A Leap Day 2016 Mathematics A To Z: Lagrangian
It’s another of my handful of free choice days today. I’ll step outside the abstract algebra focus I’ve somehow gotten lately to look instead at mechanics.
So, you likely know Newton’s Laws of Motion. At least you know of them. We build physics out of them. So a lot of applied mathematics relies on them. There’s a law about bodies at rest staying at rest. There’s one about bodies in motion continuing in a straight line. There’s one about the force on a body changing its momentum. Something about F equalling m a. There’s something about equal and opposite forces. That’s all good enough, and that’s all correct. We don’t use them anyway.
I’m overstating for the sake of a good hook. They’re all correct. And if the problem’s simple enough there’s not much reason to go past this F and m a stuff. It’s just that once you start looking at complicated problems this gets to be an awkward tool. Sometimes a system is just hard to describe using forces and accelerations. Sometimes it’s impossible to say even where to start.
For example, imagine you have one of those pricey showpiece globes. The kind that’s a big ball that spins on an axis, and whose axis in on a ring that can tip forward or back. And it’s an expensive showpiece globe. That axis is itself in another ring that rotates clockwise and counterclockwise. Give the globe a good solid spin so it won’t slow down anytime soon. Then nudge the frame, so both the horizontal ring and the ring the axis is on wobble some. The whole shape is going to wobble and move in some way. We ought to be able to model that. How? Force and mass and acceleration barely seem to even exist.
The Lagrangian we get from Joseph-Louis Lagrange, who in the 18th century saw a brilliant new way to understand physics. It doesn’t describe how things move in response to forces, at least not directly. It describes how things move using energy. In particular, it uses on potential energy and kinetic energy.
This is brilliant on many counts. The biggest is in switching from forces to energy. Forces are vectors; they carry information about their size and their direction. Energy is a scalar; it’s just a number. A number is almost always easier to work with than a number alongside a direction.
The second big brilliance is that the Lagrangian gives us freedom in choosing coordinate systems. We have to know where things are and how they’re changing. The first obvious guess for how to describe things is their position in space. And that works fine until we look at stuff such as this spinning, wobbling globe. That never quite moves, although the spinning and the wobbling is some kind of motion. The problem begs us to think of the globe’s rotation around three different axes. Newton doesn’t help us with that. The Lagrangian, though —
The Lagrangian lets us describe physics using “generalized coordinates”. By this we mean coordinates that make sense for the problem even if they don’t directly relate to where something or other is in space. Any pick of coordinates is good, as long as we can describe the potential energy and the kinetic energy of the system using them.
I’ve been writing about this as if the Lagrangian were the cure for all hard work ever. It’s not, alas. For example, we often want to study big bunches of particles that all attract (or repel) each other. That attraction (or repulsion) we represent as potential energy. This is easier to deal with than forces, granted. But that’s easier, which is not the same as easy.
Still, the Lagrangian is great. We can do all the physics we used to. And we have a new freedom to set up problems in convenient ways. And the perspective of looking at energy instead of forces gives us a fruitful view on physics problems.