It’s that time of year again, when we count down the days to Isaac Newton’s birthday (according to the Julian calendar, anyway), and how better to mark this than with mathematics? Thus, I’ll post an equation a day until either Christmas Eve or I run out of ideas, and talk about what it means and why it’s important for physics.
Since this is, after all, a celebration of Sir Isaac, let’s kick things off with arguably his most famous equation:
OK, it might not look familiar in this form, but this is, in fact, the full and correct statement of Newton’s Second Law (written in modern notation), which most people know as F=ma, force equals mass times acceleration. Newton himself expressed it thusly:
Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.
It’s in Latin, because it was the 1600’s (and also because he was kind of a dick), but this translates to (according to the 1729 translation quoted by Wikipedia, anyway):
Law II: The alteration of motion is ever proportional to the motive force impress’d; and is made in the direction of the right line in which that force is impress’d.
By “motion” Newton meant what is know known as momentum, which gets the symbol “p” in physics for reasons that passeth all understanding. Thus, this translates to the equation above: The time derivative of the momentum of an object is equal to the net force acting upon it. The derivative is the mathematical expression for the “alteration of motion” as time goes by, and the net force is, well, the “motive force impress’d.”
Why is this important? Because it’s the foundation of physics as a mathematical science.
OK, if you want to be picky, it’s his second law of motion, but the first doesn’t have a convenient mathematical representation, so if you want to do anything quantitative, you start with the second law. And this is the absolute cornerstone of what’s now known as classical mechanics: It tells you that if you know the forces acting on an object, then you can predict the resulting change in its momentum. Given the momentum, you can predict the future position, and with that, you can do anything you want.
The F=ma form that everybody learns in grade school (in an ideal world, anyway) is an approximation to the full expression, assuming an object with constant mass and low velocity. If you want to know how high you can throw a baseball, or how quickly you can stop a car, this is the form to use.
But the full equation encompasses much more than that. It works for systems whose mass is changing– for example, a rocket, which propels itself through the air by burning fuel and expelling the hot exhaust out the back of the rocket. As it goes along, the mass of the rocket decreases, so you can’t use F=ma to find the acceleration (unless you do it in tiny little steps), but you can use the derivative form with no problem. So, rocket science starts with this equation right here.
The full expression also works for any speed you like. If you’re talking about something like a baseball or a car, moving at speeds very slow compared to the speed of light, you can get away with defining momentum as mass times velocity. If you look at things moving at really high speeds, though, like a proton in a particle accelerator, or a really clever dog with access to alien propulsion technology, you need to use the full relativistic definition of momentum, which is more complicated. But Newton’s second law, in the derivative form above, will still work: if you know the forces that act on an object moving at speeds close to the speed of light, you can still predict its future momentum, and thus its future position.
So there’s a whole lot packed into that tiny little equation. There’s a lot behind it, too, since in order to come up with that, not only did Newton have to shake off thousands of years of Aristotelian thinking about the motion of objects (a process started by Galileo Galilei, who died the year Newton was born), but he needed to invent vector calculus in order to make it work.
So, as we start our countdown to Sir Isaac’s birthday, take a moment to appreciate the power and beauty of his second law. And come back tomorrow to see the next equation of the season.