nLab
Liouville-Poincaré 1-form

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 XX a smooth manifold and T *XXT^* X \to X its cotangent bundle, there is a unique differential 1-form on T *XT^* X itself,

θΩ 1(T *X) \theta \in \Omega^1(T^* X)

with the property that under the isomorphism

j:Γ(T *X)Ω 1(X) 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 σΓ(T *X)\sigma \in \Gamma(T^* X) the identification

σ *θ=j(σ) \sigma^* \theta = j(\sigma)

between the pullback of θ\theta along σ\sigma and the 1-form corresponding to σ\sigma under jj.

This unique differential 1-form θΩ 1(T *X)\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 ωdθ\omega \coloneqq d \theta is a symplectic form. Hence every cotangent bundle is canonically a symplectic manifold.

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

θ=p idx i \theta = p_i d x^i

and hence the symplectic form is

ω=dp idq i. \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.