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| In [[differential geometry]], the '''jet bundle''' is a certain construction which makes a new [[smooth manifold|smooth]] [[fiber bundle]] out of a given smooth fiber bundle. It makes it possible to write [[differential equation]]s on [[Fiber bundle#Sections|section]]s of a fiber bundle in an invariant form. Jets may also be seen as the coordinate free versions of [[Taylor expansions]].
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| Historically, jet bundles are attributed to [[Ehresmann]], and were an advance on the method ([[prolongation (mathematics)|prolongation]]) of [[Élie Cartan]], of dealing ''geometrically'' with [[derivative|higher derivatives]], by imposing [[differential form]] conditions on newly introduced formal variables. Jet bundles are sometimes called '''sprays''', although [[spray (mathematics)|sprays]] usually refer more specifically to the associated vector field induced on the corresponding bundle (''e.g.'', the [[geodesic spray]] on [[Finsler manifold]]s.)
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| 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 [[covariant classical field theory|geometrical covariant field theory]] and much work is done in [[general relativity|general relativistic]] formulations of fields using this approach.
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| ==Jets==
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| {{main|Jet (mathematics)}}
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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
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| :<math>|I| := \sum_{i=1}^{m} I(i)</math>
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| :<math>\frac{\partial^{|I|}}{\partial x^{I}} := \prod_{i=1}^{m} \left( \frac{\partial}{\partial x^{i}} \right)^{I(i)}.</math>
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| Define the local sections σ, η ∈ Γ(π) to have the same '''''r''-jet''' at ''p'' if
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| :<math>\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. </math>
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| 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 <math>j^r_p\sigma </math>. 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==
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| The '''''r''-th jet manifold of π''' is the set
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| :<math>\{j^{r}_{p}\sigma:p \in M, \sigma \in \Gamma(\pi)\}</math>
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| and is denoted ''J<sup>r</sup>(π)''. We may define projections π<sub>''r''</sub> and π<sub>''r'',0</sub> called the '''source and target projections''' respectively, by
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| :<math>\begin{cases} \pi_r: J^{r}(\pi) \to M \\ j^{r}_{p}\sigma \mapsto p \end{cases} </math>
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| :<math>\begin{cases} \pi_{r, 0}: J^{r}(\pi) \to E \\ j^{r}_{p}\sigma \mapsto \sigma(p) \end{cases}</math>
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| If 1 ≤ ''k'' ≤ ''r'', then the '''''k''-jet projection''' is the function π<sub>''r,k''</sub> defined by
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| :<math>\begin{cases} \pi_{r, k}: J^{r}(\pi) \to J^{k}(\pi)\\ j^{r}_{p}\sigma \mapsto j^{k}_{p}\sigma \end{cases}</math>
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| From this definition, it is clear that π<sub>''r''</sub> = π <small> o </small> π<sub>''r'',0</sub> and that if 0 ≤ ''m'' ≤ ''k'', then π<sub>''r,m''</sub> = π<sub>''k,m''</sub> <small> o </small> π<sub>''r,k''</sub>. It is conventional to regard π<sub>''r,r''</sub> = id<sub>''J<sup>r</sup>(π)</sub>, the [[identity function|identity map]] on ''J<sup>r</sup>(π)'' and to identify ''J<sup>0</sup>(π)'' with ''E''.
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| The functions π<sub>''r,k''</sub>, π<sub>''r'',0</sub> and π<sub>''r''</sub> are [[Smooth function#Smoothness|smooth]] [[surjective]] [[submersion (mathematics)|submersion]]s.
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| [[File:Jet Bundle Image FbN.png|500px|center]]
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| A [[coordinate system]] on ''E'' will generate a coordinate system on ''J<sup>r</sup>(π)''. Let ''(U, u)'' be an adapted [[coordinate chart]] on ''E'', where ''u'' = ''(x<sup>i</sup>, u<sup>α</sup>). The '''induced coordinate chart ''(U<sup>r</sup>, u<sup>r</sup>)''''' on ''J<sup>r</sup>(π)'' is defined by
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| :<math> U^{r} = \{ j^{r}_{p}\sigma: \sigma(p) \in U \} \,</math>
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| :<math> u^{r} = (x^{i}, u^{\alpha}, u^{\alpha}_{I})\,</math>
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| where
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| :<math>x^{i}(j^{r}_{p}\sigma) = x^{i}(p)</math>
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| :<math>u^{\alpha}(j^{r}_{p}\sigma) = u^{\alpha}(\sigma(p))</math>
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| and the <math>n \left( {}^{m+r}C_{r} -1\right)\,</math> functions
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| :<math>u^{\alpha}_{I}:U^{k} \to \mathbf{R}\,</math>
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| are specified by
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| :<math>u^{\alpha}_{I}(j^{r}_{p}\sigma) = \left.\frac{\partial^{|I|} \sigma^{\alpha}}{\partial x^{I}}\right|_{p}</math>
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| and are known as the '''derivative coordinates'''.
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| Given an atlas of adapted charts ''(U, u)'' on ''E'', the corresponding collection of charts ''(U<sup>r</sup>, u<sup>r</sup>)'' is a [[finite-dimensional]] ''C''<sup>∞</sup> atlas on ''J<sup>r</sup>(π)''.
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| ==Jet bundles==
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| Since the atlas on each ''J<sup>r</sup>(π)'' defines a manifold, the triples ''(J<sup>r</sup>(π), π<sub>r,k</sub>, J<sup>k</sup>(π))'', ''(J<sup>r</sup>(π), π<sub>r,0</sub>, E)'' and ''(J<sup>r</sup>(π), π<sub>r</sub>, M)'' all define fibered manifolds. In particular, if ''(E, π, M)'' is a fiber bundle, the triple ''(J<sup>r</sup>(π), π<sub>r</sub>, M)'' defines the '''''r''-th jet bundle of π'''.
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| If ''W'' ⊂ ''M'' is an open submanifold, then
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| :<math> J^{r}\left(\pi|_{\pi^{-1}(W)}\right) \cong \pi^{-1}_{r}(W).\,</math>
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| If ''p'' ∈ ''M'', then the fiber <math>\pi^{-1}_{r}(p)\,</math> is denoted <math>J^{r}_{p}(\pi)</math>.
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| Let σ be a local section of π with domain ''W'' ⊂ ''M''. The '''''r''-th jet prolongation of σ''' is the map ''j<sup>r</sup>σ'': ''W'' → ''J<sup>r</sup>(π)'' defined by
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| :<math> (j^{r}\sigma)(p) = j^{r}_{p}\sigma. \,</math>
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| Note that π<sub>''r''</sub> <small> o </small> ''j<sup>r</sup>σ'' = id<sub>''W''</sub>, so ''j<sup>r</sup>σ'' really is a section. In local coordinates, ''j<sup>r</sup>σ'' is given by
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| :<math> \left(\sigma^{\alpha}, \frac{\partial^{|I|} \sigma^{\alpha}}{\partial x^{|I|}}\right) \qquad 1 \leq |I| \leq r. \,</math>
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| We identify ''j<sup>0</sup>σ'' with σ.
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| ===Example===
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| If π is the [[trivial bundle]] (''M'' × '''R''', pr<sub>1</sub>, ''M''), then there is a canonical [[diffeomorphism]] between the first jet bundle ''J<sup>1</sup>(π)'' and ''T*M'' × '''R'''. To construct this diffeomorphism, for each σ in Γ<sub>''M''</sub>(π) write <math>\bar{\sigma} = pr_{2} \circ \sigma \in C^{\infty}(M)\,</math>.
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| Then, whenever ''p'' ∈ ''M''
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| :<math> j^{1}_{p}\sigma = \{ \psi : \psi \in \Gamma_{p}(\pi); \bar{\psi}(p) = \bar{\sigma}(p); d\bar{\psi}_{p} = d\bar{\sigma}_{p} \}. \,</math>
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| Consequently, the mapping
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| :<math>\begin{cases} J^{1}(\pi) \to T^*M \times \mathbf{R} \\ j^{1}_{p}\sigma \mapsto (d\bar{\sigma}_{p},\bar{\sigma}(p)) \end{cases}</math>
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| is well-defined and is clearly [[injective]]. Writing it out in coordinates shows that it is a diffeomorphism, because if ''(x<sup>i</sup>, u)'' are coordinates on ''M'' × '''R''', where ''u'' = id<sub>'''R'''</sub> is the identity coordinate, then the derivative coordinates ''u<sub>i</sub>'' on ''J<sup>1</sup>(π)'' correspond to the coordinates ∂<sub>''i''</sub> on ''T*M''.
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| Likewise, if π is the trivial bundle ('''R''' × ''M'', pr<sub>1</sub>, '''R'''), then there exists a canonical diffeomorphism between ''J<sup>1</sup>(π)'' and '''R''' × ''TM''.
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| ==Contact forms==
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| A [[differential 1-form]] θ on the space ''J<sup>r</sup>(π)'' is called a '''[[contact form]]''' (i.e. <math>\theta \in \Lambda_{C}^{r}\pi\,</math>) if it is [[pullback (differential geometry)|pulled back]] to the zero form on ''M'' by all prolongations. In other words, if <math>\theta \in \Lambda^{1}J^{r+1}\pi\,</math>, then <math>\theta \in \Lambda_{C}^{1}\pi_{r+1,r}\,</math> [[if and only if]], for every open submanifold ''W'' ⊂ ''M'' and every σ in Γ<sub>''M''</sub>(π)
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| :<math>(j^{k+1}\sigma)^{*}\theta = 0.\,</math>
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| The [[distribution (differential geometry)|distribution]] on ''J<sup>r</sup>(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 equation]]s. The Cartan distributions are not [[distribution (differential geometry)|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<sup>∞</sup>'' this distribution is involutive and finite dimensional. Its dimension coinciding with the dimension of the base manifold ''M''.
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| ===Example===
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| Let us consider the case ''(E, π, M)'', where ''E'' ≃ '''R'''<sup>2</sup> and ''M'' ≃ '''R'''. Then, ''(J<sup>1</sup>(π), π, M)'' defines the first jet bundle, and may be coordinated by ''(x, u, u<sub>1</sub>)'', where
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| :{|
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| |-
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| |align=right|<math>x(j^{1}_{p}\sigma) </math>
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| |align=left|<math>= x(p) = x\,</math>
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| |-
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| |align=right|<math>u(j^{1}_{p}\sigma) </math>
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| |align=left|<math>= u(\sigma(p)) = u(\sigma(x)) = \sigma(x) \,</math>
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| |-
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| |align=right|<math>u_{1}(j^{1}_{p}\sigma) </math>
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| |align=left|<math>= \left.\frac{\partial \sigma}{\partial x}\right|_{p} = \sigma'(x)</math>
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| |-
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| |}
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| for all ''p'' ∈ ''M'' and σ in Γ<sub>''p''</sub>(π). A general 1-form on ''J<sup>1</sup>(π)'' takes the form
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| :<math>\theta = a(x, u, u_{1})dx + b(x, u, u_{1})du + c(x, u,u_{1})du_{1}\,</math>
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| A section σ in Γ<sub>''p''</sub>(π) has first prolongation
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| :<math> j^{1}\sigma = (u,u_{1}) = \left(\sigma(p), \left. \frac{\partial \sigma}{\partial x} \right|_{p} \right).</math>
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| Hence, ''(j<sup>1</sup>σ)*θ'' can be calculated as
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| :{|
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| |<math>(j^{1}_{p}\sigma)^{*} \theta \,</math>
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| |<math>= \theta \circ j^{1}_{p}\sigma \, </math>
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| |<math>= 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)) \,</math>
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| |<math>= 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 \,</math>
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| |<math>= [\, a(x, \sigma(x), \sigma'(x)) + b(x, \sigma(x), \sigma'(x))\sigma'(x) + c(x, \sigma(x),\sigma'(x))\sigma''(x)\, ]dx \, </math>
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| |}
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| This will vanish for all sections σ if and only if ''c'' = 0 and ''a'' = −''bσ′(x)''. Hence, θ = ''b(x, u, u<sub>1</sub>)θ<sub>0</sub>'' must necessarily be a multiple of the basic contact form θ<sub>0</sub> = ''du'' − ''u<sub>1</sub>dx''. Proceeding to the second jet space ''J<sup>2</sup>(π)'' with additional coordinate ''u<sub>2</sub>'', such that
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| :<math>u_{2}(j^{2}_{p}\sigma)=\left.\frac{\partial^{2} \sigma}{\partial x^{2}}\right|_{p} = \sigma''(x)\,</math>
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| a general 1-form has the construction
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| :<math> \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}\,</math>
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| This is a contact form [[if and only if]]
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| :{|
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| |<math> (j^{2}_{p}\sigma)^{*} \theta \,</math>
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| |<math>= \theta \circ j^{2}_{p}\sigma \,</math>
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| |<math>= a(x, \sigma(x), \sigma'(x),\sigma''(x))dx + b(x, \sigma(x),\sigma'(x),\sigma''(x))d(\sigma(x))+ \,</math>
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| | <math>+ c(x, \sigma(x),\sigma'(x),\sigma'(x))d(\sigma'(x)) + e(x, \sigma(x), \sigma'(x),\sigma''(x))d(\sigma''(x)) \,</math>
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| |<math>= adx + b\sigma'(x)dx + c\sigma''(x)dx + e\sigma'''(x)dx\,</math>
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| |<math>= [\, a + b\sigma'(x) + c\sigma''(x) + e\sigma'''(x)\,]dx\,</math>
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| |<math>= 0\,</math>
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| |-
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| |}
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| which implies that ''e'' = 0 and ''a'' = −''bσ′(x)'' − ''cσ′′(x)''. Therefore, θ is a contact form if and only if
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| :<math>\theta = b(x, \sigma(x), \sigma'(x))\theta_{0} + c(x, \sigma(x), \sigma'(x))\theta_{1}\,</math>
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| where θ<sub>1</sub> = ''du<sub>1</sub> − u<sub>2</sub>dx'' is the next basic contact form (Note that here we are identifying the form θ<sub>0</sub> with its pull-back <math>(\pi_{2,1})^{*}\theta_{0}\,</math> to ''J<sup>2</sup>(π)'').
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| In general, providing ''x, u'' ∈ '''R''', a contact form on ''J<sup>r+1</sup>(π)'' can be written as a [[linear combination]] of the basic contact forms
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| :<math>\theta_{k} = du_{k} - u_{k+1}dx \qquad k=0, \ldots, r-1\,</math>
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| where <math> u_{k}(j^{k}\sigma)= \left.\frac{\partial^{k} \sigma}{\partial x^{k}}\right|_{p}\,</math>.
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| Similar arguments lead to a complete characterization of all contact forms.
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| In local coordinates, every contact one-form on ''J<sup>r+1</sup>(π)'' can be written as a linear combination
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| :<math>\theta = \sum_{|I|=0}^{r} P_{\alpha}^{I}\theta_{I}^{\alpha}\,</math>
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| with smooth coefficients <math>P^{\alpha}_{I}(x^{i},u^{\alpha})\,</math> of the basic contact forms
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| :<math>\theta_{I}^{\alpha} = du^{\alpha}_{I} - u^{\alpha}_{I,i}dx^{i}\,</math>
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| ''|I|'' is known as the '''order''' of the contact form <math>\theta_{I}^{\alpha}</math>. Note that contact forms on ''J<sup>r+1</sup>(π)'' have orders at most ''r''. Contact forms provide a characterization of those local sections of ''π<sub>r+1</sub>'' which are prolongations of sections of π.
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| Let ψ ∈ Γ<sub>''W''</sub>(''π<sub>r+1</sub>''), then ψ = ''j<sup>r+1</sup>''σ where σ ∈ Γ<sub>''W''</sub>(π) if and only if <math>\psi^{*}(\theta|_{W})=0, \forall \theta \in \Lambda_{C}^{1}\pi_{r+1,r}.\,</math>
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| ==Vector fields==
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| A general [[vector field]] on the total space ''E'', coordinated by <math>(x,u) \ \stackrel{\mathrm{def}}{=}\ (x^{i},u^{\alpha})\,</math>, is
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| :<math>V \ \stackrel{\mathrm{def}}{=}\ \rho^{i}(x,u)\frac{\partial}{\partial x^{i}} + \phi^{\alpha}(x,u)\frac{\partial}{\partial u^{\alpha}}.\,</math> | |
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| A vector field is called '''horizontal''', meaning that all the vertical coefficients vanish, i.e. ''φ<sup>α</sup>'' = 0.
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| A vector field is called '''vertical''', meaning that all the horizontal coefficients vanish, i.e. ''ρ<sup>i</sup>'' = 0.
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| For fixed ''(x, u)'', we identify
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| :<math> V_{(xu)} \ \stackrel{\mathrm{def}}{=}\ \rho^{i}(x,u) \frac{\partial}{\partial x^{i}} + \phi^{\alpha}(x,u) \frac{\partial}{\partial u^{\alpha}}\,</math>
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| having coordinates ''(x, u, ρ<sup>i</sup>, φ<sup>α</sup>)'', with an element in the fiber ''T<sub>xu</sub>E'' of ''TE'' over ''(x,u)'' in ''E'', called '''a [[tangent vector]] in ''TE'''''. A section
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| :<math> \begin{cases} \psi : E \to TE \\ (x,u) \mapsto \psi(x,u) = V \end{cases}</math>
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| is called '''a vector field on ''E'''''' with
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| :<math> V = \rho^{i}(x,u) \frac{\partial}{\partial x^{i}} + \phi^{\alpha}(x,u) \frac{\partial}{\partial u^{\alpha}}\,</math>
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| and ψ in ''Γ(TE)''.
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| The jet bundle ''J<sup>r</sup>(π)'' is coordinated by <math>(x,u,w) \ \stackrel{\mathrm{def}}{=}\ (x^{i},u^{\alpha},w_{i}^{\alpha})\,</math>. For fixed ''(x,u,w)'', identify
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| :{|
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| |<math>V_{(xuw)} \ \stackrel{\mathrm{def}}{=}\ \,</math>
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| |<math>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}} +\,</math>
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| |<math>\qquad + \ V^{\alpha}_{i_{1}i_{2}}(x,u,w) \frac{\partial}{\partial w^{\alpha}_{i_{1}i_{2}}} + \cdots \ + \ \cdots + V^{\alpha}_{i_{1}i_{2} \cdots i_{r}}(x,u,w) \frac{\partial}{\partial w^{\alpha}_{i_{1}i_{2} \cdots i_{r}}}\,</math>
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| |}
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| having coordinates <math>(x,u,w,v^{\alpha}_{i}, v^{\alpha}_{i_{1} i_{2}},\ldots,v^{ \alpha}_{i_{1}i_{2} \cdots i_{r}})\,</math>, with an element in the fiber <math>T_{xuw}(J^{r}\pi)\,</math> of ''TJ<sup>r</sup>(π)'' over ''(x, u, w)'' ∈ ''J<sup>r</sup>(π)'', called '''a tangent vector in ''TJ<sup>r</sup>(π)'''''. Here,
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| :<math>v^{\alpha}_{i}, v^{\alpha}_{i_{1}i_{2}},\ldots,v^{\alpha}_{i_{1}i_{2} \cdots i_{r}}\,</math>
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| are real-valued functions on ''J<sup>r</sup>(π)''. A section | |
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| :<math> \begin{cases} \Psi : J^{r}(\pi) \to TJ^{r}(\pi) \\ (x,u,w) \mapsto \Psi(u,w) = V \end{cases} </math>
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| is '''a vector field on ''J<sup>r</sup>(π)''''', and we say <math>\Psi \in \Gamma(T(J^{r}\pi))\,</math>.
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| ==Partial differential equations==
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| Let ''(E, π, M)'' be a fiber bundle. An '''''r''-th order [[partial differential equation]]''' on π is a [[closed manifold|closed]] [[embedding|embedded]] submanifold ''S'' of the jet manifold ''J<sup>r</sup>(π)''. A solution is a local section σ ∈ Γ<sub>''W''</sub>(π) satisfying <math>j^{r}_{p}\sigma \in S</math>, forall ''p'' in ''M''.
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| Let us consider an example of a first order partial differential equation.
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| ===Example===
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| Let π be the trivial bundle ('''R'''<sup>2</sup> × '''R''', pr<sub>1</sub>, '''R'''<sup>2</sup>) with global coordinates ''(x<sup>1</sup>, x<sup>2</sup>, u<sup>1</sup>)''. Then the map ''F : ''J<sup>1</sup>(π)'' → '''R''' defined by
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| :<math>F = u^{1}_{1}u^{1}_{2} - 2x^{2}u^{1}\,</math>
| |
| | |
| gives rise to the differential equation
| |
| | |
| :<math>S = \{ j^{1}_{p}\sigma \in J^{1}\pi : (u^{1}_{1}u^{1}_{2} - 2x^{2}u^{1})(j^{1}_{p}\sigma)=0 \} \,</math>
| |
| | |
| which can be written
| |
| | |
| :<math>\frac{\partial \sigma}{\partial x^{1}}\frac{\partial \sigma}{\partial x^{2}} - 2x^{2}\sigma = 0. \,</math>
| |
| | |
| The particular section σ: '''R'''<sup>2</sup> → '''R'''<sup>2</sup> × '''R''' defined by
| |
| | |
| :<math>\sigma(p_{1},p_{2}) = (p^{1},p^{2},p^{1}(p^{2})^{2}) \,</math>
| |
| | |
| has first prolongation given by
| |
| | |
| :<math> j^{1}\sigma(p_{1},p_{2}) = (p^{1},p^{2},p^{1}(p^{2})^{2},(p^{2})^{2},2p^{1}p^{2}) \,</math>
| |
| | |
| and is a solution of this differential equation, because
| |
| | |
| :{|
| |
| |-
| |
| |<math>(u^{1}_{1}u^{1}_{2} - 2x^{2}u^{1})(j^{1}_{p}\sigma) \,</math>
| |
| |<math>= u^{1}_{1}(j^{1}_{p}\sigma)u^{1}_{2}(j^{1}_{p}\sigma) - 2x^{2}(j^{1}_{p}\sigma)u^{1}(j^{1}_{p}\sigma) \,</math>
| |
| |-
| |
| |
| |
| |<math>= (p^{2})^{2} \cdot 2p^{1}p^{2} - 2 \cdot p^{2} \cdot p^{1}(p^{2})^{2} \,</math>
| |
| |-
| |
| |
| |
| |<math>= 2p^{1}(p^{2})^3 - 2p^{1}(p^{2})^3 \,</math>
| |
| |-
| |
| |
| |
| |<math>= 0 \,</math>
| |
| |-
| |
| |}
| |
| | |
| and so <math>j^{1}_{p}\sigma \in S</math> for ''every'' ''p'' ∈ '''R'''<sup>2</sup>.
| |
| | |
| ==Jet Prolongation==
| |
| A local diffeomorphism ψ: ''J<sup>r</sup>(π)'' → ''J<sup>r</sup>(π)'' defines a contact transformation of order ''r'' if it preserves the contact ideal, meaning that if θ is any contact form on ''J<sup>r</sup>(π)'', then ψ*θ is also a contact form.
| |
| | |
| The flow generated by a vector field ''V<sup>r</sup>'' on the jet space ''J<sup>r</sup>(π)'' forms a one-parameter group of contact transformations if and only if the [[Lie derivative]] <math>\mathcal{L}_{V^{r}}(\theta)</math> of any contact form θ preserves the contact ideal.
| |
| | |
| Let us begin with the first order case. Consider a general vector field ''V<sup>1</sup>'' on ''J<sup>1</sup>(π)'', given by
| |
| | |
| :<math>V^1 \ \stackrel{\mathrm{def}}{=}\ \rho^{i}(u^{1})\frac{\partial}{\partial x^{i}} + \phi^{\alpha}(u^{1})\frac{\partial}{\partial u^{\alpha}} + \chi^{\alpha}_{i}(u^{1})\frac{\partial}{\partial u^{\alpha}_{i}}.</math>
| |
| | |
| We now apply <math>\mathcal{L}_{V^{1}}</math> to the basic contact forms <math>\theta^{\alpha} = du^{\alpha} - u_{i}^{\alpha}dx^{i}\,</math>, and obtain
| |
| | |
| :{|
| |
| |-
| |
| |<math>\mathcal{L}_{V^{1}}(\theta^{\alpha}) </math>
| |
| |<math>= \mathcal{L}_{V^{1}}(du^{\alpha} - u_{i}^{\alpha}dx^{i}) </math>
| |
| |-
| |
| |
| |
| |<math>= \mathcal{L}_{V^{1}}du^{\alpha} - (\mathcal{L}_{V^{1}}u_{i}^{\alpha})dx^{i} - u_{i}^{\alpha}(\mathcal{L}_{V^{1}}dx^{i}) \,</math>
| |
| |-
| |
| |
| |
| |<math>= d(V^{1}u^{\alpha}) - V^{1}u_{i}^{\alpha}dx^{i} - u_{i}^{\alpha}d(V^{1}x^{i}) \,</math>
| |
| |-
| |
| |
| |
| |<math>= d\phi^{\alpha} - \chi^{\alpha}_{i}dx^{i} - u_{i}^{\alpha}d\rho^{i} \,</math>
| |
| |-
| |
| |
| |
| |<math>= \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 ] \,</math>
| |
| |-
| |
| |}
| |
| | |
| where we have expanded the [[exterior derivative]] of the functions in terms of their coordinates. Next, we note that
| |
| | |
| :<math> \theta^{k} = du^{k} - u_{i}^{k}dx^{i} \quad \Longrightarrow \quad du^{k} = \theta^{k} + u_{i}^{k}dx^{i} \,</math>
| |
| | |
| and so we may write
| |
| | |
| :{|
| |
| |-
| |
| |<math>\mathcal{L}_{V^{1}}(\theta^{\alpha}) \,</math>
| |
| |<math>= \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^{k}_{i}}\, du^{k}_{i} - \chi^{\alpha}_{i}dx^{i} - \,</math>
| |
| |-
| |
| |
| |
| |<math>-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^{k}_{i}}\, du^{k}_{i} \right ] \,</math>
| |
| |-
| |
| |
| |
| |<math>= \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} + \,</math>
| |
| |-
| |
| |
| |
| | <math>+ \left( \frac{\partial \phi^{\alpha}}{\partial u^{k}} - u_{l}^{\alpha}\frac{\partial \rho^{l}}{\partial u^{k}} \right)\theta^{k}.\,</math>
| |
| |-
| |
| |}
| |
| | |
| Therefore, ''V<sup>1</sup>'' determines a contact transformation if and only if the coefficients of ''dx<sup>i</sup>'' and <math>du^{k}_{i}\,</math> in the formula vanish. The latter requirements imply the '''contact conditions'''
| |
| | |
| :<math>\frac{\partial \phi^{\alpha}}{\partial u^{k}_{i}} - u^{\alpha}_{l} \frac{\partial \rho^{l}}{\partial u^{k}_{i}} = 0\,</math>
| |
| | |
| The former requirements provide explicit formulae for the coefficients of the first derivative terms in ''V<sup>1</sup>'':
| |
| | |
| :<math>\chi^{\alpha}_{i} = \widehat{D}_{i} \phi^{\alpha} - u^{\alpha}_{l}(\widehat{D}_{i}\rho^{l})</math>
| |
| | |
| where
| |
| | |
| :<math>\widehat{D}_{i} = \frac{\partial}{\partial x^{i}} + u^{k}_{i}\frac{\partial}{\partial u^{k}} </math>
| |
| | |
| denotes the zeroth order truncation of the total derivative ''D<sub>i</sub>''.
| |
| | |
| Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if <math>\mathcal{L}_{V^{r}}\,</math> satisfies these equations, ''V<sup>r</sup>'' is called the '''''r''-th prolongation of ''V'' to a vector field on ''J<sup>r</sup>(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'' ≅ '''R'''<sup>2</sup> and ''M'' ≃ '''R'''. Then, ''(J<sup>1</sup>(π), π, E)'' defines the first jet bundle, and may be coordinated by ''(x, u, u<sub>1</sub>)'', where
| |
| | |
| :{|
| |
| |-
| |
| |align=right|<math>x(j^{1}_{p}\sigma) \,</math>
| |
| |<math>= x(p) = x \,</math>
| |
| |-
| |
| |align=right|<math>u(j^{1}_{p}\sigma) \,</math>
| |
| |<math>= u(\sigma(p)) = u(\sigma(x)) = \sigma(x) \,</math>
| |
| |-
| |
| |align=right|<math>u_{1}(j^{1}_{p}\sigma) \,</math>
| |
| |<math>= \left.\frac{\partial \sigma}{\partial x}\right|_{p} = \dot{\sigma}(x) \,</math>
| |
| |-
| |
| |}
| |
| | |
| for all ''p'' ∈ ''M'' and σ in Γ<sub>''p''</sub>(π). A contact form on ''J<sup>1</sup>(π)'' has the form
| |
| | |
| :<math>\theta = du - u_{1}dx \,</math>
| |
| | |
| Let us consider a vector ''V'' on ''E'', having the form
| |
| | |
| :<math>V = x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} \,</math>
| |
| | |
| Then, the first prolongation of this vector field to ''J<sup>1</sup>(π)'' is
| |
| | |
| :{|
| |
| |-
| |
| |<math>V^{1} \,</math>
| |
| |<math>= V + Z \,</math>
| |
| |-
| |
| |
| |
| |<math>= x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + Z \,</math>
| |
| |-
| |
| |
| |
| |<math>= x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + \rho(x,u,u_{1})\frac{\partial}{\partial u_{1}} \,</math>
| |
| |-
| |
| |}
| |
| | |
| If we now take the Lie derivative of the contact form with respect to this prolonged vector field, <math>\mathcal{L}_{V^{1}}(\theta)\,</math>, we obtain
| |
| | |
| :{|
| |
| |-
| |
| |<math>\mathcal{L}_{V^{1}}(\theta) \,</math>
| |
| |<math>= \mathcal{L}_{V^{1}}(du - u_{1}dx) \,</math>
| |
| |-
| |
| |
| |
| |<math>= \mathcal{L}_{V^{1}}du - (\mathcal{L}_{V^{1}}u_{1})dx - u_{1}(\mathcal{L}_{V^{1}}dx) \,</math>
| |
| |-
| |
| |
| |
| |<math>= d(V^{1}u) - V^{1}u_{1}dx - u_{1}d(V^{1}x) \,</math>
| |
| |-
| |
| |
| |
| |<math>= dx - \rho(x,u,u_{1})dx + u_{1}du \,</math>
| |
| |-
| |
| |
| |
| |<math>= (1 - \rho(x,u,u_{1}) )dx + u_{1}du \,</math>
| |
| |-
| |
| |}
| |
| | |
| But, we may identify ''du'' = θ + ''u<sub>1</sub>dx''. Thus, we get
| |
| | |
| :{|
| |
| |-
| |
| |<math>\mathcal{L}_{V^{1}}(\theta) \,</math>
| |
| |<math>= [\,1 - \rho(x,u,u_{1})\,]dx + u_{1}(\theta + u_{1}dx) \,</math>
| |
| |-
| |
| |
| |
| |<math>= [\,1 + u_{1}u_{1} - \rho(x,u,u_{1})\,]dx + u_{1}\theta \,</math>
| |
| |-
| |
| |}
| |
| | |
| Hence, for <math>\mathcal{L}_{V^{1}}(\theta)\,</math> to preserve the contact ideal, we require
| |
| | |
| :{|
| |
| |-
| |
| |
| |
| |<math>1 + u_{1}u_{1} - \rho(x,u,u_{1}) = 0 \,</math>
| |
| |-
| |
| |<math>\Longrightarrow \quad \,</math>
| |
| |<math>\rho(x,u,u_{1}) = 1 + u_{1}u_{1}\,</math>
| |
| |-
| |
| |}
| |
| | |
| And so the first prolongation of ''V'' to a vector field on ''J<sup>1</sup>(π)'' is
| |
| | |
| :<math> V^{1} = x \frac{\partial}{\partial u} - u \frac{\partial}{\partial x} + (1 + u_{1}u_{1})\frac{\partial}{\partial u_{1}} \,</math>
| |
| | |
| Let us also calculate the second prolongation of ''V'' to a vector field on ''J<sup>2</sup>(π)''. We have <math>\{x,u,u_{1}, y_{2}\}\,</math> as coordinates on ''J<sup>2</sup>(π)''. Hence, the prolonged vector has the form
| |
| | |
| :<math> 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}} \,</math>
| |
| | |
| The contacts forms are
| |
| | |
| :{|
| |
| |-
| |
| |align=right|<math>\theta \,</math>
| |
| |<math>= du - u_{1}dx \,</math>
| |
| |-
| |
| |<math>\theta_{1} \,</math>
| |
| |<math>= du_{1} - u_{2}dx \,</math>
| |
| |-
| |
| |}
| |
| | |
| To preserve the contact ideal, we require
| |
| | |
| :{|
| |
| |-
| |
| |align=right|<math>\mathcal{L}_{V^{2}}(\theta) \,</math>
| |
| |<math>= 0\,</math>
| |
| |-
| |
| |<math>\mathcal{L}_{V^{2}}(\theta_{1}) \,</math>
| |
| |<math>= 0 \,</math>
| |
| |-
| |
| |}
| |
| | |
| Now, θ has no ''u<sub>2</sub>'' dependency. Hence, from this equation we will pick up the formula for ρ, which will necessarily be the same result as we found for ''V<sup>1</sup>''. Therefore, the problem is analogous to prolonging the vector field ''V<sup>1</sup>'' to ''J<sup>2</sup>(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
| |
| | |
| :<math> \rho(x,u,u_{1}) = 1 + u_{1}u_{1} \,</math>
| |
| | |
| and so
| |
| | |
| :{|
| |
| |-
| |
| |<math>V^{2} \,</math>
| |
| |<math>= V^{1} + \phi(x,u,u_{1},u_{2})\frac{\partial}{\partial u_{2}} \,</math>
| |
| |-
| |
| |
| |
| |<math>= 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}} \,</math>
| |
| |-
| |
| |}
| |
| | |
| Therefore, the Lie derivative of the second contact form with respect to ''V<sup>2</sup>'' is
| |
| | |
| :{|
| |
| |-
| |
| |<math>\mathcal{L}_{V^{2}}(\theta_{1}) \,</math>
| |
| |<math>= \mathcal{L}_{V^{2}}(du_{1} - u_{2}dx) \,</math>
| |
| |-
| |
| |
| |
| |<math>= \mathcal{L}_{V^{2}}du_{1} - (\mathcal{L}_{V^{2}}u_{2})dx - u_{2}(\mathcal{L}_{V^{2}}dx) \,</math>
| |
| |-
| |
| |
| |
| |<math>= d(V^{2}u_{1}) - V^{2}u_{2}dx - u_{2}d(V^{2}x) \,</math>
| |
| |-
| |
| |
| |
| |<math>= d(1-u_{1}u_{1}) - \phi(x,u,u_{1},u_{2})dx + u_{2}du \,</math>
| |
| |-
| |
| |
| |
| |<math>= 2u_{1}du_{1} - \phi(x,u,u_{1},u_{2})dx + u_{2}du \,</math>
| |
| |-
| |
| |}
| |
| | |
| Again, let us identify ''du'' = θ + ''u<sub>1</sub>dx'' and ''du<sub>1</sub>'' = θ<sub>1</sub> + ''u<sub>2</sub>dx''. Then we have
| |
| | |
| :{|
| |
| |-
| |
| |<math>\mathcal{L}_{V^{2}}(\theta_{1}) \,</math>
| |
| |<math>= 2u_{1}(\theta_{1} + u_{2}dx) - \phi(x,u,u_{1},u_{2})dx + u_{2}(\theta + u_{1}dx) \,</math>
| |
| |-
| |
| |
| |
| |<math>= [\, 3u_{1}u_{2} - \phi(x,u,u_{1},u_{2})\,]dx + u_{2}\theta + 2u_{1}\theta_{1} \,</math>
| |
| |-
| |
| |}
| |
| | |
| Hence, for <math>\mathcal{L}_{V^{2}}(\theta_{1})\,</math> to preserve the contact ideal, we require
| |
| | |
| :{|
| |
| |-
| |
| |
| |
| |<math>3u_{1}u_{2} - \phi(x,u,u_{1},u_{2}) = 0 \,</math>
| |
| |-
| |
| |<math>\Longrightarrow \quad \,</math>
| |
| |<math>\phi(x,u,u_{1},u_{2}) = 3u_{1}u_{2} \,</math>
| |
| |-
| |
| |}
| |
| | |
| And so the second prolongation of ''V'' to a vector field on ''J<sup>2</sup>(π)'' is
| |
| | |
| :<math> 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}} \, </math>
| |
| | |
| Note that the first prolongation of ''V'' can be recovered by omitting the second derivative terms in ''V<sup>2</sup>'', or by projecting back to ''J<sup>1</sup>(π)''.
| |
| | |
| ==Infinite Jet Spaces==
| |
| The [[inverse limit]] of the sequence of projections <math>\pi_{k+1,k}:J^{k+1}(\pi)\to J^k(\pi)</math> gives rise to the '''infinite jet space''' ''J<sup>∞</sup>(π)''. A point <math>j_p^\infty(\sigma)</math> 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<sub>∞</sub> maps <math>j_p^\infty(\sigma)</math> into ''p''. | |
| | |
| Just by thinking in terms of coordinates, ''J<sup>∞</sup>(π)'' appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure on ''J<sup>∞</sup>(π)'', not relying on differentiable charts, is given by the [[differential calculus over commutative algebras]]. Dual to the sequence of projections <math>\pi_{k+1,k}:J^{k+1}(\pi)\to J^k(\pi)</math> of manifolds is the sequence of injections <math>\pi_{k+1,k}^*: C^\infty(J^{k}(\pi))\to C^\infty(J^{k+1}(\pi))</math> of commutative algebras. Let's denote <math>C^\infty(J^{k}(\pi))</math> simply by <math>\mathcal{F}_k(\pi)</math>. Take now the [[direct limit]] <math>\mathcal{F}(\pi)</math> of the <math>\mathcal{F}_k(\pi)</math>'s. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric object ''J<sup>∞</sup>(π)''. Observe that <math>\mathcal{F}(\pi)</math>, being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.
| |
| | |
| Roughly speaking, a concrete element <math>\varphi\in\mathcal{F}(\pi)</math> will always belong to some <math>\mathcal{F}_k(\pi)</math>, so it is a smooth function on the finite-dimensional manifold ''J<sup>k</sup>''(π) in the usual sense.
| |
| | |
| ===Infinitely prolonged PDEs===
| |
| Given a ''k''-th order system of PDEs ''E'' ⊆ ''J<sup>k</sup>(π)'', the collection ''I(E)'' of vanishing on ''E'' smooth functions on ''J<sup>∞</sup>(π)'' is an [[ideal]] in the algebra <math>\mathcal{F}_k(\pi)</math>, and hence in the direct limit <math>\mathcal{F}(\pi)</math> too.
| |
| | |
| Enhance ''I(E)'' by adding all the possible compositions of [[total derivative]]s applied to all its elements. This way we get a new ideal ''I'' of <math>\mathcal{F}(\pi)</math> which is now closed under the operation of taking total derivative. The submanifold ''E''<sub>(∞)</sub> of ''J''<sup>∞</sup>(π) cut out by ''I'' is called the '''infinite prolongation''' of ''E''.
| |
| | |
| Geometrically, ''E''<sub>(∞)</sub> is the manifold of '''formal solutions''' of ''E''. A point <math>j_p^\infty(\sigma)</math> of ''E''<sub>(∞)</sub> can be easily seen to be represented by a section σ whose ''k''-jet's graph is tangent to ''E'' at the point <math>j_p^k(\sigma)</math> 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 <math>\varphi\circ j^k(\sigma)</math> at the point ''p''.
| |
| | |
| Most importantly, the closure properties of ''I'' imply that ''E''<sub>(∞)</sub> is tangent to the '''infinite-order contact structure''' <math>\mathcal{C}</math> on ''J<sup>∞</sup>(π)'', so that by restricting <math>\mathcal{C}</math> to ''E''<sub>(∞)</sub> one gets the [[diffiety]] <math>(E_{(\infty)}, \mathcal{C}|_{E_{(\infty)}})</math>, 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<sub>f</sub>: M'' → ''M'' × ''N''
| |
| :''gr<sub>f</sub>(p)'' = ''(p, f(p))''
| |
| | |
| (''gr<sub>f</sub>'' is known as the '''graph of the function ''f''''') of the trivial bundle (''M'' × ''N'', π<sub>1</sub>, ''M''). However, this restriction does not simplify the theory, as the global triviality of π does not imply the global triviality of π<sub>1</sub>.
| |
| | |
| ==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., ''[http://www.emis.de/monographs/KSM/ 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
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| * Giachetta, G., Mangiarotti, L., [[Gennadi Sardanashvily|Sardanashvily, G.]], "Advanced Classical Field Theory", World Scientific, 2009, ISBN 978-981-283-895-7
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| * [[Gennadi Sardanashvily|Sardanashvily, G.]], Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory", Lambert Academic Publishing, 2013, ISBN 978-3-659-37815-7; [http://xxx.lanl.gov/abs/0908.1886 arXiv: 0908.1886]
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| [[Category:Differential topology]]
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| [[Category:Differential equations]]
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| [[Category:Fiber bundles]]
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