What’s the deal with tautological 1-forms?

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 \theta) 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 \mathcal{M} and \mathcal{N} be smooth manifolds, and \Phi : \mathcal{M} \longrightarrow \mathcal{N} be a smooth map, not necessarily invertible. Furthermore, let f : \mathcal{N} \longrightarrow \mathbb{R} be a smooth function, let X : \mathcal{M} \longrightarrow \mathcal{TM} be a vector field on \mathcal{M}, and let \eta : \mathcal{N} \longrightarrow \mathcal{T^*N} be a covector field on \mathcal{N}.

We can use \Phi to ‘pullback’ functions on \mathcal{N} into functions on \mathcal{M}, like so:

\Phi f : \mathcal{M} \longrightarrow \mathbb{R}, \:\:\:\:\:\:\:\: (\Phi f)(x) = f(\Phi(x)),

and taking advantage of that, we now have a way to ‘pushforward’ vector fields on \mathcal{M} into vector fields on \mathcal{N}:

\Phi_* X : \mathcal{N} \longrightarrow \mathcal{TN}, \:\:\:\:\:\:\:\: \left.(\Phi_* X)(f)\right|_{\Phi(p)} = \left.X(\Phi f)\right|_{p}

which then also gives a way to ‘pullback’ covector fields on \mathcal{N} into covector fields on \mathcal{M}:

\Phi^*\eta : \mathcal{M} \longrightarrow \mathcal{T^*M}, \:\:\:\:\:\:\:\: \left.(\Phi^*\eta)(X)\right|_{p} = \left.\eta(\Phi_* X)\right|_{\Phi(p)}.

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 \mathcal{TM}) and a vector field (a function \mathcal{M} \longrightarrow \mathcal{TM}), and similarly for covectors. Typically, the various kinds of product are evaluated pointwise, e.g. if f, g, h are functions then \left.fgh\right|_x = f(x)g(x)h(x).

The tautological 1-form itself

Now let \mathcal{Q} be a smooth manifold, and specialise the above discussion to the case \mathcal{N} = \mathcal{Q}, \mathcal{M} = \mathcal{T^*Q}. Let q be coordinates on \mathcal{Q}, and (p,q) be coordinates on \mathcal{T^*Q}; that is, points on \mathcal{T^*Q} are 1-forms associated to a particular point in \mathcal{Q}: (p, q) \equiv \left.p_i dq^i\right|_q. 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

\pi : \mathcal{T^*Q} \longrightarrow \mathcal{Q}, \:\:\:\:\:\:\:\: \pi(p, q) = q,

that is, simply the projection map from \mathcal{T^*Q} ‘down’ to \mathcal{Q} – it just tells us the point on \mathcal{Q} that the covector was living at.

Now for a mystical statement: the tautological 1-form is both the pullback \pi^* interpreted as a 1-form, and also has the coordinate expression \theta = p_i dq^i. 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 \mathcal{T^*Q} \longrightarrow \mathcal{Q} induces a pullback \mathcal{T^*Q} \longrightarrow \mathcal{T^*T^*Q}, but this map has exactly the domain and range of a covector field on \mathcal{T^*Q}! Of course, this requires swapping between the perspectives of p as a coordinate on \mathcal{T^*Q} and p as a 1-form in its own right.

We can use (\eta,p,q) for coordinates on \mathcal{T^*T^*Q}, equivalently writing \eta = \eta^i dp_i + \eta_i dq^i where in a slight abuse of notation we’ve written \eta^i for the first n coordinates (the dp components) and \eta_i for the second n coordinates (the dq components).

Now to investigate \pi and its induced pullbacks and pushforwards. Let f : \mathcal{Q} \longrightarrow \mathbb{R} be a function on \mathcal{Q}, and let X : \mathcal{T^*Q} \longrightarrow \mathcal{TT^*Q} be a vector field on \mathcal{T^*Q}, which in coordinates (using the same abuse of notation as before) we will write X = X_i \partial_{p_i} + X^i \partial_{q^i}.

Recalling \pi(p,q) = q, the pushforward of X under \pi is then

\left.(\pi_* X)(f)\right|_{\pi(p,q)} = \left.X(f(\pi(p,q)))\right|_{(p,q)} \\ = \underbrace{\left.X_i \partial_{p_i}(f(q))\right|_{(p,q)}}_{=0} + \left.X^i \partial_{q_i}(f(q))\right|_{(p,q)},

so the coordinates of our pushed-forward vector field on \mathcal{TQ} are (\pi_* X)^i = X^i, i.e. just the q components of X.

Now we can look at how \pi^* acts on covector fields \eta : \mathcal{Q} \longrightarrow \mathcal{T^*Q} (in coordinates \eta = \eta_i dq^i):

\left.(\pi^* \eta)(X)\right|_{(p,q)} = \left.\eta(\pi_* X)\right|_{\pi(p,q)} \\ = \left.\eta_i dq^i\right|_q \left(\left.X^j \partial_{q^j}\right|_{(p,q)}\right) = \left.\eta_i\right|_q\left.X^i\right|_{(p,q)}.

This means that the action of \pi_* is basically to place \eta straight into \mathcal{T^*T^*Q} unchanged, with all dp components set to zero:

\pi^* : \mathcal{T^*Q} \longrightarrow \mathcal{T^*T^*Q}, \\ \pi^* : (p,q) \longmapsto (p, p, q), \\ \left.\pi^*\eta\right|_{(p,q)} = \left.\eta_i\right|_q \left.dq^i\right|_{(p,q)}.

And this is the source of the coordinate expression for \pi^* that I quoted above – it means that if we take \pi^* to be a covector field on \mathcal{T^*Q}, denoted \theta, then

\left.\theta\right|_{(p,q)} = \left.p_i\right|_q \left.dq^i\right|_{(p,q)},

or \theta = p_i dq^i for short. If you like you can think of the action of \pi^* as stripping off the dq components that belong to \mathcal{T^*Q} and replacing them with dq components that belong to \mathcal{T^*T^*Q}.

How does it ‘cancel’ pullbacks?

Now look at a general covector field \alpha : \mathcal{Q} \longrightarrow \mathcal{T^*Q}. Treating \alpha as a map (a similar trick to above) means it induces a pushforward \alpha_* : \mathcal{TQ} \longrightarrow \mathcal{TT^*Q} and a pullback \alpha^* : \mathcal{T^*T^*Q} \longrightarrow \mathcal{T^*Q}.

Let F : \mathcal{T^*Q} \longrightarrow \mathbb{R} be a function, and Y : \mathcal{Q} \longrightarrow \mathcal{TQ} a vector field with coordinate expression Y = Y^i \partial_{q^i}, which is pushed-forward like so:

(\alpha_* Y)(F) = Y(\alpha F) = Y^i \partial_{q^i} (F(\alpha, q)) \\ = \underbrace{Y^i \frac{\partial F}{\partial p_j} \left. \frac{\partial p_j}{\partial q^i} \right|_{p = \alpha}}_{=0} + Y^i \frac{\partial F}{\partial q_j} \left. \frac{\partial q_j}{\partial q^i} \right|_{p = \alpha} = \left. Y^i \right|_q \left. \partial_{q^i} F(p,q) \right|_{(\alpha, q)}.

We can use this to find how \alpha^* acts on covector fields \beta : \mathcal{T^*Q} \longrightarrow \mathcal{T^*T^*Q}:

\left. (\alpha^* \beta)(Y) \right|_q = \left. \beta(\alpha_* Y) \right|_{(\alpha,q)} = \left. \beta_i \right|_{(\alpha,q)} \left. Y^i \right|_q.

But what if we specialise to \beta = \theta = p_i dq^i, the interesting 1-form we were looking at above? We get

\left. \alpha^* \theta \right|_q = \left. p_i dq^i \right|_{(\alpha,q)} = \left. \alpha_i dq^i \right|_q = \left. \alpha \right|_q,

which is exactly the 1-form \alpha that we started off with! This is the reason that \theta 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 \mathcal{M} along with an 2-form called \omega that satisfies various properties; notably that d\omega = 0, so that at least locally we can find an \alpha such that d\alpha = \omega. Abstractly, we want to find paths \Gamma : \mathbb{R} \longrightarrow \mathcal{M} such that the action integral I is minimised:

I(\Gamma) \equiv \int_\Gamma \alpha.

Now, there are various ways to come up with symplectic manifolds, but the relevant one for physicists is ‘phase space’, i.e. \mathcal{M} = \mathcal{T^*Q}, the cotangent bundle of some manifold \mathcal{Q} (where we think of \mathcal{Q} 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 \omega is to take \omega \equiv -d\theta (the minus sign being a mere convention) where \theta = p_i dq^i. 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 S that takes curves \gamma : \mathbb{R} \longrightarrow \mathcal{Q} and gives a real number, and they would then find the curve \gamma that minimises that number (S is called a functional because it also depends on the first derivatives of \gamma). By parametrising \gamma 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. \gamma : [0,1] \longrightarrow \mathcal{Q}.

Let \gamma_q be a class of curves in \mathcal{Q} parametrised by arc length with fixed starting point \gamma_q(0) = q_0 and ending point \gamma_q(1) = q. Now define W(q) \equiv S[\gamma_q] + W_0 for some constant W_0 \equiv W(q_0), where \gamma_q is defined to be the curve with endpoints (q_0,q) that minimises S[\gamma_q]. This function W : \mathcal{Q} \longrightarrow \mathbb{R} is called Hamilton’s principal function, and note that it depends only on positions, and not the momenta! We now calculate

S[\gamma_q] = W(q) - W(q_0) \\ \:\:\:\: = \int_{\partial \gamma_q} W \\ \ \:\:\:\: = \int_{\gamma_q} dW \:\:\:\:\:\:\:\: (1) \\ \ \:\:\:\: = \int_{\gamma_q} (dW)^* \theta \:\:\:\:\:\:\:\: (2) \\ \ \:\:\:\: = \int_{dW(\gamma_q)} \theta \:\:\:\:\:\:\:\: (3) \\ \ \:\:\:\: = \int_\Gamma \theta.

The numbering refers to the following results:

  1. Generalised Stokes’ theorem. Here the ‘boundary’ of \gamma_q is just its two endpoints.
  2. The ‘cancelling’ property described above: \eta^*(\theta) = \eta
  3. A standard property of integrals of pullbacks: \int_U \Phi^*(\eta) = \int_{\Phi(U)} \eta

So we see that the process of minimising S is just a special case of the general theory of minimisation problems on symplectic manifolds. To lift the path \gamma from \mathcal{Q} to the symplectic phase space \mathcal{T^*Q}, we used the 1-form dW as a pullback, similarly to the trick we pulled above with \pi^*. That means that the momenta along \gamma are

p_i = \frac{\partial W}{\partial q^i}.

Ignoring any details of the expression for the action S, how do we derive a more familiar set of differential equations that determine \gamma? We can pick out one of the coordinates, say q^0, and call it time t, and similarly one of the momenta, say p_0, and call it energy -H (again, minus sign by convention), so that

H = -\frac{\partial W}{\partial t}.

Our aim is to eliminate W in favour of the other coordinates (we will still write q^i and p_i for the other (n-1) coordinates).

Recall that d^2W = 0 by the definition of the exterior derivative, so we can immediately write down the first of Hamilton’s equations:

\frac{\partial}{\partial t}\left(\frac{\partial W}{\partial q^i}\right) = \frac{\partial}{\partial q^i}\left(\frac{\partial W}{\partial t}\right) \\ \frac{\partial p_i}{\partial t} = -\frac{\partial H}{\partial q^i}.

And since subtracting a total differential from dW retains the d^2 = 0 property, we can define a quantity that gives us the second of Hamilton’s equations:

dA \equiv dW - d(p_iq^i) \\ \frac{\partial}{\partial t}\left(\frac{\partial A}{\partial p_i}\right) = \frac{\partial}{\partial p_i}\left(\frac{\partial A}{\partial t}\right) \\ \frac{\partial q_i}{\partial t} = \frac{\partial H}{\partial p_i}.

Note that we now have explicitly

\theta = p_i dq^i - Hdt \\ \omega = dq^i \wedge dp_i - dt \wedge dH

for our symplectic structure, and we ended up with the familiar Hamilton’s equations

\frac{\partial p_i}{\partial t} = -\frac{\partial H}{\partial q^i} \\ \frac{\partial q_i}{\partial t} = \frac{\partial H}{\partial p_i}.

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:

S[\gamma] = \int_\gamma \left.g(X,X)\right|_{\gamma(s)}ds

where s is arc length, g the metric, and X the 4-vector tangent to \gamma.

See also

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 W (called the ‘Hamilton-Jacobi equation’) which depends explicitly on the time coordinate and the Hamiltonian H. The associated time-agnostic (‘non-autonomous’) method is fairly difficult, and this article discusses all the details.

About ejlflop

Intrepid explorer of music, mathematics, computer programming. physics (an unordered list). Enthusiastic semi-lay-person.
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