$ objects.

For example, what is the $100^{\rm th}$ triangular number? Well, we just plug in $n = 100$ .

$T_{100} = \dfrac{(100)(101)}{2} = \dfrac{10100}{2} = 5050$

We just summed up all the numbers from 1 to 100 without breaking a sweat. You may be thinking, “Well, that’s cool and all, but are there any applications of this?” Well, yes, there are. The triangular numbers give us a way of figuring out how many elements are in each row of the periodic table. Each row is determined by what is called the principal quantum number, which is called $n$ . This number can be any integer from 1 to $\infty$ . The energy corresponding to $n$ has $n$ angular momentum values $(l = 0, 1,\ldots, n-1)$ which the electron can possess, each of which has a total of $2l + 1$ orbitals for an electron to inhabit, and two electrons can inhabit a given orbital. Summing up all the places an electron can be in for a given $n$ involves summing up all these possible orbitals, which takes on the form of a triangular number.

$\begin{aligned} \sum_{k=0}^{n-1}(2l +1) &= 2 \sum_{k=0}^{n-1} k + n \\ &= 2T_{n-1} + n \\ &= n(n-1) + n \\ &= n^2 \end{aligned}$

The end result of this calculation is that there are $n^2$ orbitals for a given $n$ , and two electrons can occupy each orbital; this leads to each row of the periodic table having $2\lceil (n+1)/2 \rceil^2$ elements in the $n^{\rm th}$ row, where $\lceil x \rceil$ is the ceiling function. (This complication is due to the Aufbau principle, which dictates how energy levels fill up.) They also crop up in quantum mechanics again in the quantization of angular momentum for a spherically symmetric potential (a potential that is determined only by the distance between two objects). The total angular momentum for such a particle is given by

$L^2 = \hbar^2 l(l+1) = 2 \hbar^2 T_l$

What I find fascinating is that this connection is almost never mentioned in physics courses on quantum mechanics, and I find that kind of sad. The mathematical significance of the triangular numbers in quantum mechanics is, at the very least, cute, and I wish it would just be mentioned in passing for those of us who enjoy these little hidden mathematical gems.

There are more cool properties of triangular numbers, which I encourage you to read about, and other so-called “figurate numbers,” like hexagonal numbers, tetrahedral numbers, pyramidal numbers, and so on, which have really cool properties as well.

-A

Equation of the Day #2: Cardioid

February 24, 2017February 1, 2018 AlexLeave a comment

I made the above figure in Inkscape. A cardioid is the envelope formed by a set of circles whose centers lie on a circle and which pass through one common point in space. This image shows the circle on which the centers of the circles in the above image lie. A cardioid is also the path traced by a point on a circle which is rolling along the surface of another circle when both circles have the same radius (here is a cool animation of that).

What is the cardioid’s significance? Well, it looks like a heart, which is kind of cool. It’s also the (2D) pickup pattern of certain microphones (I have a USB cardioid microphone). If a sound is produced at a given point in space, the pickup pattern shows an equal intensity curve. So, if I place a microphone at the intersection point of all those circles, the outside boundary is where a speaker producing, say, a 440 Hz tone would have to be to be heard at a given intensity. So, the best place to put it would be on the side where the curve is most round (the bottom in this picture) without being too far away from the microphone.

Another interesting fact about the cardioid is that it is the reflection of a parabola through the unit circle $(r = 1)$ . Here’s what I mean; in polar coordinates, the equation of the above cardioid is given by

$r = 2a(1-\sin\theta)$

where a is a scaling factor, and theta is the angle relative to the positive x-axis. The origin is at the intersection of the circles. The equation of a parabola opening upwards and whose focus is at the origin in polar coordinates is just

$r = [2a(1-\sin\theta)]^{-1}$

which is an inversion of the cardioid equation through $r = 1$ , or the unit circle.

♥

-A

Equation of the Day #1: Complex Numbers

February 22, 2017February 1, 2018 Alex1 Comment

Most of us are used to the real numbers. Real numbers consist of the whole numbers (0, 1, 2, 3, 4, …), the negative numbers (–1, –2, –3, …), the rational numbers (1/2, 2/3, 3/4, 44/7, …), and the irrational numbers (numbers that cannot be represented by fractions of integers, such as the golden ratio, $\sqrt{2}$ , or $\tau$ ). All of these can be written in decimal format, even though they may have infinite decimal places. But, when we use this number system, there are some numbers we can’t write. For instance, what is the square root of –1? In math class, you may have been told that you can’t take the square root of a negative number. That’s only half true, as you can’t take the square root of a negative number and write it as a real number. This is because the square root is not part of the set of real numbers.

This is where the complex numbers come in. Suppose I define a new number, let’s call it i, where

$i^2 = -1.$

We’ve now “invented” a value for the square root of –1. Now, what are its properties? If I take i³, I get –i, since i³ = i²i. If I take i⁴, then I get i²i² = +1. If I multiply this by i again, I get i. So the powers of i are cyclic through i, –1, –i, and 1.

This is interesting, but what is the magnitude of $i$ , i.e. how far is $i$ from zero? Well, the way we take the absolute value in the real number system is by squaring the number and taking the positive square root. This won’t work for i, though, because we just get back i. Let’s redefine the absolute value by taking what’s called the complex conjugate of i and multiplying the two together, then taking the positive square root. The complex conjugate of i is obtained by negating the imaginary part of i. Since i is purely imaginary (there are no real numbers that make up i), the complex conjugate is –i. Multiply them together, and you get that –i*i = –i² = 1, and the positive square root of 1 is simply 1. Therefore, the number i has a magnitude of 1. It is for this reason that i is known as the imaginary unit!

Now that we have defined this new unit, i, we can now create a new set of numbers called the complex numbers, which take the form

$z = a + bi,$

where a and b are real numbers. We can now take the square root of any real number, e.g. the square root of –4 can be written as

$\sqrt{-4} = \pm 2i,$

and we can make complex numbers with real and imaginary parts, like 3 + 4i.

How do we plot complex numbers? Well, complex numbers have a real part and an imaginary part, so the best way to do this is to create a graph where the abscissa (x-value) is the real part of the number and the ordinate (y-axis) is the imaginary part. This is known as the complex plane. For instance, 3 + 4i would have its coordinate be (3,4) in this coordinate system.

What is the magnitude of this complex number? Well, it would be the square root of itself multiplied by its complex conjugate, or the square root of

$(3 + 4i)(3 - 4i) = 9 + 12i - 12i +16 = 25.$

The positive square root of 25 is 5, so the magnitude of 3 + 4i is 5.

We can think of points on the complex plane being represented by a vector which points from the origin to the point in question. The magnitude of this vector is given by the absolute value of the point, which we can denote as r. The x-value of this vector is given by the magnitude multiplied by the cosine of the angle made by the vector with the positive part of the real axis. This angle we can denote as $\phi$ . The y-value of the vector is going to be the imaginary unit, i, multiplied by the magnitude of the vector times the sine of the angle $\phi$ . So, we get that our complex number, z, can be written as

$z = r(\cos\phi + i\sin\phi).$

The Swiss mathematician Leonhard Euler discovered a special identity relating to this equation, known now as Euler’s Formula, that reads as follows:

$e^{i\phi} = \cos \phi + i\sin\phi$

Where e is the base of the natural logarithm. So, we can then write our complex number as

$z = re^{i\phi}.$

What is the significance of this? Well, for one, you can derive one of the most beautiful equations in mathematics, known as Euler’s Identity:

$e^{i\tau} = 1 + 0$

This equation contains the most important constants in mathematics: e, Euler’s number, the base of the natural logarithm; i, the imaginary unit which I’ve spent this whole time blabbing about; $\tau$ , the irrational ratio of a circle’s circumference to its radius, which appears all over the place in trigonometry; 1, the real unit and multiplicative identity; and 0, the additive identity.

So, what bearing does this have in real life? A lot. Imaginary and complex numbers are used in solving many differential equations that model real physical situations, such as waves propagating through a medium, wave functions in quantum mechanics, electromagnetic phenomena, and fractals, which in and of themselves have a wide range of real life application.

-A