Jet bundle

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

Indifferential topology,thejet bundleis a certain construction that makes a newsmoothfiber bundleout of a given smooth fiber bundle. It makes it possible to writedifferential equationsonsectionsof a fiber bundle in an invariant form.Jetsmay also be seen as the coordinate free versions ofTaylor expansions.

Historically, jet bundles are attributed toCharles Ehresmann,and were an advance on the method (prolongation) ofÉlie Cartan,of dealinggeometricallywithhigher derivatives,by imposingdifferential formconditions on newly introduced formal variables. Jet bundles are sometimes calledsprays,althoughspraysusually refer more specifically to the associatedvector fieldinduced on the corresponding bundle (e.g., thegeodesic sprayonFinsler 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 thecalculus of variations.Consequently, the jet bundle is now recognized as the correct domain for ageometrical covariant field theoryand much work is done ingeneral relativisticformulations of fields using this approach.

Jets

Main article:Jet (mathematics)

SupposeMis anm-dimensionalmanifoldand that (E,π,M) is afiber bundle.Forp∈M,let Γ(p) denote the set of all local sections whose domain containsp.Let⁠$I=(I(1),I(2),...,I(m))$⁠be amulti-index(anm-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, η ∈ Γ(p) to have the samer-jetatpif

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

The relation that two maps have the samer-jet is anequivalence relation.Anr-jet is anequivalence classunder this relation, and ther-jet with representative σ is denoted$j_{p}^{r}\sigma$.The integerris also called theorderof the jet,pis itssourceand σ(p) is itstarget.

Jet manifolds

Ther-th jet manifold of πis the set

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

We may define projectionsπ_randπ_r,0called thesource and target projectionsrespectively, 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 thek-jet projectionis the functionπ_r,kdefined 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 thatπ_r=πoπ_r,0and that if 0 ≤m≤k,thenπ_r,m=π_k,moπ_r,k.It is conventional to regardπ_r,ras theidentity maponJ ^r(π) and to identifyJ ⁰(π) withE.

The functionsπ_r,k,π_r,0andπ_raresmoothsurjectivesubmersions.

Acoordinate systemonEwill generate a coordinate system onJ ^r(π). Let (U,u) be an adaptedcoordinate chartonE,whereu= (xⁱ,u^α). Theinduced coordinate chart (U^r,u^r)onJ ^r(π) 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 thederivative 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) onE,the corresponding collection of charts (U ^r,u ^r) is afinite-dimensionalC^∞atlas onJ ^r(π).

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 ther-th jet bundle of π.

IfW⊂Mis an open submanifold, then

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

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

Let σ be a local section of π with domainW⊂M.Ther-th jet prolongation of σis 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 alocally ringed spaceby$C^{k}(U)$for each open$U$.Let${\mathcal {I}}$be theideal sheafof$\Delta _{n}(M)$,equivalently let${\mathcal {I}}$be thesheafof smoothgermswhich vanish on$\Delta _{n}(M)$for all$0<n\leq k$.Thepullbackof thequotient 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.

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

Example

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

Then, wheneverp∈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 clearlyinjective.Writing it out in coordinates shows that it is a diffeomorphism, because if(xⁱ,u)are coordinates onM×R,whereu= id_Ris the identity coordinate, then the derivative coordinatesu_ionJ¹(π)correspond to the coordinates ∂_ionT*M.

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

Contact structure

The spaceJ^r(π) carries a naturaldistribution,that is, a sub-bundle of thetangent bundleTJ^r(π)), called theCartan distribution.The Cartan distribution is spanned by all tangent planes to graphs of holonomic sections; that is, sections of the formj^rφforφa section of π.

The annihilator of the Cartan distribution is a space ofdifferential one-formscalledcontact forms,onJ^r(π). The space of differential one-forms onJ^r(π) 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 itspullbackalong 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 of π overM.

The Cartan distribution is the main geometrical structure on jet spaces and plays an important role in the geometric theory ofpartial differential equations.The Cartan distributions are completely non-integrable. In particular, they are notinvolutive.The dimension of the Cartan distribution grows with the order of the jet space. However, on the space of infinite jetsJ^∞the Cartan distribution becomes involutive and finite-dimensional: its dimension coincides with the dimension of the base manifoldM.

Example

Consider the case(E, π, M),whereE≃R²andM≃R.Then,(J¹(π), π, 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 allp∈Mand σ in Γ_p(π). A general 1-form onJ¹(π)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 in Γ_p(π) has first prolongation

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

Hence,(j¹σ)*θcan 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 if and only ifc= 0 anda= −bσ′(x).Hence, θ =b(x, u, u₁)θ₀must necessarily be a multiple of the basic contact form θ₀=du−u₁dx.Proceeding to the second jet spaceJ²(π)with additional coordinateu₂,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 thate= 0 anda= −bσ′(x)−cσ′′(x).Therefore, θ 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 θ₁=du₁−u₂dxis the next basic contact form (Note that here we are identifying the form θ₀with its pull-back$\left(\pi _{2,1}\right)^{*}\theta _{0}$toJ²(π)).

In general, providingx, u∈R,a contact form onJ^r+1(π)can be written as alinear combinationof 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 onJ^r+1(π)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 theorderof the contact form$\theta _{i}^{\ Alpha }$.Note that contact forms onJ^r+1(π)have orders at mostr.Contact forms provide a characterization of those local sections ofπ_r+1which are prolongations of sections of π.

Let ψ ∈ Γ_W(π_r+1), thenψ=j^r+1σ where σ ∈ Γ_W(π) if and only if$\psi ^{*}(\theta |_{W})=0,\forall \theta \in \Lambda _{C}^{1}\pi _{r+1,r}.\,$

Vector fields

A generalvector fieldon the total spaceE,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 calledhorizontal,meaning that all the vertical coefficients vanish, if$\phi ^{\ Alpha }$= 0.

A vector field is calledvertical,meaning that all the horizontal coefficients vanish, ifρⁱ= 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, ρⁱ,φ^α),with an element in the fiberT_xuEofTEover(x, u)inE,calledatangent vectorinTE.A section

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

is calleda vector field onEwith

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

and ψ inΓ(TE).

The jet bundleJ^r(π)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$T_{xuw}(J^{r}\pi )$ofTJ^r(π)over(x, u, w)∈J^r(π),calleda tangent vector inTJ^r(π).Here,

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

are real-valued functions onJ^r(π).A section

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

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

Partial differential equations

Let(E, π, M)be a fiber bundle. Anr-th orderpartial differential equationon π is aclosedembeddedsubmanifoldSof the jet manifoldJ^r(π).A solution is a local section σ ∈ Γ_W(π) satisfying$j_{p}^{r}\sigma \in S$,for allpinM.

Consider an example of a first order partial differential equation.

Example

Let π be the trivial bundle (R²×R,pr₁,R²) with global coordinates (x¹,x²,u¹). Then the mapF:J¹(π) →Rdefined 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$j_{p}^{1}\sigma \in S$foreveryp∈R².

Jet prolongation

A local diffeomorphismψ:J^r(π) →J^r(π) defines a contact transformation of orderrif it preserves the contact ideal, meaning that if θ is any contact form onJ^r(π), thenψ*θis also a contact form.

The flow generated by a vector fieldV^ron the jet spaceJ^r(π)forms a one-parameter group of contact transformations if and only if theLie derivative${\mathcal {L}}_{V^{r}}(\theta )$of any contact form θ preserves the contact ideal.

Let us begin with the first order case. Consider a general vector fieldV¹onJ¹(π), 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 theexterior derivativeof 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,V¹determines a contact transformation if and only if the coefficients ofdxⁱand$du_{i}^{k}$in the formula vanish. The latter requirements imply thecontact 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 inV¹:

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

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 derivativeD_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^ris called ther-th prolongation ofVto a vector field onJ^r(π).

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

Example

Consider the case(E, π, M),whereE≅R²andM≃R.Then,(J¹(π), π, 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 allp∈Mandσin Γ_p(π). A contact form onJ¹(π)has the form

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

Consider a vectorVonE,having the form

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

Then, the first prolongation of this vector field toJ¹(π)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 ofVto a vector field onJ¹(π)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 ofVto a vector field onJ²(π).We have$\{x,u,u_{1},u_{2}\}$as coordinates onJ²(π).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,θhas nou₂dependency. Hence, from this equation we will pick up the formula forρ,which will necessarily be the same result as we found forV¹.Therefore, the problem is analogous to prolonging the vector fieldV¹toJ²(π). That is to say, we may generate ther-th prolongation of a vector field by recursively applying the Lie derivative of the contact forms with respect to the prolonged vector fields,rtimes. 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 toV²is

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

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 ofVto a vector field onJ²(π) 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 ofVcan be recovered by omitting the second derivative terms inV²,or by projecting back toJ¹(π).

Infinite jet spaces

Theinverse limitof the sequence of projections$\pi _{k+1,k}:J^{k+1}(\pi )\to J^{k}(\pi )$gives rise to theinfinite jet spaceJ^∞(π).A point$j_{p}^{\infty }(\sigma )$is the equivalence class of sections of π that have the samek-jet inpas σ for all values ofk.The natural projection π_∞maps$j_{p}^{\infty }(\sigma )$intop.

Just by thinking in terms of coordinates,J^∞(π)appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure onJ^∞(π),not relying on differentiable charts, is given by thedifferential 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 thedirect 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 objectJ^∞(π).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 manifoldJ^k(π) in the usual sense.

Infinitely prolonged PDEs

Given ak-th order system of PDEsE⊆J^k(π),the collectionI(E)of vanishing onEsmooth functions onJ^∞(π)is anidealin the algebra${\mathcal {F}}_{k}(\pi )$,and hence in the direct limit${\mathcal {F}}(\pi )$too.

EnhanceI(E)by adding all the possible compositions oftotal derivativesapplied to all its elements. This way we get a new idealIof${\mathcal {F}}(\pi )$which is now closed under the operation of taking total derivative. The submanifoldE_(∞)ofJ^∞(π) cut out byIis called theinfinite prolongationofE.

Geometrically,E_(∞)is the manifold offormal solutionsofE.A point$j_{p}^{\infty }(\sigma )$ofE_(∞)can be easily seen to be represented by a section σ whosek-jet's graph is tangent toEat the point$j_{p}^{k}(\sigma )$with arbitrarily high order of tangency.

Analytically, ifEis given by φ = 0, a formal solution can be understood as the set of Taylor coefficients of a section σ in a pointpthat make vanish theTaylor seriesof$\varphi \circ j^{k}(\sigma )$at the pointp.

Most importantly, the closure properties ofIimply thatE_(∞)is tangent to theinfinite-order contact structure${\mathcal {C}}$onJ^∞(π),so that by restricting${\mathcal {C}}$toE_(∞)one gets thediffiety$(E_{(\infty )},{\mathcal {C}}|_{E_{(\infty )}})$,and can study the associatedVinogradov (C-spectral) sequence.

Remark

This article has defined jets of local sections of a bundle, but it is possible to define jets of functionsf: M→N,whereMandNare manifolds; the jet offthen just corresponds to the jet of the section

gr_f:M→M×N

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

(gr_fis known as thegraph of the functionf) of the trivial bundle (M×N,π₁,M). However, this restriction does not simplify the theory, as the global triviality of π does not imply the global triviality of π₁.

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.ISBN3-540-56235-4,ISBN0-387-56235-4.
• Saunders, D. J., "The Geometry of Jet Bundles", Cambridge University Press, 1989,ISBN0-521-36948-7
• 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,ISBN0-8218-0958-X.
• Olver, P. J.,"Equivalence, Invariants and Symmetry", Cambridge University Press, 1995,ISBN0-521-47811-1