Tautological 1-form

See also symplectic form#Discussion.
Aliases: Poincaré 1-form, Liouville 1-form, canonical 1-form, symplectic potential

Physical Intuition: The "Action" of a Nudge

The tautological 1-form, $\lambda$, provides the fundamental link between the geometry of phase space and the physical concept of action.
Let's use the simplest physical example: a 1D mass on a spring.

• The configuration space is $Q=\mathbb{R}$ (position $q$).
• The cotangent bundle (phase space) is $T^{\ast }Q$, with coordinates $(q,p)$, where $p$ is the momentum.
  In this simple case, the tautological 1-form is written:

This expression tells us:

"Given the system's current momentum ($p$), how much action would it accumulate under an infinitesimal (virtual) displacement ($dq$)?"

And recall that we have accepted the principle of max/min action, so we will appreciate curves $\gamma$ such that

is maximum/minimum. See action#What is the action, intuitively?.
It is essential for Hamiltonian systems in contact geometry.
This is similar but different from the concept of virtual work, which would pair force with displacement ($\delta W=Fdq$) to give units of energy. The tautological form pairs momentum with displacement to give units of action (Energy $\times$ Time).
Notice the structure:

• Work = force × displacement
• Action = momentum × displacement

Formal Definition

The tautological 1-form $\lambda$ is a canonical 1-form (a section of $T^{\ast }(T^{\ast }Q)$) on any cotangent bundle $T^{\ast }Q$.

In Local Coordinates

We are looking for a map $\lambda \in {\mathrm{\Omega }}^1(T^{\ast }Q)$. At any point $(q,p)\in T^{\ast }Q$, we want to define ${\lambda }_{(q,p)}$ which maps tangent vectors of $T^{\ast }Q$ to $\mathbb{R}$:

A tangent vector $V\in T_{(q,p)}(T^{\ast }Q)$ describes an infinitesimal displacement in phase space. In local coordinates, this vector has a "position part" $v$ and a "momentum part" $\eta$. We can write $V=(q,p,v,\eta )$.
The definition of the 1-form is to ignore the change in momentum ($\eta$) and pair the momentum ($p$) with the change in position ($v$):

This precisely matches our physical intuition $\lambda =pdq$.

Coordinate-Free Definition

We can define $\lambda$ without resorting to coordinates.

1. Consider the bundle projection $\pi :T^{\ast }Q\to Q$, which just forgets the momentum and tells you what point $q$ you are at.
2. Consider its differential (or pushforward) $d\pi :T(T^{\ast }Q)\to TQ$. This map takes a tangent vector $V$ in phase space and returns just its "position part" $v$.
3. A point in the cotangent bundle, let's call it $\alpha \in T^{\ast }Q$, is a 1-form at the point $q=\pi (\alpha )$. (This is the same as our $p$ from before).

We can now define the tautological 1-form $\lambda$ at the point $\alpha$ acting on the vector $V$ as:

This definition perfectly captures the idea:

• Take the vector $V$ (living in $T_{\alpha }(T^{\ast }Q)$).
• Push it forward with $d\pi$ to get its "position part" $v=d\pi (V)$ (living in $TQ$).
• Evaluate the 1-form $\alpha$ (which is the point you're at in $T^{\ast }Q$) on that vector $v$.
  This is identical to the local coordinate definition ${\lambda }_{(q,p)}(V)=p(v)$.

Significance

The tautological 1-form is "tautological" because it's the most natural 1-form you can build on a cotangent bundle. Its true power is that its exterior derivative

(or $\omega =d\lambda$ by convention) defines the canonical symplectic form on the phase space. This 2-form $\omega$ is the geometric object that underpins all of Hamiltonian mechanics.