nLab evolutionary vector field

Idea

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

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 $v$ on the jet bundle $J^\infty_\Sigma(E)$ which is vertical with respect to the projection $\pi_\infty: J^\infty_\Sigma E \to \Sigma$ such that the following equivalent conditions hold:

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

   1. $\mathcal{L}_v d = d \mathcal{L}_v$,

   2. $\mathcal{L}_v = \iota_v \delta + \delta \iota_v$,

   where $d$ is the horizontal derivative on differential forms on $J^\infty E$, while $\delta$ is the vertical derivative.

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

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

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

Definition

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

As a generalized vector field on the fiber bundle

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

Definition

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

$$\nu \circ w = \pi_{\infty,0} \phantom{AAAAAAA} \array{ J^\infty_\Sigma(E) && \overset{w}{\longrightarrow} && V_\Sigma E \\ & {}_{\mathllap{\pi_{\infty,0}}}\searrow && \swarrow_{\mathllap{\nu}} \\ && E } \,,$$

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

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

Definition

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

1. $w$ and $pr w$ agree on functions on $E$

2. $pr w$ preserves the contact ideal, i.e., $\mathcal{L}_v\theta$ is a contact form whenever $\theta$ is a contact form.

Equivalence of the definitions

Proposition

(equivalence of the two definitions)

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

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

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

Related concepts

• source form

• evolutionary derivative

• point symmetry

• conserved current

• Noether's theorem

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)