Setting up Differential Equations of Motion

[tomkeus]

No ti is not totally bull, it is very standard trick you can se in lot of books. In Lagrangian formalism you don't take Lagrangian as a function of only t (at the bottom, everything in physics is function of time only, but in most cases, that is not the most productive way to look at things), but you take it as $$L(\dot{q_i}, q_i, t)$$ where $$\dot{q_i}$$, and $$q_i$$ are general speeds, and coordinates. Examine this step $$\frac{d\dot{\theta}}{d\theta}\frac{d\theta}{dt}=\frac{d\dot{\theta}}{d\theta}\dot{\theta}$$. In it you just replace $$\frac{d\theta}{dt}$$ with $$\dot{\theta}$$ because $$\dot{\theta}$$ is in Lagrangian formalism coordinate the same way $$\theta$$ is coordinate.

 

[sach1tb]

Also can you explain what each of these terms represents from a Netwonian point of view?

$$r\ddot{\theta}+2\dot{r}\dot{\theta}+g\sin\theta=0$$

I know that the last one comes from the gravitational force. But what are the other two terms? I'm thinking that they might be related to torque, or angular momentum but I can't quite figure out what's going on here.

I'd really like to fully understand the mathematical description of this system perfectly before I move on to to the next system in the book which is far more complex.

The first term is the centrifugal force, and the second one is Coriolis force. Feynman's lectures Vol 1. pages 19-8, 19-9 have an excellent (non-mathematical) explanation for Coriolis force.

"This other force is called Coriolis force, and it has a strange property that when we move something in a rotating system, it seems to be pushed sidewise. Like the centrifugal force, it is an apparent force. But if we live in a system that is rotating, and move something radially, we find that we must also push it sidewise to move it radially. This sidewise push which we have to exert is what turned our body around"​