Vis consili expers mole ruit sua.
- Quintus Horatius Flaccus
The words of Horace above attain a spare and austere beauty in Latin, but the meaning is carried equally well in English. Force without wisdom falls of its own weight. Two different sets of words carry his millenia-old thoughts to the present time. There are different ways of saying the same thing.
In physics, force permeates the first few semesters of study and continues to be a useful tool in higher physics. But in many respects force is not the most elegant tool in our arsenal. It’s a vector; not only do we have to worry about magnitude but we also have to deal with direction. With some exceptions, vectors are significantly more difficult to deal with than regular numbers. But look at Newton’s First Law, which we all know:
F and a are both vectors. Force in the x-direction produces acceleration in the x-direction, etc. Easy enough for basic problems, but for complicated ones things get rapidly very difficult. Other coordinate systems can make the situation even worse. Non-classical physics is worse still; the concept of a force doesn’t even necessarily have much meaning in quantum mechanics.
But like translating from one language to another, we have alternate ways to work with a given physical situation. Instead of examining the forces, we can look at the energy. This is just a conceptual introduction so we’ll skip all the derivation for now, and present the final result. Define a quantity L, called the Lagrangian. It’s a measure of the energy of the system; specifically it’s the measure of the kinetic energy K minus the potential energy U. Thus by definition L = K - U.
Now pick coordinates: maybe x, y, and z, or θ and φ, dimensionless quantities, or whatever you feel like and in any combination. Pick and choose as many coordinates as it takes to describe the system, and call them qi, for however many i’s you need. It turns out that the following equation completely describes the classical time evolution of the system:
In this differential equation, you have
- Energy specified by L
- Coordinates specified by q. The dot above q just means the time derivative of q, to save space.
- A time relationship connecting them
And that’s all you need. No forces to be seen. No vectors of any kind. You can specify the coordinate system in whatever way most easily describes the configuration of the system without having to worry about whether that same coordinate system also can easily describe the forces within that system. Now let’s pick the easiest example in the world to test it out. In the future we’ll do harder things that really show the power of Lagrangian dynamics, but since this is just dipping our toes into the water we’ll just use a very simple rock falling straight downward.
The potential energy of that rock is just U = mgy, where y is the height above the ground.
The kinetic energy of that rick is just K = (1/2)mv2, where v is the velocity and thus the time derivative of y, which we write as y with a dot over it.
That makes the Lagrangian L = K - U become:
Plug that into the differential equation above with q = y (because as we said before the various q are any coordinate you want), and you get:
Now carry out the y and y-dot differentiation formally, resulting in:
Finally doing the time derivative:
Which lo and behold finally gives us:
The rock has an acceleration g vertically downward, independent of its mass exactly as we expect from Newton’s laws. We never touched a force or a vector. “But Matt,” you say, “isn’t that actually a lot harder than doing things with forces?” Yes it is, for this problem. But this problem is just a toy example to show that this does give us the right answer for a problem we can already do trivially. For instance, try to characterize the motion of a marble placed on top of a basketball using forces. If you can do it at all, you’re a much more patient physicist than me. Using energy methods it’s still not quite a simple calculation but it’s thoroughly tractable. But that’s a story for another day.