Convergence of Fourier series

In differential geometry, the jet bundle is a certain construction which 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 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.)

More recently, 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

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Suppose M is an m-dimensional manifold and that (E, π, M) is a fiber bundle. For p ∈ M, let Γ(π) denote the set of all local sections whose domain contains p. Let I = (I(1), I(2), ..., I(m)) be a multi-index (an ordered m-tuple of integers), then

$|I|:=\sum _{i=1}^{m}I(i)$

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

Define the local sections σ, η ∈ Γ(π) 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 is denoted $j_{p}^{r}\sigma$. The integer r is also called the order of the jet, p is its source and σ(p) is its target.

Jet manifolds

The r-th jet manifold of π is the set

$\{j_{p}^{r}\sigma :p\in M,\sigma \in \Gamma (\pi )\}$

and is denoted J^r(π). We may define projections π_r and π_r,0 called the source and target projections respectively, by

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

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

If 1 ≤ k ≤ r, then the k-jet projection is the function π_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_r = π o π_r,0 and that if 0 ≤ m ≤ k, then π_r,m = π_k,m o π_r,k. It is conventional to regard π_r,r = id_J^r(π), the identity map on J^r(π) and to identify J⁰(π) with E.

The functions π_r,k, π_r,0 and π_r are smooth surjective submersions.

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

$U^{r}=\{j_{p}^{r}\sigma :\sigma (p)\in U\}\,$

$u^{r}=(x^{i},u^{\alpha },u_{I}^{\alpha })\,$

where

$x^{i}(j_{p}^{r}\sigma )=x^{i}(p)$

$u^{\alpha }(j_{p}^{r}\sigma )=u^{\alpha }(\sigma (p))$

and the $n\left({}^{m+r}C_{r}-1\right)\,$ functions

$u_{I}^{\alpha }:U^{k}\to \mathbf {R} \,$

are specified by

$u_{I}^{\alpha }(j_{p}^{r}\sigma )=\left.{\frac {\partial ^{|I|}\sigma ^{\alpha }}{\partial x^{I}}}\right|_{p}$

and are known as the derivative coordinates.

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).

Jet bundles

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

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 be a local section of π with domain W ⊂ M. The r-th jet prolongation of σ is the map j^rσ: W → J^r(π) defined by

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

Note that π_r o j^rσ = id_W, so j^rσ really is a section. In local coordinates, j^rσ is given by

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

We identify j⁰σ with σ.

Example

If π is the trivial bundle (M × R, pr₁, M), then there is a canonical diffeomorphism between the first jet bundle J¹(π) and T*M × R. To construct this diffeomorphism, for each σ in Γ_M(π) write ${\bar {\sigma }}=pr_{2}\circ \sigma \in C^{\infty }(M)\,$.

Then, whenever p ∈ M

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

Consequently, the mapping

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

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) correspond to the coordinates ∂_i on T*M.

Likewise, if π is the trivial bundle (R × M, pr₁, R), then there exists a canonical diffeomorphism between J¹(π) and R × TM.

Contact forms

A differential 1-form θ on the space J^r(π) is called a contact form (i.e. $\theta \in \Lambda _{C}^{r}\pi \,$) if it is pulled back to the zero form on M by all prolongations. In other words, if $\theta \in \Lambda ^{1}J^{r+1}\pi \,$, then $\theta \in \Lambda _{C}^{1}\pi _{r+1,r}\,$ if and only if, for every open submanifold W ⊂ M and every σ in Γ_M(π)

$(j^{k+1}\sigma )^{*}\theta =0.\,$

The distribution on J^r(π) generated by the contact forms is called the Cartan distribution. It 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 not involutive and are of growing dimension when passing to higher order jet spaces. Surprisingly though, when passing to the space of infinite order jets J^∞ this distribution is involutive and finite dimensional. Its dimension coinciding with the dimension of the base manifold M.

Example

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

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

for all p ∈ M and σ in Γ_p(π). A general 1-form on J¹(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)*θ can be calculated as

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

This will vanish for all sections σ if and only if c = 0 and a = −bσ′(x). Hence, θ = b(x, u, u₁)θ₀ must necessarily be a multiple of the basic contact form θ₀ = du − u₁dx. Proceeding to the second jet space J²(π) 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

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

which implies that e = 0 and a = −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 − u₂dx is the next basic contact form (Note that here we are identifying the form θ₀ with its pull-back $(\pi _{2,1})^{*}\theta _{0}\,$ to J²(π)).

In general, providing x, u ∈ R, a contact form on J^r+1(π) 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 $u_{k}(j^{k}\sigma )=\left.{\frac {\partial ^{k}\sigma }{\partial x^{k}}}\right|_{p}\,$.

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

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

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

with smooth coefficients $P_{I}^{\alpha }(x^{i},u^{\alpha })\,$ 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) have orders at most r. Contact forms provide a characterization of those local sections of π_r+1 which are prolongations of sections of π.

Let ψ ∈ Γ_W(π_r+1), then ψ = 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 general vector field on the total space E, coordinated by $(x,u)\ {\stackrel {\mathrm {def} }{=}}\ (x^{i},u^{\alpha })\,$, is

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

A vector field is called horizontal, meaning that all the vertical coefficients vanish, i.e. φ^α = 0.

A vector field is called vertical, meaning that all the horizontal coefficients vanish, i.e. ρⁱ = 0.

For fixed (x, u), we identify

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

having coordinates (x, u, ρⁱ, φ^α), with an element in the 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 in Γ(TE).

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

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

having coordinates $(x,u,w,v_{i}^{\alpha },v_{i_{1}i_{2}}^{\alpha },\ldots ,v_{i_{1}i_{2}\cdots i_{r}}^{\alpha })\,$, with an element in the fiber $T_{xuw}(J^{r}\pi )\,$ of TJ^r(π) over (x, u, w) ∈ J^r(π), called a tangent vector in TJ^r(π). Here,

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

are real-valued functions on J^r(π). 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), and we say $\Psi \in \Gamma (T(J^{r}\pi ))\,$.

Partial differential equations

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

Let us 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 map F : J¹(π) → R defined by

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

gives rise to the differential equation

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

which can be written

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

The particular section σ: R² → R² × R defined by

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

has first prolongation given by

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

and is a solution of this differential equation, because

$(u_{1}^{1}u_{2}^{1}-2x^{2}u^{1})(j_{p}^{1}\sigma )\,$	$=u_{1}^{1}(j_{p}^{1}\sigma )u_{2}^{1}(j_{p}^{1}\sigma )-2x^{2}(j_{p}^{1}\sigma )u^{1}(j_{p}^{1}\sigma )\,$
	$=(p^{2})^{2}\cdot 2p^{1}p^{2}-2\cdot p^{2}\cdot p^{1}(p^{2})^{2}\,$
	$=2p^{1}(p^{2})^{3}-2p^{1}(p^{2})^{3}\,$
	$=0\,$

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

Jet Prolongation

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

The flow generated by a vector field V^r on the jet space J^r(π) 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 preserves the contact ideal.

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

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

We now apply ${\mathcal {L}}_{V^{1}}$ to the basic contact forms $\theta ^{\alpha }=du^{\alpha }-u_{i}^{\alpha }dx^{i}\,$, and obtain

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

where we have expanded the exterior derivative of the functions in terms of their coordinates. Next, we note that

$\theta ^{k}=du^{k}-u_{i}^{k}dx^{i}\quad \Longrightarrow \quad du^{k}=\theta ^{k}+u_{i}^{k}dx^{i}\,$

and so we may write

${\mathcal {L}}_{V^{1}}(\theta ^{\alpha })\,$	$={\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}\,dx^{i}+{\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}\,(\theta ^{k}+u_{i}^{k}dx^{i})+{\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}\,du_{i}^{k}-\chi _{i}^{\alpha }dx^{i}-\,$
	$-u_{l}^{\alpha }\left[{\frac {\partial \rho ^{l}}{\partial x^{i}}}\,dx^{i}+{\frac {\partial \rho ^{l}}{\partial u^{k}}}\,(\theta ^{k}+u_{i}^{k}dx^{i})+{\frac {\partial \rho ^{l}}{\partial u_{i}^{k}}}\,du_{i}^{k}\right]\,$
	$=\left[{\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}+{\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}u_{i}^{k}-u_{l}^{\alpha }\left({\frac {\partial \rho ^{l}}{\partial x^{i}}}+{\frac {\partial \rho ^{l}}{\partial u^{k}}}u_{i}^{k}\right)-\chi _{i}^{\alpha }\right]\,dx^{i}+\left[{\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}-u_{l}^{\alpha }{\frac {\partial \rho ^{l}}{\partial u_{i}^{k}}}\right]\,du_{i}^{k}+\,$
	$+\left({\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}-u_{l}^{\alpha }{\frac {\partial \rho ^{l}}{\partial u^{k}}}\right)\theta ^{k}.\,$

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:

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

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).

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

Example

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

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

for all p ∈ M and σ in Γ_p(π). A contact form on J¹(π) has the form

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

Let us 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) is

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

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

${\mathcal {L}}_{V^{1}}(\theta )\,$	$={\mathcal {L}}_{V^{1}}(du-u_{1}dx)\,$
	$={\mathcal {L}}_{V^{1}}du-({\mathcal {L}}_{V^{1}}u_{1})dx-u_{1}({\mathcal {L}}_{V^{1}}dx)\,$
	$=d(V^{1}u)-V^{1}u_{1}dx-u_{1}d(V^{1}x)\,$
	$=dx-\rho (x,u,u_{1})dx+u_{1}du\,$
	$=(1-\rho (x,u,u_{1}))dx+u_{1}du\,$

But, we may identify du = θ + u₁dx. Thus, we get

${\mathcal {L}}_{V^{1}}(\theta )\,$	$=[\,1-\rho (x,u,u_{1})\,]dx+u_{1}(\theta +u_{1}dx)\,$
	$=[\,1+u_{1}u_{1}-\rho (x,u,u_{1})\,]dx+u_{1}\theta \,$

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

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

And so the first prolongation of V to a vector field on J¹(π) 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). We have $\{x,u,u_{1},y_{2}\}\,$ as coordinates on J²(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 contacts forms are

$\theta \,$	$=du-u_{1}dx\,$
$\theta _{1}\,$	$=du_{1}-u_{2}dx\,$

To preserve the contact ideal, we require

${\mathcal {L}}_{V^{2}}(\theta )\,$	$=0\,$
${\mathcal {L}}_{V^{2}}(\theta _{1})\,$	$=0\,$

Now, θ has no u₂ dependency. Hence, from this equation we will pick up the formula for ρ, 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). 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

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

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

${\mathcal {L}}_{V^{2}}(\theta _{1})\,$	$={\mathcal {L}}_{V^{2}}(du_{1}-u_{2}dx)\,$
	$={\mathcal {L}}_{V^{2}}du_{1}-({\mathcal {L}}_{V^{2}}u_{2})dx-u_{2}({\mathcal {L}}_{V^{2}}dx)\,$
	$=d(V^{2}u_{1})-V^{2}u_{2}dx-u_{2}d(V^{2}x)\,$
	$=d(1-u_{1}u_{1})-\phi (x,u,u_{1},u_{2})dx+u_{2}du\,$
	$=2u_{1}du_{1}-\phi (x,u,u_{1},u_{2})dx+u_{2}du\,$

Again, let us identify du = θ + u₁dx and du₁ = θ₁ + u₂dx. Then we have

${\mathcal {L}}_{V^{2}}(\theta _{1})\,$	$=2u_{1}(\theta _{1}+u_{2}dx)-\phi (x,u,u_{1},u_{2})dx+u_{2}(\theta +u_{1}dx)\,$
	$=[\,3u_{1}u_{2}-\phi (x,u,u_{1},u_{2})\,]dx+u_{2}\theta +2u_{1}\theta _{1}\,$

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

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

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

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). A point $j_{p}^{\infty }(\sigma )$ is the equivalence class of sections of π that have the same k-jet in p as σ for all values of k. The natural projection π_∞ maps $j_{p}^{\infty }(\sigma )$ into p.

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 on J^∞(π), 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 $\pi _{k+1,k}^{*}:C^{\infty }(J^{k}(\pi ))\to C^{\infty }(J^{k+1}(\pi ))$ 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). 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) in the usual sense.

Infinitely prolonged PDEs

Given a k-th order system of PDEs E ⊆ J^k(π), the collection I(E) of vanishing on E smooth functions on J^∞(π) 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) 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 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 = 0, a formal solution can be understood as the set of Taylor coefficients of a section σ 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), so that by restricting ${\mathcal {C}}$ to E_(∞) one gets the diffiety $(E_{(\infty )},{\mathcal {C}}|_{E_{(\infty )}})$, and can study the associated 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, 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

References

• 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
• Giachetta, G., Mangiarotti, L., Sardanashvily, G., "Advanced Classical Field Theory", World Scientific, 2009, ISBN 978-981-283-895-7
• Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory", Lambert Academic Publishing, 2013, ISBN 978-3-659-37815-7; arXiv: 0908.1886