We need to define the Jet space on a fiber bundle. This is pretty simple. This is just the space of pairs (where M is as in the last post) where we identify and if and all order derivatives agree as well. I could probably do more than a whole post just on this space, and that is beside the point, so I’ll just leave you in confusion about why this space…
Let’s be more careful this time around. Let’s separate the concepts of Lagrangian, Lagrangian density, and action. The Lagrangian is a function on the jet space that needs to somehow be related to the base space (mostly going to be space-time) so that if we integrate we get an action. Since this space is a manifold, we need the Lagrangian to actually be a differential form. This is not the case as the Lagrangian is only a function on the jet space. Luckily, we can make a form by multiplying the Lagrangian by a form to make the “Lagrangian density.” We try to do this in the most natural way. So if our manifold is nice (i.e. orientable), then we just multiply by the volume form. Since this is often the case, people don’t worry about whether they are talking about densities or not since they can be equated.
Now we can talk about the action. This is where lower level physics differs greatly from the mathematical formulation. So we have a Lagrangian and we formulate the action . We have a problem, we are trying to do something globally that most likely won’t converge (try this if M isn’t compact). Let’s look at this locally on compact sets then. If we have two fields and that differ on a compact set K, then the difference of the two actions over that set is known as the variation. When the variation is zero on a compact set, the action is said to be stationary. This is “the principal of least action,” and if you solve that differential equation, you get the equations of motion in classical mechanics.
So everything is in place. We just need to recap and look at an example to pin this down more solidly. Our manifold is a Lorentzian manifold and the fiber space is the Hermitian scalar field, i.e. . At this point we have the setup for many field theories and the difference will be the Lagrangian. For instance, we could setup Maxwell fields on this, or even quantum fields.
Now the most basic that we could go to really see what is going on and be as explicit as possible would be a point particle in 1-d space. Note that the Lagrangian is a function, the Lagrangian density is a differential form (not necessarily the same with just the volume form tacked on, but in our cases it will be), and the action is a functional. In this case the Lagrangian (nonrelativistically) is . Hmm…I’m thinking about actually finding the variation, setting it equal to zero and solving to show you get the typical equations of motion, but since I do at some point want to get to QFT, I won’t.
I guess as a last thing, I should note that, even though I didn’t talk about it, a very common fiber space is in order to get proper gauge symmetries. This could come up later.