# nLab Liouville-Poincaré 1-form

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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.

The 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

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

Last revised on November 23, 2017 at 06:28:57. See the history of this page for a list of all contributions to it.