Built on Facts

Yesterday I completely forgot to mention the anniversary of the most amazing thing humanity has ever done.  July 20, 1969, Neil Armstrong, Buzz Aldrin, and Michael Collins successfully completed our first steps onto another world.  Though I was born long after their success, I am in absolute awe of them and those who helped make the mission possible.  They are an inspiration to me, and will be inspiring future generations forever.  Let’s do some classical mechanics in their honor:

This a cute trick that I first saw in my undergrad classical mechanics class. You grad students and above will already know it, but hey, everyone has to start somewhere.

Let’s say you have some force which is a function of position. A gravitational potential, for example. You have some initial velocity and you’d like to know how velocity varies as a function of distance. Say you want to find the maximum height of a projectile shot directly upwards, or similar. You’re not worried directly about velocity as a function of time. Odds are the equation you’re given is the equation for force, and thus the equation for acceleration up to a factor of 1/mass.

Hmm. There’s time involved in that, and if possible maybe getting rid of it will shed some light on whatever our problem happens to be. How about the chain rule, which we learned in calculus?

Does that help any? Yes, it does! The last term dx/dt is just the velocity v, and so we have

Now there’s a version of Newton’s second law you may not have seen before. No time dependence at all, not directly. And that can make it quite useful for various forces F which depend on v but not t directly, chiefly through rearranging into the integral

For an example of throwing a rock in a gravitational potential F = -mg, (taking g positive) we get

With initial and final positions and velocities subscripted appropriately. Performing the integrations and canceling m gives

Now we might as well solve for the final position so the equation is a little more useful

You could set the final velocity and initial position to 0 if you wanted to find the maximum height attained by a rock thrown upward if you wanted, and you’d get exactly the usual result.

“So Matt, this is probably the third time you’ve treated the falling rock with some different weird method. What gives? Isn’t 1/2 g t² good enough?” Yes it is, but the rock is just an example which I’m using precisely because we know what should happen and thus we can verify that we’re not misunderstanding the new method. There’s plenty of problems where this method will give you a pretty easy solution where using the time dependence directly would be much harder. In fact I’m going to treat that rock at least two more ways; we haven’t yet done the classical Hamiltonian formulation or the quantum mechanical propagator.

Professional carpenters have a lot of tools in their workshops for a reason. Physicists learn a lot of math for precisely the same reason.