# nLab evolutionary vector field

Contents

### Context

#### Variational calculus

variational calculus

# Contents

## 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.