nLab
evolutionary vector field

Contents

Contents

Idea

In variational calculus an evolutionary vector field is a particular type of vertical vector field on the infinite-order jet bundle J Σ (E)J^\infty_\Sigma(E) of a fiber bundle EfbΣE \overset{fb}{\to} \Sigma, which may be interpreted as the generator of a transformation of the space of sections Γ Σ(E)\Gamma_\Sigma(E) of EE.

Evolutionary vector fields describe infinitesimal symmetries of Lagrangian field theories in the formulation of the variational bicomplex, and as such form an ingredient of Noether's theorem.

Definition

There are (at least) two different definitions, def. and def. below, which are equivalent (prop. below).

As a vector field on the jet bundle

Definition

(as a vector field on the jet bundle)

An evolutionary vector field is a vector field vv on the jet bundle J Σ (E)J^\infty_\Sigma(E) which is vertical with respect to the projection π :J Σ EΣ\pi_\infty: J^\infty_\Sigma E \to \Sigma such that the following equivalent conditions hold:

  1. The Lie derivative v\mathcal{L}_v satisfies

    1. vd=d v\mathcal{L}_v d = d \mathcal{L}_v,

    2. v=ι vδ+δι v\mathcal{L}_v = \iota_v \delta + \delta \iota_v,

    where dd is the horizontal derivative on differential forms on J EJ^\infty E, while δ\delta is the vertical derivative.

  2. vv preserves the contact ideal?, i.e., for which vθ\mathcal{L}_v\theta is a contact form? whenever θ\theta is a contact form?.

We denote the subspace of evolutionary vector fields as 𝔛 ev(J Σ (E))𝔛 V(J Σ (E))\mathfrak{X}_{ev}(J^\infty_\Sigma(E)) \subset \mathfrak{X}_V(J^\infty_\Sigma(E)).

Note that every evolutionary vector field is uniquely defined by its action on the pullbacks of functions on EE to J Σ (E)J^\infty_\Sigma(E). To formalize this, one may use the following notion.

Definition

Let v𝔛 ev(J E)v \in \mathfrak{X}_{ev}(J^\infty E) be an evolutionary vector field. Its characteristic is the map π ,0*v:J EVE\pi_{\infty,0*} \circ v: J^\infty E \to V E.

As a generalized vector field on the fiber bundle

Alternatively, a generalized vector field on the fiber bundle EE may be seen as a vector field on EE whose coefficients are functions on the jet bundle J EJ^\infty E. More formally, it is a particular map from J EJ^\infty E to TET E. Here we are interested only in a particular class of vertical vector fields.

Definition

An evolutionary vector field is a map w:J Σ (E)V ΣEw \colon J^\infty_\Sigma(E) \to V_\Sigma E from the jet bundle J EJ^\infty E to the vertical tangent bundle V ΣEV_\Sigma E, such that

νw=π ,0AAAAAAAJ Σ (E) w V ΣE π ,0 ν E, \nu \circ w = \pi_{\infty,0} \phantom{AAAAAAA} \array{ J^\infty_\Sigma(E) && \overset{w}{\longrightarrow} && V_\Sigma E \\ & {}_{\mathllap{\pi_{\infty,0}}}\searrow && \swarrow_{\mathllap{\nu}} \\ && E } \,,

where ν:V ΣEE\nu: V_\Sigma E \to E is the bundle map of V ΣEV_\Sigma E and π ,0:J Σ (E)E\pi_{\infty,0}: J^\infty_\Sigma(E) \to E is the target map of the jet bundle.

Every evolutionary vector field has a unique prolongation to a vector field on J Σ (E)J^\infty_\Sigma(E).

Definition

Let ww be an evolutionary vector field in the sense of def. . Its prolongation to J Σ EJ^\infty_\Sigma E is the unique vertical vector field prwpr w on J Σ EJ^\infty_\Sigma E such that

  1. ww and prwpr w agree on functions on EE

  2. prwpr w preserves the contact ideal, i.e., vθ\mathcal{L}_v\theta is a contact form? whenever θ\theta is a contact form?.

Equivalence of the definitions

Proposition

(equivalence of the two definitions)

If vv is an evolutionary vector field in the sense of def. , then its characteristic π ,0*v\pi_{\infty,0*} \circ v (def. ) is an evolutionary vector field in the sense of def. .

Conversely, if ww is an evolutionary vector field in the sense of def. , then its prolongation prwpr w (def. ) is an evolutionary vector field in the sense of def. .

Furthermore, pr(π ,0*v)=vpr (\pi_{\infty,0*} \circ v) = v and π ,0*(prw)=w\pi_{\infty,0*} \circ (pr w) = w, so that this establishes a bijection between both definitions.

References

See also

  • Ian Anderson, Charles Torre, section 2A of Classification of Local Generalized Symmetries for the Vacuum Einstein Equations, Communications in Mathematical Physics 176, 479-539 (1996)

  • Joseph Krasil'shchik, Alexander Verbovetsky, section 1.4 of Geometry of jet spaces and integrable systems, J.Geom.Phys.61:1633-1674, 2011 (arXiv:1002.0077)

Last revised on November 30, 2017 at 07:35:08. See the history of this page for a list of all contributions to it.