Epistemic status: All pretty standard derivations, except the last section on mechanics which is a bit hand-wavy.
When formulating mechanics on cotangent bundles, one comes across an object called the ‘tautological 1-form’ (often denoted ) which is supposedly key to the whole process. Here I will attempt to describe what this 1-form is, why it is useful, and the role it plays in the fundamentals of classical mechanics.
Pullbacks and Pushforwards
First a word about smooth maps between manifolds, and the operations derived from them. Let and be smooth manifolds, and be a smooth map, not necessarily invertible. Furthermore, let be a smooth function, let be a vector field on , and let be a covector field on .
We can use to ‘pullback’ functions on into functions on , like so:
and taking advantage of that, we now have a way to ‘pushforward’ vector fields on into vector fields on :
which then also gives a way to ‘pullback’ covector fields on into covector fields on :
I have written a bunch of vertical “evaluate-here” bars for clarification. It is common to be rather casual about the difference between a vector (lives in ) and a vector field (a function ), and similarly for covectors. Typically, the various kinds of product are evaluated pointwise, e.g. if are functions then .
The tautological 1-form itself
Now let be a smooth manifold, and specialise the above discussion to the case , . Let be coordinates on , and be coordinates on ; that is, points on are 1-forms associated to a particular point in : . Having these two equivalent ways to look at points on a cotangent bundle is an important point which we shall return to later.
For the current purpose we will study the map
that is, simply the projection map from ‘down’ to – it just tells us the point on that the covector was living at.
Now for a mystical statement: the tautological 1-form is both the pullback interpreted as a 1-form, and also has the coordinate expression . How on earth can both these things be true, and besides, how can one ‘interpret a pullback as a 1-form’?!
That last claim is actually not too bad: a map induces a pullback , but this map has exactly the domain and range of a covector field on ! Of course, this requires swapping between the perspectives of as a coordinate on and as a 1-form in its own right.
We can use for coordinates on , equivalently writing where in a slight abuse of notation we’ve written for the first coordinates (the components) and for the second coordinates (the components).
Now to investigate and its induced pullbacks and pushforwards. Let be a function on , and let be a vector field on , which in coordinates (using the same abuse of notation as before) we will write .
Recalling , the pushforward of under is then
so the coordinates of our pushed-forward vector field on are , i.e. just the components of .
Now we can look at how acts on covector fields (in coordinates ):
This means that the action of is basically to place straight into unchanged, with all components set to zero:
And this is the source of the coordinate expression for that I quoted above – it means that if we take to be a covector field on , denoted , then
or for short. If you like you can think of the action of as stripping off the components that belong to and replacing them with components that belong to .
How does it ‘cancel’ pullbacks?
Now look at a general covector field . Treating as a map (a similar trick to above) means it induces a pushforward and a pullback .
Let be a function, and a vector field with coordinate expression , which is pushed-forward like so:
We can use this to find how acts on covector fields :
But what if we specialise to , the interesting 1-form we were looking at above? We get
which is exactly the 1-form that we started off with! This is the reason that is said to ‘cancel’ a pullback, as it gives us back the 1-form that we were using to create the pullback in the first place.
The basics of mechanics
How does this all link into physics? For mechanics you need a symplectic manifold along with an 2-form called that satisfies various properties; notably that , so that at least locally we can find an such that . Abstractly, we want to find paths such that the action integral is minimised:
Now, there are various ways to come up with symplectic manifolds, but the relevant one for physicists is ‘phase space’, i.e. , the cotangent bundle of some manifold (where we think of as being the ‘real’ physical space that we see around us, perhaps 4D space-time or similar). And it turns out that the logical choice of is to take (the minus sign being a mere convention) where . Thanks to the discussion above we now know exactly what this object is (spoiler: it’s the tautological 1-form!).
Traditionally a physicist would have something called an action functional that takes curves and gives a real number, and they would then find the curve that minimises that number ( is called a functional because it also depends on the first derivatives of ). By parametrising by the time coordinate the normal Euler-Lagrange equations are derived. However, we wish to stay agnostic about which coordinate represents time! So we will keep our paths parametrised by arc length, i.e. .
Let be a class of curves in parametrised by arc length with fixed starting point and ending point . Now define for some constant , where is defined to be the curve with endpoints that minimises . This function is called Hamilton’s principal function, and note that it depends only on positions, and not the momenta! We now calculate
The numbering refers to the following results:
- Generalised Stokes’ theorem. Here the ‘boundary’ of is just its two endpoints.
- The ‘cancelling’ property described above:
- A standard property of integrals of pullbacks:
So we see that the process of minimising is just a special case of the general theory of minimisation problems on symplectic manifolds. To lift the path from to the symplectic phase space , we used the 1-form as a pullback, similarly to the trick we pulled above with . That means that the momenta along are
Ignoring any details of the expression for the action , how do we derive a more familiar set of differential equations that determine ? We can pick out one of the coordinates, say , and call it time , and similarly one of the momenta, say , and call it energy (again, minus sign by convention), so that
Our aim is to eliminate in favour of the other coordinates (we will still write and for the other (n-1) coordinates).
Recall that by the definition of the exterior derivative, so we can immediately write down the first of Hamilton’s equations:
And since subtracting a total differential from retains the property, we can define a quantity that gives us the second of Hamilton’s equations:
Note that we now have explicitly
for our symplectic structure, and we ended up with the familiar Hamilton’s equations
And so, as if by magic, we’ve recovered the traditional formalism of Hamiltonian mechanics as a special case of minimisation procedures on symplectic manifolds.
Of course, we didn’t have to use the 0th coordinate to represent time/energy. Really, time and position are distinguished from each other by the form of the Lorentzian metric, which has not yet entered into our method. It’s true that non-relativistic mechanics will inevitably privilege a time variable; but, the action for a free relativistic point particle is nicely Lorentz-invariant:
where is arc length, the metric, and the 4-vector tangent to .
This blog post was inspired by
- John Baez’s two posts on parallels between thermodynamics and mechanics
- The fact that the Wikipedia page on the tautological 1-form is so abstruse
You may also be interested in
- A 1-page summary of ‘Abstract Hamiltonian mehanics’, which describes an approach which is agnostic about time coordinates (it does not discuss minimisation procedures though)
- An article on how to do geometric Hamilton-Jacobi mechanics properly. It’s possible to derive a single nonlinear differential equation for (called the ‘Hamilton-Jacobi equation’) which depends explicitly on the time coordinate and the Hamiltonian . The associated time-agnostic (‘non-autonomous’) method is fairly difficult, and this article discusses all the details.