Jet bundle

 ( Differential Topology ) 

In differential topology, the jet bundle is a certain construction that makes a new smooth manifold 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 derivative, 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 .)

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.

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

$\backslash begin\{align\}$

                                  |I| &:= \sum_{i=1}^m I(i) \\
  \frac{\partial^}{\partial x^I} &:= \prod_{i=1}^m \left( \frac{\partial}{\partial x^i} \right)^{I(i)}.
     

\end{align}

Define the local sections σ, η ∈ Γ(p) to have the same r-jet at p if

$$

 \left.\frac{\partial^ \sigma^\alpha}{\partial x^I}\right|_{p} =
 \left.\frac{\partial^ \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 $j^r\_p\backslash sigma$. The integer r is also called the order of the jet, p is its source and σ( p) is its target.

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Jet manifolds The r-th jet manifold of π is the set

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

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

$\backslash begin\{cases\}$

 \pi_r: J^r(\pi) \to M \\
 j^r_p\sigma \mapsto p
     

\end{cases}, \qquad \begin{cases}

 \pi_{r, 0}: J^r(\pi) \to E \\
 j^r_p\sigma \mapsto \sigma(p)
     

\end{cases}

If 1 ≤ k ≤ r, then the k-jet projection is the function π_r,k defined by

$$\backslash begin\{cases\}$$

 \pi_{r, k}: J^r(\pi) \to J^{k}(\pi) \\
          j^r_p\sigma \mapsto j^{k}_p\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 as the identity map on J ^r( π) and to identify J ⁰( π) with E.

The functions π_r,k, π _r,0 and π_r are Smooth function 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

$$\backslash begin\{align\}$$

 U^r &= \left\{j^r_p \sigma: p \in M, \sigma(p) \in U\right\} \\
 u^r &= \left(x^i, u^\alpha, u^\alpha_I\right)
     

\end{align}

where

$$\backslash begin\{align\}$$

      x^i\left(j^r_p\sigma\right) &= x^i(p) \\
 u^\alpha\left(j^r_p\sigma\right) &= u^\alpha(\sigma(p))
     

\end{align}

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

$$\backslash begin\{cases\}$$

 u^\alpha_I:U^k \to \mathbf{R} \\
 u^\alpha_I\left(j^r_p\sigma\right) = \left.\frac{\partial^ \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).

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

If W ⊂ M is an open submanifold, then

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

If p ∈ M, then the fiber $\backslash pi^\{-1\}\_r(p)\backslash ,$ is denoted $J^r\_p(\backslash pi)$.

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

$(j^r\; \backslash sigma)(p)\; =\; j^r\_p\; \backslash sigma.\; \backslash ,$

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

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

We identify $j^\; 0\backslash sigma$ with $\backslash sigma$ .

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Algebro-geometric perspective An independently motivated construction of the sheaf of sections $\backslash Gamma\; J^k\backslash left(\backslash pi\_\{TM\}\backslash right)$ is given.''

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

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

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Example If π is the trivial bundle ( M × R, pr₁, M), then there is a canonical diffeomorphism between the first jet bundle $J^1(\backslash pi)$ and T*M × R. To construct this diffeomorphism, for each σ in $\backslash Gamma\_M(\backslash pi)$ write $\backslash bar\{\backslash sigma\}\; =\; pr\_2\; \backslash circ\; \backslash sigma\; \backslash in\; C^\backslash infty(M)\backslash ,$.

Then, whenever p ∈ M

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

Consequently, the mapping

$\backslash begin\{cases\}$

 J^1(\pi) \to T^*M \times \mathbf{R} \\
 j^1_p\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) 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 $J^1(\backslash pi)$and R × TM.

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Contact structure The space J^r(π) carries a natural distribution, that is, a sub-bundle of the tangent bundle TJ^r(π)), 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 for φ a section of π.

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

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

for all local sections σ of π 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.

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Example 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

$\backslash begin\{align\}$

   x\left(j^1_p\sigma\right) &= x(p) = x \\
   u\left(j^1_p\sigma\right) &= u(\sigma(p)) = u(\sigma(x)) = \sigma(x) \\
 u_1\left(j^1_p\sigma\right) &= \left.\frac{\partial \sigma}{\partial x}\right|_p = \sigma'(x)
     

\end{align}

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

$\backslash theta\; =\; a(x,\; u,\; u\_1)dx\; +\; b(x,\; u,\; u\_1)du\; +\; c(x,\; u,\; u\_1)du\_1\backslash ,$

A section σ in Γ _p(π) has first prolongation

$j^1\backslash sigma\; =\; (u,\; u\_1)\; =\; \backslash left(\backslash sigma(p),\; \backslash left.\; \backslash frac\{\backslash partial\; \backslash sigma\}\{\backslash partial\; x\}\; \backslash right|\_p\; \backslash right).$

Hence, (j¹σ)*θ can be calculated as

$\backslash begin\{align\}$

 \left(j^1_p\sigma\right)^* \theta
   &= \theta \circ j^1_p\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{align}

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

$u\_2(j^2\_p\backslash sigma)\; =\; \backslash left.\backslash frac\{\backslash partial^2\; \backslash sigma\}\{\backslash partial\; x^2\}\backslash right|\_p\; =\; \backslash sigma\text{'}\text{'}(x)\backslash ,$

a general 1-form has the construction

$\backslash theta\; =\; a(x,\; u,\; u\_1,u\_2)dx\; +\; b(x,\; u,\; u\_1,u\_2)du\; +\; c(x,\; u,\; u\_1,u\_2)du\_1\; +\; e(x,\; u,\; u\_1,u\_2)du\_2\backslash ,$

This is a contact form if and only if

$\backslash begin\{align\}$

 \left(j^2_p\sigma\right)^* \theta
   &= \theta \circ j^2_p\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{align}

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

$\backslash theta\; =\; b(x,\; \backslash sigma(x),\; \backslash sigma\text{'}(x))\backslash theta\_\{0\}\; +\; c(x,\; \backslash sigma(x),\; \backslash sigma\text{'}(x))\backslash theta\_1,$

where θ₁ = du₁ − u₂ dx is the next basic contact form (Note that here we are identifying the form θ₀ with its pull-back $\backslash left(\backslash pi\_\{2,1\}\backslash right)^\{*\}\backslash theta\_\{0\}$ 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

$\backslash theta\_k\; =\; du\_k\; -\; u\_\{k+1\}dx\; \backslash qquad\; k\; =\; 0,\; \backslash ldots,\; r\; -\; 1\backslash ,$

where

$u\_k\backslash left(j^k\; \backslash sigma\backslash right)\; =\; \backslash left.\backslash frac\{\backslash partial^k\; \backslash sigma\}\{\backslash partial\; x^k\}\backslash right|\_p.$

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

$\backslash theta\; =\; \backslash sum\_\{|I|=0\}^r\; P\_\backslash alpha^I\; \backslash theta\_I^\backslash alpha$

with smooth coefficients $P^\backslash alpha\_i(x^i,\; u^\backslash alpha,\; u^\backslash alpha\_I)$ of the basic contact forms

$\backslash theta\_I^\backslash alpha\; =\; du^\backslash alpha\_I\; -\; u^\backslash alpha\_\{I,i\}\; dx^i\backslash ,$

|I| is known as the order of the contact form $\backslash theta\_i^\backslash alpha$. 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 $\backslash psi^*\; (\backslash theta|\_\{W\})\; =\; 0,\; \backslash forall\; \backslash theta\; \backslash in\; \backslash Lambda\_C^1\; \backslash pi\_\{r+1,r\}.\backslash ,$

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

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

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

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

For fixed (x, u), we identify

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

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

$\backslash begin\{cases\}$

 \psi : E \to TE \\
   (x, u) \mapsto \psi(x, u) = V
     

\end{cases}

is called a vector field on E with

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

and ψ in Γ(TE).

The jet bundle J^r(π) is coordinated by $(x,\; u,\; w)\; \backslash mathrel\backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\; \backslash left(x^i,\; u^\backslash alpha,\; w\_i^\backslash alpha\backslash right)\backslash ,$. 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^\alpha_i(x, u, w) \frac{\partial}{\partial w^\alpha_i} +
     V^\alpha_{i_1 i_2}(x, u, w) \frac{\partial}{\partial w^\alpha_{i_1 i_2}} + \cdots +
     V^\alpha_{i_1 \cdots i_r}(x, u, w) \frac{\partial}{\partial w^\alpha_{i_1 \cdots i_r}}
     

having coordinates

$\backslash left(x,\; u,\; w,\; v^\backslash alpha\_i,\; v^\backslash alpha\_\{i\_1\; i\_2\},\; \backslash cdots,\; v^\backslash alpha\_\{i\_1\; \backslash cdots\; i\_r\}\backslash right),$

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

$v^\backslash alpha\_i,\; v^\backslash alpha\_\{i\_1\; i\_2\},\; \backslash ldots,\; v^\backslash alpha\_\{i\_1\; \backslash cdots\; i\_r\}$

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

$\backslash begin\{cases\}$

 \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 $\backslash Psi\; \backslash in\; \backslash Gamma(T\backslash left(J^r\backslash pi\backslash right)).$

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Partial differential equations Let (E, π, M) be a fiber bundle. An r-th order partial differential equation on π is a closed manifold embedding submanifold S of the jet manifold J^r(π). A solution is a local section σ ∈ Γ _W(π) satisfying $j^\{r\}\_p\backslash sigma\; \backslash in\; S$, for all p in M.

Consider an example of a first order partial differential equation.

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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

$F\; =\; u^1\_1\; u^1\_2\; -\; 2x^2\; u^1$

gives rise to the differential equation

$S\; =\; \backslash left\backslash \{j^1\_p\backslash sigma\; \backslash in\; J^1\backslash pi\backslash \; :\backslash \; \backslash left(u^1\_1u^1\_2\; -\; 2x^2u^1\backslash right)\backslash left(j^1\_p\backslash sigma\backslash right)\; =\; 0\backslash right\backslash \}$

which can be written

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

The particular

$\backslash begin\{cases\}$

 \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

$j^1\backslash sigma\backslash left(p\_1,\; p\_2\backslash right)\; =\; \backslash left(\; p^1,\; p^2,\; p^1\backslash left(p^2\backslash right)^2,\; \backslash left(p^2\backslash right)^2,\; 2p^1\; p^2\; \backslash right)$

and is a solution of this differential equation, because

$\backslash begin\{align\}$

 \left(u^1_1 u^1_2 - 2x^2 u^1 \right)\left(j^1_p\sigma\right)
   &= u^1_1\left(j^1_p\sigma\right)u^1_2\left(j^1_p\sigma\right) - 2x^2\left(j^1_p\sigma\right)u^1\left(j^1_p\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{align}

and so $j^1\_p\backslash sigma\; \backslash in\; S$ for every p ∈ R².

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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 $\backslash mathcal\{L\}\_\{V^r\}(\backslash theta)$ 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

$V^1\backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash rho^i\backslash left(x^i,\; u^\backslash alpha,\; u\_I^\backslash alpha\backslash right)\backslash frac\{\backslash partial\}\{\backslash partial\; x^i\}\; +\; \backslash phi^\{\backslash alpha\}\backslash left(x^i,\; u^\backslash alpha,\; u\_I^\backslash alpha\backslash right)\backslash frac\{\backslash partial\}\{\backslash partial\; u^\{\backslash alpha\}\}\; +\; \backslash chi^\{\backslash alpha\}\_i\backslash left(x^i,\; u^\backslash alpha,\; u\_I^\backslash alpha\backslash right)\backslash frac\{\backslash partial\}\{\backslash partial\; u^\{\backslash alpha\}\_i\}.$

We now apply $\backslash mathcal\{L\}\_\{V^1\}$ to the basic contact forms $\backslash theta^\{\backslash alpha\}\_0\; =\; du^\{\backslash alpha\}\; -\; u\_i^\{\backslash alpha\}dx^i,$ and expand the exterior derivative of the functions in terms of their coordinates to obtain:

$\backslash begin\{align\}$

 \mathcal{L}_{V^1}\left(\theta^{\alpha}_0\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^1u^{\alpha}\right) - V^1u_i^{\alpha}dx^i - u_i^{\alpha}d\left(V^1 x^i\right) \\
   &= d\phi^{\alpha} - \chi^{\alpha}_idx^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^k_i} du^k_i - \chi^{\alpha}_i 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^k_m} du^k_m \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^k_i} du^k_i -
      \chi^{\alpha}_i 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^k_i} du^k_i
      \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^{\alpha}_i
      \right] dx^i +
      \left[ \frac{\partial \phi^{\alpha}}{\partial u^k_i} - u_l^{\alpha}\frac{\partial \rho^l}{\partial u^k_i}\right] du^k_i +
      \left( \frac{\partial \phi^{\alpha}}{\partial u^k} - u_l^{\alpha}\frac{\partial \rho^l}{\partial u^k} \right)\theta^k
     

\end{align}

Therefore, V¹ determines a contact transformation if and only if the coefficients of dxⁱ and $du^k\_i$ in the formula vanish. The latter requirements imply the contact conditions

$\backslash frac\{\backslash partial\; \backslash phi^\{\backslash alpha\}\}\{\backslash partial\; u^k\_i\}\; -\; u^\{\backslash alpha\}\_l\; \backslash frac\{\backslash partial\; \backslash rho^l\}\{\backslash partial\; u^k\_i\}\; =\; 0$

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

$\backslash chi^\{\backslash alpha\}\_i\; =\; \backslash widehat\{D\}\_i\; \backslash phi^\{\backslash alpha\}\; -\; u^\{\backslash alpha\}\_l\backslash left(\backslash widehat\{D\}\_i\backslash rho^l\backslash right)$

where

$\backslash widehat\{D\}\_i\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; x^i\}\; +\; u^k\_i\backslash frac\{\backslash partial\}\{\backslash partial\; u^k\}$

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 $\backslash mathcal\{L\}\_\{V^r\}$ 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.

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Example 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

$\backslash begin\{align\}$

   x(j^1_{p}\sigma) &= x(p) = x  \\
   u(j^1_{p}\sigma) &= u(\sigma(p)) = u(\sigma(x)) = \sigma(x) \\
 u_1(j^1_{p}\sigma) &=  \left.\frac{\partial \sigma}{\partial x}\right|_{p} = \dot{\sigma}(x)
     

\end{align}

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

$\backslash theta\; =\; du\; -\; u\_1\; dx$

Consider a vector V on E, having the form

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

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

$\backslash begin\{align\}$

 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{align}

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

$\backslash begin\{align\}$

 \mathcal{L}_{V^1}(\theta)
   &= \mathcal{L}_{V^1}(du - u_1dx) \\
   &= \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^1u\right) - V^1 u_1 dx - u_1 d\left(V^1x\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_1u_1 - \rho(x, u, u_1)]dx + u_1\theta
     

\end{align}

Hence, for preservation of the contact ideal, we require

$1\; +\; u\_1\; u\_1\; -\; \backslash rho(x,\; u,\; u\_1)\; =\; 0\; \backslash quad\; \backslash Leftrightarrow\; \backslash quad\; \backslash rho(x,\; u,\; u\_1)\; =\; 1\; +\; u\_1\; u\_1.$

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

$V^1\; =\; x\; \backslash frac\{\backslash partial\}\{\backslash partial\; u\}\; -\; u\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; +\; (1\; +\; u\_1u\_1)\backslash frac\{\backslash partial\}\{\backslash partial\; u\_1\}.$

Let us also calculate the second prolongation of V to a vector field on J²(π). We have $\backslash \{x,\; u,\; u\_1,\; u\_2\backslash \}$ as coordinates on J²(π). Hence, the prolonged vector has the form

$V^2\; =\; x\; \backslash frac\{\backslash partial\}\{\backslash partial\; u\}\; -\; u\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; +\; \backslash rho(x,\; u,\; u\_1,\; u\_2)\backslash frac\{\backslash partial\}\{\backslash partial\; u\_1\}\; +\; \backslash phi(x,\; u,\; u\_1,\; u\_2)\backslash frac\{\backslash partial\}\{\backslash partial\; u\_2\}.$

The contact forms are

$\backslash begin\{align\}$

   \theta &= du - u_1dx \\
 \theta_1 &= du_1 - u_2dx
     

\end{align}

To preserve the contact ideal, we require

$\backslash begin\{align\}$

   \mathcal{L}_{V^2}(\theta) &= 0 \\
 \mathcal{L}_{V^2}(\theta_1) &= 0
     

\end{align}

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

$\backslash rho(x,\; u,\; u\_1)\; =\; 1\; +\; u\_1\; u\_1$

and so

$\backslash begin\{align\}$

 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{align}

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

$\backslash begin\{align\}$

 \mathcal{L}_{V^2}(\theta_1)
   &= \mathcal{L}_{V^2}(du_1 - u_2dx) \\
   &=  \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^2u_2dx - u_2d(V^2x) \\
   &= d(1 + u_1 u_1) - \phi(x, u, u_1, u_2)dx + u_2du \\
   &= 2u_1du_1 - \phi(x, u, u_1, u_2)dx + u_2du \\
   &= 2u_1du_1 - \phi(x, u, u_1, u_2)dx + u_2 (\theta + u_1dx)              &   du &= \theta + u_1 dx \\
   &= 2u_1(\theta_1 + u_2dx) - \phi(x, u, u_1, u_2)dx + u_2(\theta + u_1dx) & du_1 &= \theta_1 + u_2 dx \\
   &= [3u_1u_2 - \phi(x, u, u_1, u_2)]dx + u_2\theta + 2u_1\theta_1
     

\end{align}

Hence, for $\backslash mathcal\{L\}\_\{V^2\}(\backslash theta\_1)$ to preserve the contact ideal, we require

$3u\_1\; u\_2\; -\; \backslash phi(x,\; u,\; u\_1,\; u\_2)\; =\; 0\; \backslash quad\; \backslash Leftrightarrow\; \backslash quad\; \backslash phi(x,\; u,\; u\_1,\; u\_2)\; =\; 3u\_1\; u\_2.$

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

$V^2\; =\; x\; \backslash frac\{\backslash partial\}\{\backslash partial\; u\}\; -\; u\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; +\; (1\; +\; u\_1\; u\_1)\backslash frac\{\backslash partial\}\{\backslash partial\; u\_1\}\; +\; 3u\_1\; u\_2\backslash frac\{\backslash partial\}\{\backslash partial\; u\_2\}.$

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).

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Infinite jet spaces The inverse limit of the sequence of projections $\backslash pi\_\{k+1,k\}:J^\{k+1\}(\backslash pi)\backslash to\; J^k(\backslash pi)$ gives rise to the infinite jet space J^∞(π). A point $j\_p^\backslash infty(\backslash sigma)$ 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 $j\_p^\backslash infty(\backslash sigma)$ 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 $\backslash pi\_\{k+1,k\}:\; J^\{k+1\}(\backslash pi)\; \backslash to\; J^k(\backslash pi)$ of manifolds is the sequence of injections $\backslash pi\_\{k+1,k\}^*:\; C^\backslash infty(J^\{k\}(\backslash pi))\; \backslash to\; C^\backslash infty\backslash left(J^\{k+1\}(\backslash pi)\backslash right)$ of commutative algebras. Let's denote $C^\backslash infty(J^\{k\}(\backslash pi))$ simply by $\backslash mathcal\{F\}\_k(\backslash pi)$. Take now the direct limit $\backslash mathcal\{F\}(\backslash pi)$ of the $\backslash mathcal\{F\}\_k(\backslash pi)$'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 $\backslash mathcal\{F\}(\backslash pi)$, being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.

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

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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 $\backslash mathcal\{F\}\_k(\backslash pi)$, and hence in the direct limit $\backslash mathcal\{F\}(\backslash pi)$ too.

Enhance I(E) by adding all the possible compositions of applied to all its elements. This way we get a new ideal I of $\backslash mathcal\{F\}(\backslash pi)$ 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 $j\_p^\backslash infty(\backslash sigma)$ 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 $j\_p^k(\backslash sigma)$ 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 $\backslash varphi\backslash circ\; j^k(\backslash sigma)$ at the point p.

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

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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.

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See also

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

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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. , .
• Saunders, D. J., "The Geometry of Jet Bundles", Cambridge University Press, 1989,
• Krasil'shchik, I. S., Vinogradov, A. M., et, "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, .
• Olver, P. J., "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995,

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