nLab Liouville-Poincaré 1-form

Definition

For $X$ a smooth manifold and $T^* X \to X$ its cotangent bundle, there is a unique differential 1-form on $T^* X$ itself,

$$\theta \in \Omega^1(T^* X)$$

with the property that under the isomorphism

$$j \;\colon\; \Gamma(T^* X) \stackrel{\simeq}{\to} \Omega^1(X)$$

between differential 1-forms and smooth sections of the cotangent bundle we have for every smooth section $\sigma \in \Gamma(T^* X)$ the identification

$$\sigma^* \theta = j(\sigma)$$

between the pullback of $\theta$ along $\sigma$ and the 1-form corresponding to $\sigma$ under $j$.

This unique differential 1-form $\theta \in \Omega^1(T^* X)$ is called the Liouville form or Poincaré 1-form or canonical form or tautological form on the cotangent bundle.

Its de Rham differential $\omega \coloneqq d \theta$ is a symplectic form. Hence every cotangent bundle is canonically a symplectic manifold.

On a coordinate chart $\mathbb{R}^n$ of $X$ with canonical coordinate functions denoted $(x^i)$, the cotangent bundle over the chart is $T^\ast \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n$ with canonical coordinates $((x^i), (p_j))$. In these coordinates the canonical 1-form is (using Einstein summation convention)

$$\theta = p_i d x^i$$

and hence the symplectic form is

$$\omega = d p_i \wedge d x^i \,.$$

Related concepts

• Maurer-Cartan form

• Euler vector field