Knowlpedia

Definition

Liouville vector field

The unique vector field dual to a chosen primitive of an exact symplectic form.

Definition

Let (M,ω)(M,\omega) be an with a specified primitive λ\lambda, so ω=dλ\omega=d\lambda. The Liouville vector field associated with λ\lambda is the unique smooth ZZ satisfying

ιZω=λ.\iota_Z\omega=\lambda.

Nondegeneracy of ω\omega gives existence and uniqueness. Cartan's formula then yields

LZω=d(ιZω)+ιZdω=dλ=ω.\mathcal L_Z\omega=d(\iota_Z\omega)+\iota_Zd\omega=d\lambda=\omega.

Thus the local flow of ZZ dilates the symplectic form: wherever it is defined, φt∗ω=etω\varphi_t^*\omega=e^t\omega. The field depends on the chosen primitive, not only on ω\omega.

Equivalent characterization

A vector field ZZ on a is Liouville precisely when

LZω=ω.\mathcal L_Z\omega=\omega.

Indeed, this equation implies d(ιZω)=ωd(\iota_Z\omega)=\omega, so λ=ιZω\lambda=\iota_Z\omega is a primitive; the converse is the calculation in the core. Accordingly, the existence of a global Liouville vector field forces the symplectic form to be exact.

Dependence on the primitive

If λ′=λ+α\lambda'=\lambda+\alpha is another primitive, then α\alpha is closed and the corresponding field is Z′=Z+XZ'=Z+X, where ιXω=α\iota_X\omega=\alpha. When α=dH\alpha=dH, the correction XX is the XHX_H in the sign convention ιXHω=dH\iota_{X_H}\omega=dH. Hence changing a primitive by an exact form changes the Liouville field by a Hamiltonian field.

Canonical example

On T∗QT^*Q with coordinates (qi,pi)(q_i,p_i), take

λ=∑ipi dqi,ω=dλ=∑idpi∧dqi.\lambda=\sum_i p_i\,dq_i,\qquad \omega=d\lambda=\sum_i dp_i\wedge dq_i.

The corresponding Liouville field is

Z=∑ipi∂∂pi,Z=\sum_i p_i\frac{\partial}{\partial p_i},

whose flow multiplies cotangent vectors by ete^t. With the alternative convention ω=−dλ=∑idqi∧dpi\omega=-d\lambda=\sum_i dq_i\wedge dp_i, one also changes the defining contraction sign if one wants the same radial field.

Conventions and scope
References
  1. Kai Cieliebak and Yakov Eliashberg, From Stein to Weinstein and Back, Colloquium Publications 59, American Mathematical Society, 2012. AMS DOI record. Relevant: §2, Liouville forms and vector fields.
  2. Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2008. Springer DOI record. Relevant: pp. 69–79, contact forms and Liouville dynamics.