Graded Algebras and Classical Mechanics

Certain algebras possess an important classifying property known as a grading.  This essentially amounts to being able to express these algebras as a special kind of direct sum that respects multiplication.

Definition[Graded Algebra]:  An algebra A over a field \kappa is said to be graded if, as a vector space over \kappa, A can be written as the direct sum of a family of subspaces (A_{i})_{i\in \mathbb{N}} – i.e.

\displaystyle A = \bigoplus_{i\in \mathbb{N}}A_{i}

and such that multiplication behaves according to

A_{i}\cdot A_{j} \subseteq A_{i+j}, for all i,j\in \mathbb{N}

In classical mechanics, one deals with smooth manifolds. To understand what these are we need some definitions.

Definition[Chart]:  Let M be a manifold of dimension n.  Then a chart is a pair (U, \psi), where U\subset M is open and \psi: U\to V\subset \mathbb{R}^{n} is a homeomorphism to some open V.

The collection of all charts such that each x\in M belongs to some chart is called an atlas.  An atlas characterizes a manifold.  More formally,

Definition[Atlas]:  An atlas on a topological space T is a collection of charts \{(U_{\alpha},\varphi_{\alpha})\}_{\alpha\in I} where the U_{\alpha} are open sets that cover T, and for each index \alpha

\varphi_{\alpha}: U_{\alpha}\to \mathbb{R}^n

is a homeomorphism of U_{\alpha} onto an open subset of n-dimensional Euclidean space.

Definition[Smooth Atlas]: An atlas \mathcal{A} is called smooth if for all \varphi_{i}, \varphi_{j} we have that

\varphi_{j}\circ \varphi_{i}^{-1} : \varphi_{i}(U_{i}\cap U_{j})\to \varphi_{j}(U_{i}\cap U_{j})

is a diffeomorphism.


Definition[Diffeomorphism]: Given two manifolds M and N, a differentiable map f: M\to N is called a diffeomorphism if it is a bijection, and its inverse f^{-1}: N\to M is also differentiable.

Now, a manifold may not be generated by a unique atlas.  To account for this, we seek a preferred atlas known as a maximal atlas.  In defining such a thing, we need to know what it means for atlases to be compatible.

Definition[Compatibility]: Let \mathcal{A} and \mathcal{A}' be two smooth atlases.  Then \mathcal{A} and \mathcal{A}' are called compatible if and only if \mathcal{A}\cup \mathcal{A}' is again a smooth atlas. 

This notion of compatibility is actually an equivalence relation.  The union over an equivalence class of atlases gives us our maximal atlas.  If we denote an equivalence class by [\mathcal{A}], then we have

Definition[Maximal Atlas]:  A maximal atlas for a manifold M is the union of all smooth atlases in an equivalence class – i.e.

\mathcal{A}_{max} = \bigcup\{\mathcal{B} : \mathcal{B}\in [\mathcal{A}]\}

This maximal atlas is said to be the differentiable structure of the manifold.  We are now ready to understand the idea of a smooth manifold.

Definition[Smooth Manifold]:  A smooth manifold is a pair (M, \mathcal{A}_{max}) where M is a manifold and \mathcal{A}_{max} is a differentiable structure of M.

Smooth manifolds are important in classical mechanics because they allow us to do calculus on them.  Now, to return to the original topic, we are also interested in the graded algebra of smooth differentiable forms on a smooth manifold M with respect to the wedge product, which is denoted by

\displaystyle \mathcal{A}^{\bullet}(M) = \bigoplus_{k=0}^{n}\mathcal{A}^{k}(M)

– Differential Forms – 

For now, let’s restrict our focus to what are called differential 1-forms, or just 1-forms.  According to one source, “Informally, a differential form is what can be integrated along a path.”

[1-Forms]

Starting in \mathbb{R}^{2}, let U be some open subset of \mathbb{R}^{2} and let F, G be two real-valued functions defined on U.  Then an expression of the form

F(x,y)dx+G(x,y)dy

is called a differential 1-form on U.  A particularly important example of a differential 1-form is the total differential of a C^{1} real-valued function f defined on some open subset U.

Definition[Total Differential]:  Let f by a C^{1} real-valued function defined on an open subset U of \mathbb{R}^{2}.  Then its total differential df is defined by

\displaystyle df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy

This can also be expressed as a kind of dot product of vectors, namely \nabla f\cdot (dx,dy), where \nabla f is the gradient of f.  Note that in this case we have \displaystyle F(x,y) = \frac{\partial f}{\partial x} and \displaystyle G(x,y) = \frac{\partial f}{\partial y}.

1-forms can be defined similarly for higher dimensions.  For instance, a differential 1-form on an open subset U of \mathbb{R}^{3} is an expression of the form

F(x,y,z)dx+G(x,y,z)dy+H(x,y,z)dz

where F, G and H are real-valued functions on U.  Again, if f is a C^{1} function defined on U, then the total differential

\displaystyle df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz

is an example.

In general, a 1-form on an open subset of \mathbb{R}^n is an expression of the form

F_{1}(x_{1},x_{2},...,x_{n})dx_{1}+F_{2}(x_{1},x_{2},...,x_{n})dx_{2}+\ldots + F_{n}(x_{1},x_{2},...,x_{n})dx_{n}

or simply \displaystyle \sum_{i = 1}^{n}F_{i}(x_{1},x_{2},...,x_{n})dx_{i}.  If we let \boldsymbol{x} = (x_{1}, x_{2},..., x_{n}), then we can write a 1-form even more succinctly as

\displaystyle \varphi = \sum_{i=1}^{n}F_{i}(\boldsymbol{x})dx_{i}

So, if f is a C^{1} real-valued function defined on U, the the general total differential is

\displaystyle df = \sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}dx_{i}

Note that each dx_i is a 1-form.  These are defined by the property that for each vector \boldsymbol{v} = (v_{1},v_{2},...,v_{n})\in T_{x}\mathbb{R}^{n}

dx_{i}(\boldsymbol{v}) = v_{i}

Clearly, these form a basis for the 1-forms on \mathbb{R}^{n}.

Definition[Smooth 1-Form]:  A smooth 1-form \varphi on \mathbb{R}^{n} is a real-valued function on the set of all tangent vectors to \mathbb{R}^{n}, i.e. – 

\varphi : T\mathbb{R}^{n}\to \mathbb{R}

with the properties that

  1. \varphi is linear on the tangent space T_{x}\mathbb{R}^{n} for each x\in \mathbb{R}^{n} – i.e. if \boldsymbol{v}_{1}, \boldsymbol{v}_{2}\in T_{x}\mathbb{R}^{n} and c_{1},c_{2}\in \mathbb{R},then \varphi(c_{1}\boldsymbol{v}_{1}+c_{2}\boldsymbol{v}_{2}) = c_{1}\varphi(\boldsymbol{v}_{1})+c_{2}\varphi(\boldsymbol{v}_{2}).
  2. For any smooth vector field v = v(x), the function \varphi(v):\mathbb{R}^{n}\to \mathbb{R} is smooth.

A smooth 1-form on an n-dimensional manifold M is defined similarly.

Note:  Another source simply says that a smooth 1-form on an open subset U of \mathbb{R}^{n} is given by an expression

\displaystyle \varphi = \sum_{i=1}^{n}f_{i}dx_{i}

where f_{i}\in C^{\infty}(U).

To better understand the graded algebra of smooth differential forms on M with respect to the wedge product we need to understand that the 1-forms on \mathbb{R}^{n} are part of an algebra called the algebra of differential forms on \mathbb{R}^{n}.  Multiplication in this algebra is known as the wedge product and is denoted by the symbol “\wedge“.  It has the property of being skew-symmetric or anti-commutative:

dx_{i}\wedge dx_{j} = -dx_{j}\wedge dx_{i}

From this it follows that dx_{i}\wedge dx_{i} = -dx_{i}\wedge dx_{i}, which implies that dx_{i}\wedge dx_{i} = 0 (provided we are not working over a field with characteristic 2).  This means that the wedge product is alternating.

From here we can build up k-forms on \mathbb{R}^{n}.  If each summand of a differential form \varphi contains k dx_{i}‘s, then \varphi is called a k-form.

Note: Functions are considered to be 0-forms.

The set

\{dx_{i_{1}}\wedge \ldots \wedge dx_{i_{k}} : 1\leq i_{1} < i_{2} < \ldots < i_{k} \leq n\}

is a basis for the k-forms on \mathbb{R}^{n}.  Thus, every k-form can be expressed in the form

\displaystyle \varphi = \sum_{|I| = k}f_{I}dx_{i_{1}}\wedge \ldots \wedge dx_{i_{k}}

where I ranges over all multi-indecies I = (i_{1}, i_{2},...,i_{k}) of length k.

The Exterior Derivative

The exterior derivative is an operation that sends a k-form to a (k+1)-form.

Definition[Exterior Derivative (1-Form)]:  Let f be a 0-form (function).  Then its exterior derivative df is the 1-form

\displaystyle df = \sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}dx_{i}

Note that this corresponds to the total differential of f.

Definition[Exterior Derivative (k-form)]:  Let \varphi be a k-form.  Then its exterior derivative d\varphi is the (k+1)-form obtained from \varphi by applying $lated d$ to each of the functions involved in \varphi.

Definition[Directional Derivative]:  Let f be a differentiable function defined on a region R in \mathbb{R}^{n}.  Let \boldsymbol{v} be a vector based at the point p_{0}\in R.  Then the derivative of f along \boldsymbol{v}, denoted D_{\boldsymbol{v}}f, is defined as follows.  Let F(t) = f(p_{0}+t\boldsymbol{v}).  Then 

D_{\boldsymbol{v}}f = F'(0)

Theorem:  Let f be a differentiable function defined on a region R of \mathbb{R}^{n}.  Let \varphi be defined on tangent vectors to points of R by

\varphi(\boldsymbol{v}) = D_{\boldsymbol{v}}f

Then

\varphi = df

Proof:

Let \boldsymbol{v} be a vector based at a point p of R with coordinates (p_{1}, p_{2},..., p_{n}). Let \boldsymbol{v} have coordinates (v_{1}, v_{2}, ..., v_{n}).  Let \textbf{u}_{i} be the ith unit vector.  Then

\displaystyle df(\boldsymbol{v}) = \left(\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}dx_{i}\right)\left(\sum_{i=1}^{n}v_{i}\textbf{u}_{i}\right)

=\displaystyle \sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}f(p)v_{i}

Next, let’s evaluate \varphi(\boldsymbol{v}) = D_{\boldsymbol{v}}f = F'(0) where we recall that F(t) = f(p+t\boldsymbol{v}).  So, really, we should write \displaystyle F'(0) = \frac{\partial}{\partial t}f(0).  Now, F is actually a composition of two functions, F(t) = f(g(t)), where g(t) = p+t\boldsymbol{v} = (p_{1}+tv_{1}, p_{2}+tv_{2},...,p_{n}+tv_{n}).  If we now apply the chain rule, then we get

\displaystyle \frac{\partial}{\partial t}f(0) = \sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}f(g(0))\frac{d}{dt}x_{i}(0)

But x_{i} = p_{i}+tv_{i}, so \displaystyle \frac{d}{dt}x_{i} = v_{i}.  Therefore,

\displaystyle \frac{\partial}{\partial t}f(0) = \sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}f(p)v_{i}

 This finishes the proof.

The exterior derivative obeys both the Leibniz rule and the chain rule:

  • d(fg) = gdf+fdg         Leibniz Rule
  • d(h(f)) = h'(f)df        Chain Rule

If \phi is a p-form and \psi is  a q-form, then the Leibniz rule takes the form

d(\phi \wedge \psi) = d\phi \wedge \psi+(-1)^{p}\phi\wedge d\psi

Theorem:  For any differential form \phi,

d(d\phi) = 0

(Or more succinctly, d^2 = 0).

Proof:

First, let f be a function (i.e. 0-form).  Then

\displaystyle d(df) = d\left(\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}dx_{i}\right)

\displaystyle = \sum_{i=1}^{n}d\left(\frac{\partial f}{\partial x_{i}}\right)dx_{i}

\displaystyle = \sum_{i=1}^{n}\left(\sum_{j=1}^{n}\frac{\partial f_{x_{i}}}{\partial x_{j}}dx_{j}\right)dx_{i}

[where \displaystyle f_{x_{i}} : = \frac{\partial f}{\partial x_{i}}]

\displaystyle =\sum_{i=1}^{n}\left(\sum_{j=1}^{n}\frac{\partial^{2} f}{\partial x_{i}\partial x_{j}}dx_{j}\right)dx_{i}

\displaystyle =\sum_{i,j}\frac{\partial^{2} f}{\partial x_{i}\partial x_{j}}dx_{j}\wedge dx_{i}

\displaystyle =\sum_{i<j}\left(\frac{\partial^{2}f}{\partial x_{j}\partial x_{i}} - \frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}\right)dx_{i}\wedge dx_{j} = 0

since mixed partials commute.  Now, since dx_{i} actually means d(x_{i}), where x_{i} is the i-th coordinate function, we have that d(dx_{i}) = 0.  Let

\displaystyle \phi = \sum_{|I| = k}f_{I}dx_{i_{1}}\wedge \ldots \wedge dx_{i_{k}}

Then

\displaystyle d(d\phi) = \left(\sum_{|I|=k}df_{I}\wedge dx_{i_{1}}\wedge \ldots \wedge dx_{i_{k}}\right)

\displaystyle = \sum_{|I|=k}\Big(d(df_{I})\wedge dx_{i_{1}}\wedge \ldots \wedge dx_{i_{k}}-df_{I}\wedge d(dx_{i_{1}})\wedge \ldots \wedge dx_{i_{k}}+\ldots\Big)

[By the Leibniz Rule]

Per the reasoning above with functions, this is clearly 0.

Much of the above has been described in terms of differential forms on \mathbb{R}^{n}.  Most of it transfers fairly seamlessly to general manifolds (i.e. the concepts are defined similarly).  So, to tie this all back to the graded algebra given above, we should understand \mathcal{A}^{0}(M) to be C^{\infty}(M) (i.e. the 0-forms on M) and \mathcal{A}^{k}(M) to be the algebra of smooth differential k-forms on M. We’ll look more at graded algebras in the next post.

A Quick Intro to Manifolds

An n-dimensional manifold is intuitively described as a topological space that locally resembles \mathbb{R}^n.  In other words, if you pick a point on a manifold and “zoom in”, it starts to look more and more like a Euclidean space of the same dimension. According to Penrose a manifold is

… a space that can be thought of as ‘curved’ in various ways, but where, locally (i.e. in a small enough neighborhood of any of its points), it looks like a piece of ordinary Euclidean space.

Some examples will help to illustrate.

Example: If we start with n = 0, then \mathbb{R}^{0} = \{0\}, which is just a singleton.  This means that a 0-manifold is one in which every point has a neighborhood that looks like a single point.  Thus, a 0-manifold just is a collection of points.

Example: If n=1, then \mathbb{R}^{1} is the real line.  Thus, a 1-manifold is one in which each point has a neighborhood that looks like an open line segment.  A circle is a good example of this.  In topology, one ignores “bending” so that a small portion of the circle is treated like a small line segment.  Let’s consider this more carefully by considering the unit circle x^{2}+y^{2} = 1 and seeing how it “looks like” a line segment at each point.

LUFTOCirc

Begin with the upper half part of the circle described by y = \sqrt{1-x^{2}},~ -1<x<1.

LUFTOUpperHalfCirc

Note that the y-coordinate is always positive on this part of the circle.  Since any point on this portion of the circle can be uniquely described by its x-coordinate we can use the projection map \rho_{top}: S^{(+)}\to (-1,1) defined by

\rho_{top}(x,y) = x

to map the open upper half-circle S^{(+)} to the open interval (-1,1).

This mapping is both continuous and invertible.  Some example projections are shown in the diagram below.

LUFTOUpperHalfProj

The pair (\rho_{top}, S^{(+)}) is called a chart.  There are similar charts for the open lower half-circle, the open right half-circle and the open left half-circle.  Taken together, these charts cover the whole circle forming an atlas. 

Example:  For n=2, the space \mathbb{R}^{2} is the plane.  Thus, a 2-manifold is one in which every point has a neighborhood that looks like an open portion of the plane.  The surface of a sphere or a torus are good examples.

The formal definition of a manifold is a bit involved, so let’s start with the condensed version and unpack it from there.

Definition[Manifold]: A topological manifold is a second-countable Hausdorff  space that is locally homeomorphic to Euclidean space.

Let’s first look at what it means for a space to be “second-countable”.

Definition[Second-Countable]: A space is said to be second-countable if its topology has a countable base.

Here a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements from B.  The base B is said to generate the topology T.

That B is countable simply means that there exists an injective function f: B\to \mathbb{N}.  In other words, the elements of B can be put in one-to-one correspondence with a subset of the natural numbers.

Next, let’s consider Hausdorff spaces.

Definition[Hausdorff Space]:  A Hausdorff space H is a topological space such that for all x,y\in H with x\neq y, there exist neighborhoods U, V in the topology with x\in U, y\in V, and U\cap V = \varnothing.

Sometimes Hausdorff spaces are referred to as separated spaces.  This is illustrated below.

LUFTOHausdorff

As Penrose puts it

A Hausdorff space has the defining property that, for any two distinct points of the space, there are open sets containing each which do not intersect.

The last bit we need is to understand what it means to be locally homeomorphic to Euclidean space.

Definition[Homeomorphism]:  Let S and T be two topological spaces.  A homeomorphism h is a function h:S\to T that satisfies the following properties:

  1. h is a bijection.
  2. h is continuous.
  3. h^{-1} is continuous.

Definition[Local Homeomorphism]:  Let S and T be two topological spaces.  A function f:S\to T is called a local homeomorphism if for every point x\in S there exists an open set U containing x, such that the image f(U) is open in T, and the restriction f|_{U}: U\to f(U) is a homeomorphism.

So, to be locally homeomorphic to Euclidean space means that every point of an n-manifold has a neighborhood that is homeomorphic to an open Euclidean n-ball

B^{n} = \{(x_{1}, x_{2}, ..., x_{n}) \in \mathbb{R}^{n}: x_{1}^{2}+x_{2}^{2}+\ldots +x_{n}^{2}<1\}

In other words, if H is a second-countable Hausdorff space, then it is locally homeomorphic to Euclidean space iff there exists a function f: H\to \mathbb{R}^{n} such that for each x\in H there is a neighborhood U of x such that f|_{U}:U\to B^{n} is a homeomorphism (where f(U) = B^{n}).

References:

Differential Forms: A Complement to Vector Calculus, by Steven H. Weintraub

The Road to Reality, by Roger Penrose

Fundamentals of Topology, by Benjamin T. Sims

http://en.wikipedia.org/wiki/Second_countable

http://en.wikipedia.org/wiki/Manifold

http://en.wikipedia.org/wiki/Base_(topology)

http://en.wikipedia.org/wiki/Hausdorff_space