Chasing lambda

I’ve read that the cotangent bundle (phase space) to the configuration space (base manifold) is a more special space than the tangent bundle. This is rather interesting. The cotangent bundle is where we do Lagrangian mechanics while the cotangent bundle is where we do Hamiltonian mechanics. Why would one be any better than the other and what does ‘better’ mean anyways?

Let’s set it up. Our system of interest can be described by some number of parameters or ‘generalized coordinates’ and these coordinate patches form a manifold M called the configuration manifold. We’re generally interested in paths connecting points of M, those paths will be the evolution of the system from one state to another state.

Its reasonable to ask how the system is changing along one of these paths. Let p:I\to M be a path parameterized by t\in I. The coordinates on M are ‘charts’ mapping M to \mathbb{R}^n and so we can ‘export’ this problem of tangent vectors to euclidean space by differentiating the composition q\circ p:I\to \mathbb{R}^n. That’s a path in \mathbb{R}^n and its tangent vector is

\vec{v}=\frac{dp}{dt}|_{t=0}

where we’ve evaluated the derivative at t=0 to obtain the tangent vector to the point p(0)=x\in M. (I’m going to side step an extra step here where we should construct equivalences classes …). Notice that I left off the explicit notation for composition with the coordinate chart. In fact we’ll abuse notation with some severity. p(t) has coordinates and we’ll call these coordinates q_i(t) even though the labels q_i represent components of the coordinate chart and are independent of time. Like I said…with severity.

You get the idea. All the tangent vectors at a point form a vector space at that point equal in dimension to the manifold M. We denote that vector space by T_x M. The basis for that vector space using the local coordinates q^i is \partial/\partial q^i where i=1,...,n. Glue all these tangent spaces together and you get the tangent bundle TM, where each tangent space forms a fiber of that bundle. The local coordinates on M induce local coordinates on TM which are: (q^i, \dot{q}^i).

While we’re interested in paths in M which describe the evolution of our system we tend to work in spaces like the tangent bundle. In the Lagrangian is a function of this space: L:TM\to \mathbb{R}

The Lagrangian along with the equation \dot{q}^i=dq^i/dt creates a vector field on TM. These vectors are drawn from the tangent bundle of TM, or T(TM). Yeah there are going to be a lot of those T’s. While points of TM look like X^i \partial /\partial q^i,points in T(TM) look like X^i \partial /\partial q^i + Y^i \partial /\partial \dot{q}^i.

But let’s back up a second. Remember T_x M from a few paragraphs ago? The tangent space to x\in M? Well to each vector space there is a dual space. This isn’t a result of differential geometry but of linear algebra. We’ll call this space T^*_x M which is the cotangent space. Its the space of linear functionals on T_xM. Meaning that that if \alpha \in T^*_x M then \alpha: T_xM\to \mathbb{R}

What does the basis of this look like? Well we can construct this quick enough because we’ll need this idea later. Let’s define the differential of a function f that has x in its domain. Then we define the action of df on a vector v in T_xM as df(\vec{v})=\vec{f} which is clearly linear (its linearity comes from v’s linearity) and it maps vectors to numbers. In fact we can express a basis for T_x^*M in terms of the differentials of the local coordinates on M. This gives us a basis of dq^i, i=1,...,n. So a typical element/point of T_x^*M looks like \alpha=a_idq^i.

Once again we can glue all these spaces together to create a larger space, the cotangent space T*M. Its on this space that we’ll end up doing Hamiltonian dynamics (and having Hamiltonian dynamics as well).

A quick summary. We start with M and end up with TM and T*M. Each of these is a manifold (of twice the dimension of M). Also each of these manifolds has their own tangent and cotangent bundles T(TM), T*(TM) and T(T*M) and T*(T*M). Its these spaces (the six of them) that we tend to do our physics on, in particular the cotangent bundle and its bundles is where we’ll be spending some time. Let’s stop here however for a break.

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About because0fbeauty

Fascinated by the way mathematics and physics interact, captivated by visual and tactile mathematics and hoping to become a better expositor of these things is why I blog...occasionally...when I remember.
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