nLab Liouville-Poincaré 1-form



Differential geometry

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



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

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Cartan geometry (super, higher)



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 11:28:57. See the history of this page for a list of all contributions to it.