One way to think about a system like a flowing fluid is as a continuum. Inside every small box placed at some point in space there is some stuff, and the equations of motion for that stuff are Newton's Laws $F=ma$. In this view there is a 3D volume filled with stuff, and the properties of that stuff are described by some functions of space, for example density: $\rho(x,y,z)$. In modern Functional Analysis theory such a beast is actually a point in an infinite dimensional vector space (basically every function is a weighted infinite sum of  "basis functions" like a Fourier Series).

An alternative way to think about that same system is as a massive number of atoms. In this view the configuration of the system is something like $Q(t) = (\{q_{11}, q_{12}, q_{13}\} \ldots , \{q_{n1},q_{n2},q_{n3}\})(t)$ With the position of each atom described by 3 coordinates for a total of 3N coordinates (or 2N for a 2D problem for example). This is the perspective I'm going to take for the moment. We're going to need to figure out something about the geometry related to these 3N dimensional vectors $Q(t)$.

Suppose you're pretending that a fluid is flowing around on the surface of a sphere. Now there are only two coordinates needed to describe the position of the particle, but still they are flowing in 3D. Because it's a sphere, if we want to use 2 coordinates, we need more than one coordinate system to describe the whole sphere. Think of a coordinate system as a paper map of the earth where you measure x and y on the piece of paper. We can't put the whole earth on one piece of paper with unique positions for every point on the earth (what happens at the poles?), but we can do it on two pieces of paper. In Differential Geometry a region of space with a coordinate system for that region is called a "Chart" and a set of charts that covers the whole space is called an "Atlas".

Locally, in the region around any point, the sphere looks like a plane, globally, it's still a sphere. To describe the position of an atom on the surface of the sphere we'll need to keep track of which coordinate system each particle's coordinates refer to (which chart it's in).

Instead of thinking of $Q(t)$ as N points in 3 dimensional space, we can consider $Q(t)$ as one single point in a 3N dimensional space. The Lagrangian perspective on Classical Mechanics takes this viewpoint. In fact, since the Lagrangian depends on position and velocity, the Lagrangian depends on a single point in the tangent bundle of the 3N dimensional space.

Ok, wait, what the heck is a Tangent Bundle? Well, if you have some space like $R^3$ the euclidean volume space, then you can imagine at every point in this space you could put a vector that told you where a particle at that point was headed. The vector could go in any of the 3 directions. Now you've gone from a space that can have 3 coordinates, to a space with 6 coordinates (3 for position, and 3 for direction). This is the Tangent Bundle. In particular, you can only add vectors that are at the same point since the Tangent Space at a point is separate from the Tangent Space at other points.

The type of geometry we're talking about here is Differential Geometry and it's essential to modern physics especially General Relativity but even in Classical Mechanics Differential Geometry is important. (For a physicists perspective try the book by Bernard Schutz, or take a look at Jack Wisdom's MIT AI lab note, or the book by Sussman and Wisdom)

So we've described our entire fluid system theoretically as a single point in 3N dimensional space, and we have a function $L(Q,Q_t)$ which is the Lagrangian on the tangent bundle, now all we need to know is how that point Q moves through the tangent bundle manifold, because that will tell us the position and velocity of every particle for all of time... just what we needed.

To describe motion on a manifold we need the concept of a flow. Imagine we have a space like the surface of the sphere, and at every point we choose one of the vectors in the tangent space at that point (a velocity vector). Now we start a particle at some initial point $Q_0$ and we let it move according to the velocity vector field. At any time t the particle will have a position, and because of the position, it will have a velocity, and there will be a unique curve associated with the initial point (thanks to the existence/uniqueness theorems in ODE theory). We call this curve a flow on the manifold. The operator which takes an initial point and computes a one parameter curve from the vector field is called the exponential of the vector field. So if $x_0$ is the initial point then $\exp(\epsilon v)[x_0]$ is the curve that the vector field $v$ defines through the point $x_0$ with the parameter $\epsilon$ controlling how far along the curve we are.

A vector field defines a group action on the points of the manifold. If we're at a point x and we flow along getting to a point y, these points are both in the manifold, also for every flow field from x to y there is some flow that would return us from y to x. Since the set of points on the manifold is closed under the flow, and there is an inverse these are the main things we need to get a group.

Ok, that sets up the terminology. Next I'd like to figure out what the heck do we get from this whole mess when we want to describe how things change dynamically.

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1. November 24, 2009

A comment here. If you try to read Math books about Differential Geometry, they will be extremely abstract. Fiber bundles will be described in terms of Mobius strips and things like that. This is because abstract mathematicians generally have no idea how useful this stuff is for making models in physics.

A Tangent Bundle is a kind of fiber bundle. Think of a fiber bundle as adding an abstract dimension at each location in space such as Temperature, or Pressure, or Density. These attributes are attributes of the particular point in space. They're not tangent directions (ie. they don't tell you how to flow from one point to another) but they do tell you something useful about space at the point. The Tangent Bundle is THE fiber bundle whose fibers are the directions in which we can travel in the manifold (vectors).