Jet bundle

"Jet space" redirects here. Not to be confused with space jet.

In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Jets may also be seen as the coordinate free versions of Taylor expansions.

Historically, jet bundles are attributed to Charles Ehresmann, and were an advance on the method (prolongation) of Élie Cartan, of dealing geometrically with higher derivatives, by imposing differential form conditions on newly introduced formal variables. Jet bundles are sometimes called sprays, although sprays usually refer more specifically to the associated vector field induced on the corresponding bundle (e.g., the geodesic spray on Finsler manifolds.)

Since the early 1980s, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the calculus of variations. Consequently, the jet bundle is now recognized as the correct domain for a geometrical covariant field theory and much work is done in general relativistic formulations of fields using this approach.

Jets

Main article: Jet (mathematics)

Suppose M is an m-dimensional manifold and that (E, πぱい, M) is a fiber bundle. For p ∈ M, let Γがんま(p) denote the set of all local sections whose domain contains p. Let ⁠$I=(I(1),I(2),...,I(m))$⁠ be a multi-index (an m-tuple of non-negative integers, not necessarily in ascending order), then define:

${\begin{aligned}|I|&:=\sum _{i=1}^{m}I(i)\\{\frac {\partial ^{|I|}}{\partial x^{I}}}&:=\prod _{i=1}^{m}\left({\frac {\partial }{\partial x^{i}}}\right)^{I(i)}.\end{aligned}}$

Define the local sections σしぐま, ηいーた ∈ Γがんま(p) to have the same r-jet at p if

$\left.{\frac {\partial ^{|I|}\sigma ^{\alpha }}{\partial x^{I}}}\right|_{p}=\left.{\frac {\partial ^{|I|}\eta ^{\alpha }}{\partial x^{I}}}\right|_{p},\quad 0\leq |I|\leq r.$

The relation that two maps have the same r-jet is an equivalence relation. An r-jet is an equivalence class under this relation, and the r-jet with representative σしぐま is denoted $j_{p}^{r}\sigma$. The integer r is also called the order of the jet, p is its source and σしぐま(p) is its target.

Jet manifolds

The r-th jet manifold of πぱい is the set

$J^{r}(\pi )=\left\{j_{p}^{r}\sigma :p\in M,\sigma \in \Gamma (p)\right\}.$

We may define projections πぱい_r and πぱい_r,0 called the source and target projections respectively, by

${\begin{cases}\pi _{r}:J^{r}(\pi )\to M\\j_{p}^{r}\sigma \mapsto p\end{cases}},\qquad {\begin{cases}\pi _{r,0}:J^{r}(\pi )\to E\\j_{p}^{r}\sigma \mapsto \sigma (p)\end{cases}}$

If 1 ≤ k ≤ r, then the k-jet projection is the function πぱい_r,k defined by

${\begin{cases}\pi _{r,k}:J^{r}(\pi )\to J^{k}(\pi )\\j_{p}^{r}\sigma \mapsto j_{p}^{k}\sigma \end{cases}}$

From this definition, it is clear that πぱい_r = πぱい o πぱい_r,0 and that if 0 ≤ m ≤ k, then πぱい_r,m = πぱい_k,m o πぱい_r,k. It is conventional to regard πぱい_r,r as the identity map on J ^r(πぱい) and to identify J ⁰(πぱい) with E.

The functions πぱい_r,k, πぱい_r,0 and πぱい_r are smooth surjective submersions.

A coordinate system on E will generate a coordinate system on J ^r(πぱい). Let (U, u) be an adapted coordinate chart on E, where u = (xⁱ, u^{αあるふぁ}). The induced coordinate chart (U^r, u^r) on J ^r(πぱい) is defined by

${\begin{aligned}U^{r}&=\left\{j_{p}^{r}\sigma :p\in M,\sigma (p)\in U\right\}\\u^{r}&=\left(x^{i},u^{\alpha },u_{I}^{\alpha }\right)\end{aligned}}$

where

${\begin{aligned}x^{i}\left(j_{p}^{r}\sigma \right)&=x^{i}(p)\\u^{\alpha }\left(j_{p}^{r}\sigma \right)&=u^{\alpha }(\sigma (p))\end{aligned}}$

and the $n\left({\binom {m+r}{r}}-1\right)$ functions known as the derivative coordinates:

${\begin{cases}u_{I}^{\alpha }:U^{k}\to \mathbf {R} \\u_{I}^{\alpha }\left(j_{p}^{r}\sigma \right)=\left.{\frac {\partial ^{|I|}\sigma ^{\alpha }}{\partial x^{I}}}\right|_{p}\end{cases}}$

Given an atlas of adapted charts (U, u) on E, the corresponding collection of charts (U ^r, u ^r) is a finite-dimensional C^∞ atlas on J ^r(πぱい).

Jet bundles

Since the atlas on each $J^{r}(\pi )$ defines a manifold, the triples $(J^{r}(\pi ),\pi _{r,k},J^{k}(\pi ))$, $(J^{r}(\pi ),\pi _{r,0},E)$ and $(J^{r}(\pi ),\pi _{r},M)$ all define fibered manifolds. In particular, if $(E,\pi ,M)$is a fiber bundle, the triple $(J^{r}(\pi ),\pi _{r},M)$ defines the r-th jet bundle of πぱい.

If W ⊂ M is an open submanifold, then

$J^{r}\left(\pi |_{\pi ^{-1}(W)}\right)\cong \pi _{r}^{-1}(W).\,$

If p ∈ M, then the fiber $\pi _{r}^{-1}(p)\,$ is denoted $J_{p}^{r}(\pi )$.

Let σしぐま be a local section of πぱい with domain W ⊂ M. The r-th jet prolongation of σしぐま is the map $j^{r}\sigma :W\rightarrow J^{r}(\pi )$ defined by

$(j^{r}\sigma )(p)=j_{p}^{r}\sigma .\,$

Note that $\pi _{r}\circ j^{r}\sigma =\mathbb {id} _{W}$, so $j^{r}\sigma$ really is a section. In local coordinates, $j^{r}\sigma$ is given by

$\left(\sigma ^{\alpha },{\frac {\partial ^{|I|}\sigma ^{\alpha }}{\partial x^{I}}}\right)\qquad 1\leq |I|\leq r.\,$

We identify $j^{0}\sigma$ with $\sigma$ .

Algebro-geometric perspective

An independently motivated construction of the sheaf of sections $\Gamma J^{k}\left(\pi _{TM}\right)$ is given.

Consider a diagonal map ${\textstyle \Delta _{n}:M\to \prod _{i=1}^{n+1}M}$, where the smooth manifold $M$ is a locally ringed space by $C^{k}(U)$ for each open $U$. Let ${\mathcal {I}}$ be the ideal sheaf of $\Delta _{n}(M)$, equivalently let ${\mathcal {I}}$ be the sheaf of smooth germs which vanish on $\Delta _{n}(M)$ for all $0<n\leq k$. The pullback of the quotient sheaf ${\Delta _{n}}^{*}\left({\mathcal {I}}/{\mathcal {I}}^{n+1}\right)$ from ${\textstyle \prod _{i=1}^{n+1}M}$ to $M$ by $\Delta _{n}$ is the sheaf of k-jets.

The direct limit of the sequence of injections given by the canonical inclusions ${\mathcal {I}}^{n+1}\hookrightarrow {\mathcal {I}}^{n}$ of sheaves, gives rise to the infinite jet sheaf ${\mathcal {J}}^{\infty }(TM)$. Observe that by the direct limit construction it is a filtered ring.

Example

If πぱい is the trivial bundle (M × R, pr₁, M), then there is a canonical diffeomorphism between the first jet bundle $J^{1}(\pi )$ and T*M × R. To construct this diffeomorphism, for each σしぐま in $\Gamma _{M}(\pi )$ write ${\bar {\sigma }}=pr_{2}\circ \sigma \in C^{\infty }(M)\,$.

Then, whenever p ∈ M

$j_{p}^{1}\sigma =\left\{\psi :\psi \in \Gamma _{p}(\pi );{\bar {\psi }}(p)={\bar {\sigma }}(p);d{\bar {\psi }}_{p}=d{\bar {\sigma }}_{p}\right\}.\,$

Consequently, the mapping

${\begin{cases}J^{1}(\pi )\to T^{*}M\times \mathbf {R} \\j_{p}^{1}\sigma \mapsto \left(d{\bar {\sigma }}_{p},{\bar {\sigma }}(p)\right)\end{cases}}$

is well-defined and is clearly injective. Writing it out in coordinates shows that it is a diffeomorphism, because if (xⁱ, u) are coordinates on M × R, where u = id_R is the identity coordinate, then the derivative coordinates u_i on J¹(πぱい) correspond to the coordinates ∂_i on T*M.

Likewise, if πぱい is the trivial bundle (R × M, pr₁, R), then there exists a canonical diffeomorphism between $J^{1}(\pi )$and R × TM.

Contact structure

The space J^r(πぱい) carries a natural distribution, that is, a sub-bundle of the tangent bundle TJ^r(πぱい)), called the Cartan distribution. The Cartan distribution is spanned by all tangent planes to graphs of holonomic sections; that is, sections of the form j^rφふぁい for φふぁい a section of πぱい.

The annihilator of the Cartan distribution is a space of differential one-forms called contact forms, on J^r(πぱい). The space of differential one-forms on J^r(πぱい) is denoted by $\Lambda ^{1}J^{r}(\pi )$ and the space of contact forms is denoted by $\Lambda _{C}^{r}\pi$. A one form is a contact form provided its pullback along every prolongation is zero. In other words, $\theta \in \Lambda ^{1}J^{r}\pi$ is a contact form if and only if

$\left(j^{r+1}\sigma \right)^{*}\theta =0$

for all local sections σしぐま of πぱい over M.

The Cartan distribution is the main geometrical structure on jet spaces and plays an important role in the geometric theory of partial differential equations. The Cartan distributions are completely non-integrable. In particular, they are not involutive. The dimension of the Cartan distribution grows with the order of the jet space. However, on the space of infinite jets J^∞ the Cartan distribution becomes involutive and finite-dimensional: its dimension coincides with the dimension of the base manifold M.

Example

Consider the case (E, πぱい, M), where E ≃ R² and M ≃ R. Then, (J¹(πぱい), πぱい, M) defines the first jet bundle, and may be coordinated by (x, u, u₁), where

${\begin{aligned}x\left(j_{p}^{1}\sigma \right)&=x(p)=x\\u\left(j_{p}^{1}\sigma \right)&=u(\sigma (p))=u(\sigma (x))=\sigma (x)\\u_{1}\left(j_{p}^{1}\sigma \right)&=\left.{\frac {\partial \sigma }{\partial x}}\right|_{p}=\sigma '(x)\end{aligned}}$

for all p ∈ M and σしぐま in Γがんま_p(πぱい). A general 1-form on J¹(πぱい) takes the form

$\theta =a(x,u,u_{1})dx+b(x,u,u_{1})du+c(x,u,u_{1})du_{1}\,$

A section σしぐま in Γがんま_p(πぱい) has first prolongation

$j^{1}\sigma =(u,u_{1})=\left(\sigma (p),\left.{\frac {\partial \sigma }{\partial x}}\right|_{p}\right).$

Hence, (j¹σしぐま)*θしーた can be calculated as

${\begin{aligned}\left(j_{p}^{1}\sigma \right)^{*}\theta &=\theta \circ j_{p}^{1}\sigma \\&=a(x,\sigma (x),\sigma '(x))dx+b(x,\sigma (x),\sigma '(x))d(\sigma (x))+c(x,\sigma (x),\sigma '(x))d(\sigma '(x))\\&=a(x,\sigma (x),\sigma '(x))dx+b(x,\sigma (x),\sigma '(x))\sigma '(x)dx+c(x,\sigma (x),\sigma '(x))\sigma ''(x)dx\\&=[a(x,\sigma (x),\sigma '(x))+b(x,\sigma (x),\sigma '(x))\sigma '(x)+c(x,\sigma (x),\sigma '(x))\sigma ''(x)]dx\end{aligned}}$

This will vanish for all sections σしぐま if and only if c = 0 and a = −bσしぐま′(x). Hence, θしーた = b(x, u, u₁)θしーた₀ must necessarily be a multiple of the basic contact form θしーた₀ = du − u₁dx. Proceeding to the second jet space J²(πぱい) with additional coordinate u₂, such that

$u_{2}(j_{p}^{2}\sigma )=\left.{\frac {\partial ^{2}\sigma }{\partial x^{2}}}\right|_{p}=\sigma ''(x)\,$

a general 1-form has the construction

$\theta =a(x,u,u_{1},u_{2})dx+b(x,u,u_{1},u_{2})du+c(x,u,u_{1},u_{2})du_{1}+e(x,u,u_{1},u_{2})du_{2}\,$

This is a contact form if and only if

${\begin{aligned}\left(j_{p}^{2}\sigma \right)^{*}\theta &=\theta \circ j_{p}^{2}\sigma \\&=a(x,\sigma (x),\sigma '(x),\sigma ''(x))dx+b(x,\sigma (x),\sigma '(x),\sigma ''(x))d(\sigma (x))+{}\\&\qquad \qquad c(x,\sigma (x),\sigma '(x),\sigma ''(x))d(\sigma '(x))+e(x,\sigma (x),\sigma '(x),\sigma ''(x))d(\sigma ''(x))\\&=adx+b\sigma '(x)dx+c\sigma ''(x)dx+e\sigma '''(x)dx\\&=[a+b\sigma '(x)+c\sigma ''(x)+e\sigma '''(x)]dx\\&=0\end{aligned}}$

which implies that e = 0 and a = −bσしぐま′(x) − cσしぐま′′(x). Therefore, θしーた is a contact form if and only if

$\theta =b(x,\sigma (x),\sigma '(x))\theta _{0}+c(x,\sigma (x),\sigma '(x))\theta _{1},$

where θしーた₁ = du₁ − u₂dx is the next basic contact form (Note that here we are identifying the form θしーた₀ with its pull-back $\left(\pi _{2,1}\right)^{*}\theta _{0}$ to J²(πぱい)).

In general, providing x, u ∈ R, a contact form on J^r+1(πぱい) can be written as a linear combination of the basic contact forms

$\theta _{k}=du_{k}-u_{k+1}dx\qquad k=0,\ldots ,r-1\,$

where

$u_{k}\left(j^{k}\sigma \right)=\left.{\frac {\partial ^{k}\sigma }{\partial x^{k}}}\right|_{p}.$

Similar arguments lead to a complete characterization of all contact forms.

In local coordinates, every contact one-form on J^r+1(πぱい) can be written as a linear combination

$\theta =\sum _{|I|=0}^{r}P_{\alpha }^{I}\theta _{I}^{\alpha }$

with smooth coefficients $P_{i}^{\alpha }(x^{i},u^{\alpha },u_{I}^{\alpha })$ of the basic contact forms

$\theta _{I}^{\alpha }=du_{I}^{\alpha }-u_{I,i}^{\alpha }dx^{i}\,$

|I| is known as the order of the contact form $\theta _{i}^{\alpha }$. Note that contact forms on J^r+1(πぱい) have orders at most r. Contact forms provide a characterization of those local sections of πぱい_r+1 which are prolongations of sections of πぱい.

Let ψぷさい ∈ Γがんま_W(πぱい_r+1), then ψぷさい = j^r+1σしぐま where σしぐま ∈ Γがんま_W(πぱい) if and only if $\psi ^{*}(\theta |_{W})=0,\forall \theta \in \Lambda _{C}^{1}\pi _{r+1,r}.\,$

Vector fields

A general vector field on the total space E, coordinated by $(x,u)\mathrel {\stackrel {\mathrm {def} }{=}} \left(x^{i},u^{\alpha }\right)\,$, is

$V\mathrel {\stackrel {\mathrm {def} }{=}} \rho ^{i}(x,u){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }(x,u){\frac {\partial }{\partial u^{\alpha }}}.\,$

A vector field is called horizontal, meaning that all the vertical coefficients vanish, if $\phi ^{\alpha }$ = 0.

A vector field is called vertical, meaning that all the horizontal coefficients vanish, if ρろーⁱ = 0.

For fixed (x, u), we identify

$V_{(x,u)}\mathrel {\stackrel {\mathrm {def} }{=}} \rho ^{i}(x,u){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }(x,u){\frac {\partial }{\partial u^{\alpha }}}\,$

having coordinates (x, u, ρろーⁱ, φふぁい^{αあるふぁ}), with an element in the fiber T_xuE of TE over (x, u) in E, called a tangent vector in TE. A section

${\begin{cases}\psi :E\to TE\\(x,u)\mapsto \psi (x,u)=V\end{cases}}$

is called a vector field on E with

$V=\rho ^{i}(x,u){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }(x,u){\frac {\partial }{\partial u^{\alpha }}}$

and ψぷさい in Γがんま(TE).

The jet bundle J^r(πぱい) is coordinated by $(x,u,w)\mathrel {\stackrel {\mathrm {def} }{=}} \left(x^{i},u^{\alpha },w_{i}^{\alpha }\right)\,$. For fixed (x, u, w), identify

$V_{(x,u,w)}\mathrel {\stackrel {\mathrm {def} }{=}} V^{i}(x,u,w){\frac {\partial }{\partial x^{i}}}+V^{\alpha }(x,u,w){\frac {\partial }{\partial u^{\alpha }}}+V_{i}^{\alpha }(x,u,w){\frac {\partial }{\partial w_{i}^{\alpha }}}+V_{i_{1}i_{2}}^{\alpha }(x,u,w){\frac {\partial }{\partial w_{i_{1}i_{2}}^{\alpha }}}+\cdots +V_{i_{1}\cdots i_{r}}^{\alpha }(x,u,w){\frac {\partial }{\partial w_{i_{1}\cdots i_{r}}^{\alpha }}}$

having coordinates

$\left(x,u,w,v_{i}^{\alpha },v_{i_{1}i_{2}}^{\alpha },\cdots ,v_{i_{1}\cdots i_{r}}^{\alpha }\right),$

with an element in the fiber $T_{xuw}(J^{r}\pi )$ of TJ^r(πぱい) over (x, u, w) ∈ J^r(πぱい), called a tangent vector in TJ^r(πぱい). Here,

$v_{i}^{\alpha },v_{i_{1}i_{2}}^{\alpha },\ldots ,v_{i_{1}\cdots i_{r}}^{\alpha }$

are real-valued functions on J^r(πぱい). A section

${\begin{cases}\Psi :J^{r}(\pi )\to TJ^{r}(\pi )\\(x,u,w)\mapsto \Psi (u,w)=V\end{cases}}$

is a vector field on J^r(πぱい), and we say $\Psi \in \Gamma (T\left(J^{r}\pi \right)).$

Partial differential equations

Let (E, πぱい, M) be a fiber bundle. An r-th order partial differential equation on πぱい is a closed embedded submanifold S of the jet manifold J^r(πぱい). A solution is a local section σしぐま ∈ Γがんま_W(πぱい) satisfying $j_{p}^{r}\sigma \in S$, for all p in M.

Consider an example of a first order partial differential equation.

Example

Let πぱい be the trivial bundle (R² × R, pr₁, R²) with global coordinates (x¹, x², u¹). Then the map F : J¹(πぱい) → R defined by

$F=u_{1}^{1}u_{2}^{1}-2x^{2}u^{1}$

gives rise to the differential equation

$S=\left\{j_{p}^{1}\sigma \in J^{1}\pi \ :\ \left(u_{1}^{1}u_{2}^{1}-2x^{2}u^{1}\right)\left(j_{p}^{1}\sigma \right)=0\right\}$

which can be written

${\frac {\partial \sigma }{\partial x^{1}}}{\frac {\partial \sigma }{\partial x^{2}}}-2x^{2}\sigma =0.$

The particular

${\begin{cases}\sigma :\mathbf {R} ^{2}\to \mathbf {R} ^{2}\times \mathbf {R} \\\sigma (p_{1},p_{2})=\left(p^{1},p^{2},p^{1}(p^{2})^{2}\right)\end{cases}}$

has first prolongation given by

$j^{1}\sigma \left(p_{1},p_{2}\right)=\left(p^{1},p^{2},p^{1}\left(p^{2}\right)^{2},\left(p^{2}\right)^{2},2p^{1}p^{2}\right)$

and is a solution of this differential equation, because

${\begin{aligned}\left(u_{1}^{1}u_{2}^{1}-2x^{2}u^{1}\right)\left(j_{p}^{1}\sigma \right)&=u_{1}^{1}\left(j_{p}^{1}\sigma \right)u_{2}^{1}\left(j_{p}^{1}\sigma \right)-2x^{2}\left(j_{p}^{1}\sigma \right)u^{1}\left(j_{p}^{1}\sigma \right)\\&=\left(p^{2}\right)^{2}\cdot 2p^{1}p^{2}-2\cdot p^{2}\cdot p^{1}\left(p^{2}\right)^{2}\\&=2p^{1}\left(p^{2}\right)^{3}-2p^{1}\left(p^{2}\right)^{3}\\&=0\end{aligned}}$

and so $j_{p}^{1}\sigma \in S$ for every p ∈ R².

Jet prolongation

A local diffeomorphism ψぷさい : J^r(πぱい) → J^r(πぱい) defines a contact transformation of order r if it preserves the contact ideal, meaning that if θしーた is any contact form on J^r(πぱい), then ψぷさい*θしーた is also a contact form.

The flow generated by a vector field V^r on the jet space J^r(πぱい) forms a one-parameter group of contact transformations if and only if the Lie derivative ${\mathcal {L}}_{V^{r}}(\theta )$ of any contact form θしーた preserves the contact ideal.

Let us begin with the first order case. Consider a general vector field V¹ on J¹(πぱい), given by

$V^{1}\ {\stackrel {\mathrm {def} }{=}}\ \rho ^{i}\left(x^{i},u^{\alpha },u_{I}^{\alpha }\right){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }\left(x^{i},u^{\alpha },u_{I}^{\alpha }\right){\frac {\partial }{\partial u^{\alpha }}}+\chi _{i}^{\alpha }\left(x^{i},u^{\alpha },u_{I}^{\alpha }\right){\frac {\partial }{\partial u_{i}^{\alpha }}}.$

We now apply ${\mathcal {L}}_{V^{1}}$ to the basic contact forms $\theta _{0}^{\alpha }=du^{\alpha }-u_{i}^{\alpha }dx^{i},$ and expand the exterior derivative of the functions in terms of their coordinates to obtain:

${\begin{aligned}{\mathcal {L}}_{V^{1}}\left(\theta _{0}^{\alpha }\right)&={\mathcal {L}}_{V^{1}}\left(du^{\alpha }-u_{i}^{\alpha }dx^{i}\right)\\&={\mathcal {L}}_{V^{1}}du^{\alpha }-\left({\mathcal {L}}_{V^{1}}u_{i}^{\alpha }\right)dx^{i}-u_{i}^{\alpha }\left({\mathcal {L}}_{V^{1}}dx^{i}\right)\\&=d\left(V^{1}u^{\alpha }\right)-V^{1}u_{i}^{\alpha }dx^{i}-u_{i}^{\alpha }d\left(V^{1}x^{i}\right)\\&=d\phi ^{\alpha }-\chi _{i}^{\alpha }dx^{i}-u_{i}^{\alpha }d\rho ^{i}\\&={\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}dx^{i}+{\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}du^{k}+{\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}du_{i}^{k}-\chi _{i}^{\alpha }dx^{i}-u_{i}^{\alpha }\left[{\frac {\partial \rho ^{i}}{\partial x^{m}}}dx^{m}+{\frac {\partial \rho ^{i}}{\partial u^{k}}}du^{k}+{\frac {\partial \rho ^{i}}{\partial u_{m}^{k}}}du_{m}^{k}\right]\\&={\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}dx^{i}+{\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}\left(\theta ^{k}+u_{i}^{k}dx^{i}\right)+{\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}du_{i}^{k}-\chi _{i}^{\alpha }dx^{i}-u_{l}^{\alpha }\left[{\frac {\partial \rho ^{l}}{\partial x^{i}}}dx^{i}+{\frac {\partial \rho ^{l}}{\partial u^{k}}}\left(\theta ^{k}+u_{i}^{k}dx^{i}\right)+{\frac {\partial \rho ^{l}}{\partial u_{i}^{k}}}du_{i}^{k}\right]\\&=\left[{\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}+{\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}u_{i}^{k}-u_{l}^{\alpha }\left({\frac {\partial \rho ^{l}}{\partial x^{i}}}+{\frac {\partial \rho ^{l}}{\partial u^{k}}}u_{i}^{k}\right)-\chi _{i}^{\alpha }\right]dx^{i}+\left[{\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}-u_{l}^{\alpha }{\frac {\partial \rho ^{l}}{\partial u_{i}^{k}}}\right]du_{i}^{k}+\left({\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}-u_{l}^{\alpha }{\frac {\partial \rho ^{l}}{\partial u^{k}}}\right)\theta ^{k}\end{aligned}}$

Therefore, V¹ determines a contact transformation if and only if the coefficients of dxⁱ and $du_{i}^{k}$ in the formula vanish. The latter requirements imply the contact conditions

${\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}-u_{l}^{\alpha }{\frac {\partial \rho ^{l}}{\partial u_{i}^{k}}}=0$

The former requirements provide explicit formulae for the coefficients of the first derivative terms in V¹:

$\chi _{i}^{\alpha }={\widehat {D}}_{i}\phi ^{\alpha }-u_{l}^{\alpha }\left({\widehat {D}}_{i}\rho ^{l}\right)$

where

${\widehat {D}}_{i}={\frac {\partial }{\partial x^{i}}}+u_{i}^{k}{\frac {\partial }{\partial u^{k}}}$

denotes the zeroth order truncation of the total derivative D_i.

Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if ${\mathcal {L}}_{V^{r}}$ satisfies these equations, V^r is called the r-th prolongation of V to a vector field on J^r(πぱい).

These results are best understood when applied to a particular example. Hence, let us examine the following.

Example

Consider the case (E, πぱい, M), where E ≅ R² and M ≃ R. Then, (J¹(πぱい), πぱい, E) defines the first jet bundle, and may be coordinated by (x, u, u₁), where

${\begin{aligned}x(j_{p}^{1}\sigma )&=x(p)=x\\u(j_{p}^{1}\sigma )&=u(\sigma (p))=u(\sigma (x))=\sigma (x)\\u_{1}(j_{p}^{1}\sigma )&=\left.{\frac {\partial \sigma }{\partial x}}\right|_{p}={\dot {\sigma }}(x)\end{aligned}}$

for all p ∈ M and σしぐま in Γがんま_p(πぱい). A contact form on J¹(πぱい) has the form

$\theta =du-u_{1}dx$

Consider a vector V on E, having the form

$V=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}$

Then, the first prolongation of this vector field to J¹(πぱい) is

${\begin{aligned}V^{1}&=V+Z\\&=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+Z\\&=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+\rho (x,u,u_{1}){\frac {\partial }{\partial u_{1}}}\end{aligned}}$

If we now take the Lie derivative of the contact form with respect to this prolonged vector field, ${\mathcal {L}}_{V^{1}}(\theta ),$ we obtain

${\begin{aligned}{\mathcal {L}}_{V^{1}}(\theta )&={\mathcal {L}}_{V^{1}}(du-u_{1}dx)\\&={\mathcal {L}}_{V^{1}}du-\left({\mathcal {L}}_{V^{1}}u_{1}\right)dx-u_{1}\left({\mathcal {L}}_{V^{1}}dx\right)\\&=d\left(V^{1}u\right)-V^{1}u_{1}dx-u_{1}d\left(V^{1}x\right)\\&=dx-\rho (x,u,u_{1})dx+u_{1}du\\&=(1-\rho (x,u,u_{1}))dx+u_{1}du\\&=[1-\rho (x,u,u_{1})]dx+u_{1}(\theta +u_{1}dx)&&du=\theta +u_{1}dx\\&=[1+u_{1}u_{1}-\rho (x,u,u_{1})]dx+u_{1}\theta \end{aligned}}$

Hence, for preservation of the contact ideal, we require

$1+u_{1}u_{1}-\rho (x,u,u_{1})=0\quad \Leftrightarrow \quad \rho (x,u,u_{1})=1+u_{1}u_{1}.$

And so the first prolongation of V to a vector field on J¹(πぱい) is

$V^{1}=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+(1+u_{1}u_{1}){\frac {\partial }{\partial u_{1}}}.$

Let us also calculate the second prolongation of V to a vector field on J²(πぱい). We have $\{x,u,u_{1},u_{2}\}$ as coordinates on J²(πぱい). Hence, the prolonged vector has the form

$V^{2}=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+\rho (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{1}}}+\phi (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{2}}}.$

The contact forms are

${\begin{aligned}\theta &=du-u_{1}dx\\\theta _{1}&=du_{1}-u_{2}dx\end{aligned}}$

To preserve the contact ideal, we require

${\begin{aligned}{\mathcal {L}}_{V^{2}}(\theta )&=0\\{\mathcal {L}}_{V^{2}}(\theta _{1})&=0\end{aligned}}$

Now, θしーた has no u₂ dependency. Hence, from this equation we will pick up the formula for ρろー, which will necessarily be the same result as we found for V¹. Therefore, the problem is analogous to prolonging the vector field V¹ to J²(πぱい). That is to say, we may generate the r-th prolongation of a vector field by recursively applying the Lie derivative of the contact forms with respect to the prolonged vector fields, r times. So, we have

$\rho (x,u,u_{1})=1+u_{1}u_{1}$

and so

${\begin{aligned}V^{2}&=V^{1}+\phi (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{2}}}\\&=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+(1+u_{1}u_{1}){\frac {\partial }{\partial u_{1}}}+\phi (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{2}}}\end{aligned}}$

Therefore, the Lie derivative of the second contact form with respect to V² is

${\begin{aligned}{\mathcal {L}}_{V^{2}}(\theta _{1})&={\mathcal {L}}_{V^{2}}(du_{1}-u_{2}dx)\\&={\mathcal {L}}_{V^{2}}du_{1}-\left({\mathcal {L}}_{V^{2}}u_{2}\right)dx-u_{2}\left({\mathcal {L}}_{V^{2}}dx\right)\\&=d(V^{2}u_{1})-V^{2}u_{2}dx-u_{2}d(V^{2}x)\\&=d(1+u_{1}u_{1})-\phi (x,u,u_{1},u_{2})dx+u_{2}du\\&=2u_{1}du_{1}-\phi (x,u,u_{1},u_{2})dx+u_{2}du\\&=2u_{1}du_{1}-\phi (x,u,u_{1},u_{2})dx+u_{2}(\theta +u_{1}dx)&du&=\theta +u_{1}dx\\&=2u_{1}(\theta _{1}+u_{2}dx)-\phi (x,u,u_{1},u_{2})dx+u_{2}(\theta +u_{1}dx)&du_{1}&=\theta _{1}+u_{2}dx\\&=[3u_{1}u_{2}-\phi (x,u,u_{1},u_{2})]dx+u_{2}\theta +2u_{1}\theta _{1}\end{aligned}}$

Hence, for ${\mathcal {L}}_{V^{2}}(\theta _{1})$ to preserve the contact ideal, we require

$3u_{1}u_{2}-\phi (x,u,u_{1},u_{2})=0\quad \Leftrightarrow \quad \phi (x,u,u_{1},u_{2})=3u_{1}u_{2}.$

And so the second prolongation of V to a vector field on J²(πぱい) is

$V^{2}=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+(1+u_{1}u_{1}){\frac {\partial }{\partial u_{1}}}+3u_{1}u_{2}{\frac {\partial }{\partial u_{2}}}.$

Note that the first prolongation of V can be recovered by omitting the second derivative terms in V², or by projecting back to J¹(πぱい).

Infinite jet spaces

The inverse limit of the sequence of projections $\pi _{k+1,k}:J^{k+1}(\pi )\to J^{k}(\pi )$ gives rise to the infinite jet space J^∞(πぱい). A point $j_{p}^{\infty }(\sigma )$ is the equivalence class of sections of πぱい that have the same k-jet in p as σしぐま for all values of k. The natural projection πぱい_∞ maps $j_{p}^{\infty }(\sigma )$ into p.

Just by thinking in terms of coordinates, J^∞(πぱい) appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure on J^∞(πぱい), not relying on differentiable charts, is given by the differential calculus over commutative algebras. Dual to the sequence of projections $\pi _{k+1,k}:J^{k+1}(\pi )\to J^{k}(\pi )$ of manifolds is the sequence of injections $\pi _{k+1,k}^{*}:C^{\infty }(J^{k}(\pi ))\to C^{\infty }\left(J^{k+1}(\pi )\right)$ of commutative algebras. Let's denote $C^{\infty }(J^{k}(\pi ))$ simply by ${\mathcal {F}}_{k}(\pi )$. Take now the direct limit ${\mathcal {F}}(\pi )$ of the ${\mathcal {F}}_{k}(\pi )$'s. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric object J^∞(πぱい). Observe that ${\mathcal {F}}(\pi )$, being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.

Roughly speaking, a concrete element $\varphi \in {\mathcal {F}}(\pi )$ will always belong to some ${\mathcal {F}}_{k}(\pi )$, so it is a smooth function on the finite-dimensional manifold J^k(πぱい) in the usual sense.

Infinitely prolonged PDEs

Given a k-th order system of PDEs E ⊆ J^k(πぱい), the collection I(E) of vanishing on E smooth functions on J^∞(πぱい) is an ideal in the algebra ${\mathcal {F}}_{k}(\pi )$, and hence in the direct limit ${\mathcal {F}}(\pi )$ too.

Enhance I(E) by adding all the possible compositions of total derivatives applied to all its elements. This way we get a new ideal I of ${\mathcal {F}}(\pi )$ which is now closed under the operation of taking total derivative. The submanifold E_(∞) of J^∞(πぱい) cut out by I is called the infinite prolongation of E.

Geometrically, E_(∞) is the manifold of formal solutions of E. A point $j_{p}^{\infty }(\sigma )$ of E_(∞) can be easily seen to be represented by a section σしぐま whose k-jet's graph is tangent to E at the point $j_{p}^{k}(\sigma )$ with arbitrarily high order of tangency.

Analytically, if E is given by φふぁい = 0, a formal solution can be understood as the set of Taylor coefficients of a section σしぐま in a point p that make vanish the Taylor series of $\varphi \circ j^{k}(\sigma )$ at the point p.

Most importantly, the closure properties of I imply that E_(∞) is tangent to the infinite-order contact structure ${\mathcal {C}}$ on J^∞(πぱい), so that by restricting ${\mathcal {C}}$ to E_(∞) one gets the diffiety $(E_{(\infty )},{\mathcal {C}}|_{E_{(\infty )}})$, and can study the associated Vinogradov (C-spectral) sequence.

Remark

This article has defined jets of local sections of a bundle, but it is possible to define jets of functions f: M → N, where M and N are manifolds; the jet of f then just corresponds to the jet of the section

gr_f: M → M × N

gr_f(p) = (p, f(p))

(gr_f is known as the graph of the function f) of the trivial bundle (M × N, πぱい₁, M). However, this restriction does not simplify the theory, as the global triviality of πぱい does not imply the global triviality of πぱい₁.

See also

• Jet group
• Jet (mathematics)
• Lagrangian system
• Variational bicomplex

Further reading

• Ehresmann, C., "Introduction à la théorie des structures infinitésimales et des pseudo-groupes de Lie." Geometrie Differentielle, Colloq. Inter. du Centre Nat. de la Recherche Scientifique, Strasbourg, 1953, 97-127.
• Kolář, I., Michor, P., Slovák, J., Natural operations in differential geometry. Springer-Verlag: Berlin Heidelberg, 1993. ISBN 3-540-56235-4, ISBN 0-387-56235-4.
• Saunders, D. J., "The Geometry of Jet Bundles", Cambridge University Press, 1989, ISBN 0-521-36948-7
• Krasil'shchik, I. S., Vinogradov, A. M., [et al.], "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, ISBN 0-8218-0958-X.
• Olver, P. J., "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995, ISBN 0-521-47811-1