This week I’ve been talking to my third year mechanical engineering class about the Lagrangian approach to solving dynamical problems. OK, please don’t close your browser now, rest assured that you don’t need to know what the Lagrangian approach is to follow this post. (if you do, then click here.)
I reckon there are two ways of dealing with this with a class of students. One (my favourite, as a physicist) is to go into gory detail about the principle of least action and calculus of variations (again, don’t worry what these are, because the chances are that you will never ever need to know them). The other is to say ‘here is the formula that makes it all work’ and leave it at that.
I have taken the latter approach with my students, and concentrated on teaching them how to use the Lagrangian method to solve practical problems, rather than trying to explain why it works. Is this short-sighted of me? In one sense I think no; for example you don’t need to know what is happening in the engine of a car in order to drive it. But in another sense yes – there will be some students who want to know why this magic formula works, and I’m short-changing them if I don’t develop that curiosity. Though explaining it (which is mathematically nasty) risks alienating half the class who just want to be able to make it work. Can’t win.
P.S. The ultimate in physics magic formulas has got to be Schrodingers equation of quantum mechanics. Although there is a lot of sense in why it looks the way it does, it is not built on really solid physics ground. It just happens to work. And work really, really well.