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| {{Merge from|jet group|date=August 2011}} | | A '''toroidal mirror''' is a form of [[parabolic reflector]] which has a different focal distance depending on the angle of the mirror. The curvature is actually that of an elliptic [[paraboloid]] with <math>a \ne b</math>. If the shape were that of a [[toroid]], the mirror would exhibit [[Aberration in optical systems|spherical aberration]]. |
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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]].
| | Toroidal mirrors are used in Yolo [[telescope]]s and optical [[monochromator]]s (mirrors C and E in the diagram). In these devices, the source and detectors of the light are not located on the optic axis of the mirror, so the use of a true paraboloid of revolution would cause a distorted image. |
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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.)
| | ==See also== |
| | * [[List of telescope types]] |
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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.
| | ==External links== |
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| ==Jets==
| | * [http://bhs.broo.k12.wv.us/homepage/alumni/dstevick/erwin_c.htm Toroidal mirrors for Yolo telescopes] |
| {{main|Jet (mathematics)}}
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| Let <math>(\mathcal{E}, \pi, \mathcal{M})</math> be a [[fiber bundle]] in a category of [[manifold]]s and let <math>p \in \mathcal{M}</math>, with <math>\dim\mathcal{M}=m</math>.
| | [[Category:Mirrors]] |
| Let <math>\Gamma(\pi)\,</math> denote the set of all local sections whose domain contains <math>p\,</math>. Let <math>I=(I(1),I(2),\ldots,I(m))</math> be a [[multi-index]] (an ordered <math>m</math>-tuple of integers), then
| | [[Category:Telescope types]] |
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| :<math>|I| := \sum_{i=1}^{m} I(i)</math>
| | {{astronomy-stub}} |
| | {{optics-stub}} |
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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>
| | [[zh:環形面鏡]] |
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| Define the local sections <math>\sigma, \eta \in \Gamma(\pi)</math> to have the same '''<math>r\,</math>-jet''' at <math>p\,</math> 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 <math>r</math>-jet is an [[equivalence relation]]. An ''r''-jet is an [[equivalence class]] under this relation, and the ''r''-jet with representative <math>\sigma\,</math> is denoted <math>j^{r}_{p}\sigma</math>. The integer <math>r</math> is also called the '''order''' of the jet.
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| <math>p\,</math> is the '''source''' of <math>j^{r}_{p}\sigma</math>.
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| <math>\sigma\,(p)</math> is the '''target''' of <math>j^{r}_{p}\sigma</math>.
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| ==Jet manifolds==
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| The '''<math>r^{th}\,</math> jet manifold of <math>\pi\,</math>''' is the set
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| :<math>\{j^{r}_{p}\sigma:p \in \mathcal{M}, \sigma \in \Gamma(\pi)\}</math>
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| and is denoted <math>J^{r}\pi\,</math>. We may define projections <math>\pi_{r}\,</math> and <math>\pi_{r,0}\,</math> called the '''source and target projections''' respectively, by
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| :{|
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| |<math>\pi_{r}:J^{r}\pi\, </math>
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| |<math>\longrightarrow \mathcal{M} </math>
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| |-
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| |align=right|<math>j^{r}_{p}\sigma </math>
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| ||<math>\longmapsto p </math>
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| |-
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| |<math>\pi_{r,0}:J^{r}\pi\, </math>
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| |<math>\longrightarrow \mathcal{E} </math>
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| |-
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| |align=right|<math>j^{r}_{p}\sigma </math>
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| |<math>\longmapsto \sigma(p) </math>
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| |-
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| |}
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| If <math>1 \leq k \leq r</math>, then the '''<math>k</math>-jet projection''' is the function <math>\pi_{r,k}\,</math> defined by
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| :{|
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| |<math>\pi_{r,k}:J^{r}\pi \,</math>
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| |<math>\longrightarrow J^{k}\pi </math>
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| |-
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| |align=right|<math>j^{r}_{p}\sigma </math>
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| |<math>\longmapsto j^{k}_{p}\sigma </math>
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| |-
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| |}
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| From this definition, it is clear that <math>\pi_{r} = \pi \circ \pi_{r,0}</math> and that if <math>0 \leq m \leq k</math>, then <math>\pi_{r,m} = \pi_{k,m} \circ \pi_{r,k}</math>. It is conventional to regard <math>\pi_{r,r}=\operatorname{id}_{J^{r}\pi}\,</math>, the [[identity function|identity map]] on <math>J^{r}\pi \,</math> and to identify <math>J^{0}\pi\,</math> with <math>\mathcal{E}</math>.
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| The functions <math>\pi_{r,k}, \pi_{r,0}\,</math> and <math>\pi_{r}\,</math> are [[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 <math>\mathcal{E}</math> will generate a coordinate system on <math>J^{r}\pi\,</math>. Let <math>(U,u)\,</math> be an adapted [[coordinate chart]] on <math>\mathcal{E}</math>, where <math>u = (x^{i}, u^{\alpha})\,</math>. The '''induced coordinate chart <math>(U^{r}, u^{r})\,</math>''' on <math>J^{r}\pi\,</math> is defined by
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| :{|
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| |align=right|<math>U^{r} \,</math>
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| |<math>= \{ j^{r}_{p}\sigma: \sigma(p) \in U \} \,</math>
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| |-
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| |align=right|<math>u^{r} \,</math>
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| |<math>= (x^{i}, u^{\alpha}, u^{\alpha}_{I})\,</math>
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| |-
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| |}
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| where
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| :{|
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| |<math>x^{i}(j^{r}_{p}\sigma) \,</math>
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| |<math>= x^{i}(p) \,</math>
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| |-
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| |<math>u^{\alpha}(j^{r}_{p}\sigma) \,</math>
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| |<math>= u^{\alpha}(\sigma(p)) \, </math>
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| |-
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| |}
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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} \longrightarrow \mathbb{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 <math>(U,u)\,</math> on <math>\mathcal{E}</math>, the corresponding collection of charts <math>(U^{r},u^{r})\,</math> is a [[finite-dimensional]] <math>C^{\infty}\,</math> atlas on <math>J^{r}\pi\,</math>.
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| ==Jet bundles==
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| Since the atlas on each <math>J^{r}\pi\,</math> defines a manifold, the triples <math>(J^{r}\pi, \pi_{r,k}, J^{k}\pi), (J^{r}\pi, \pi_{r,0}, \mathcal{E})\,</math> and <math>(J^{r}\pi, \pi_{r}, \mathcal{M})\,</math> all define fibered manifolds.
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| In particular, if <math>(\mathcal{E}, \pi, \mathcal{M})\,</math> is a fiber bundle, the triple <math>(J^{r}\pi, \pi_{r}, \mathcal{M})\,</math> defines the '''<math>r^{th}\,</math> jet bundle of <math>\pi\,</math>'''.
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| If <math>W \subset \mathcal{M}\,</math> 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 <math>p \in \mathcal{M}\,</math>, then the fiber <math>\pi^{-1}_{r}(p)\,</math> is denoted <math>J^{r}_{p}\pi\,</math>.
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| Let <math>\sigma\,</math> be a local section of <math>\pi\,</math> with domain <math>W \subset \mathcal{M}\,</math>. The '''<math>r^{th}\,</math> jet prolongation of <math>\sigma\,</math>''' is the map <math>j^{r}\sigma:W \longrightarrow J^{r}\pi\,</math> defined by
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| :<math> (j^{r}\sigma)(p) = j^{r}_{p}\sigma. \,</math>
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| Note that <math>\pi_{r} \circ j^{r}\sigma = \operatorname{id}_{W} \,</math>, so <math>j^{r}\sigma\,</math> really is a section. In local coordinates, <math>j^{r}\sigma\,</math> 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 <math>j^{0}\sigma\,</math> with <math>\sigma\,</math>.
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| ===Example===
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| If <math>\pi\,</math> is the [[trivial bundle]] <math>(\mathcal{M} \times \mathbb{R}, pr_{1}, \mathcal{M})</math>, then there is a canonical [[diffeomorphism]] between the first jet bundle <math>J^{1}\pi\,</math> and <math>T^{*}\mathcal{M} \times \mathbb{R} </math>.
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| To construct this diffeomorphism, for each <math>\sigma \in \Gamma_{M}(\pi)\,</math> write <math>\bar{\sigma} = pr_{2} \circ \sigma \in C^{\infty}(M)\,</math>.
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| Then, whenever <math>p \in M \,</math>
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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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| :{|
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| |<math>J^{1}\pi \,</math>
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| |<math>\longrightarrow T^{*}\mathcal{M} \times \mathbb{R}</math>
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| |-
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| |align=right|<math>j^{1}_{p}\sigma \,</math>
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| |<math> \longmapsto (d\bar{\sigma}_{p},\bar{\sigma}(p)) \,</math>
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| |-
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| |}
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| is well-defined and is clearly [[injective]]. Writing it out in coordinates shows that it is a diffeomorphism, because if <math>(x^{i},u)\,</math> are coordinates on <math>\mathcal{M} \times \mathbb{R}</math>, where <math>u=id_{\mathbb{R}}\,</math> is the identity coordinate, then the derivative coordinates <math>u_{i}\,</math> on <math>J^{1}\pi\,</math> correspond to the coordinates <math>\partial_{i}\,</math> on <math>T^{*}\mathcal{M}\,</math>.
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| Likewise, if <math>\pi\,</math> is the trivial bundle <math>(\mathbb{R} \times \mathcal{M}, pr_{1}, \mathbb{R})</math>, then there exists a canonical diffeomorphism between <math>J^{1}\pi\,</math> and <math>\mathbb{R} \times T\mathcal{M}\,</math>
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| ==Contact forms==
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| A [[differential 1-form]] <math>\theta\,</math> on the space <math>J^{r}\pi\,</math> 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 <math>\mathcal{M}\,</math> by all prolongations.
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| 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 <math>W \subset \mathcal{M}\,</math> and every <math>\sigma \in \Gamma_{W}(\pi),\,</math>
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| :<math>(j^{k+1}\sigma)^{*}\theta = 0.\,</math>
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| The [[distribution (differential geometry)|distribution]] on <math>J^{r}\pi\,</math> 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 <math>J^\infty</math> this distribution is involutive and finite dimensional. Its dimension coinciding with the dimension of the base manifold <math>\mathcal{M}</math>.
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| ===Example===
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| Let us consider the case <math>(\mathcal{E},\pi,\mathcal{M})</math>, where <math>\mathcal{E} \simeq \mathbb{R}^{2}</math> and <math>\mathcal{M} \simeq \mathbb{R}</math>.
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| Then, <math>(J^{1}\pi, \pi, \mathcal{M})</math> defines the first jet bundle, and may be coordinated by <math>(x,u,u_{1})\,</math>, where
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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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| |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 <math> p \in \mathcal{M}</math> and <math>\sigma \in \Gamma_{p}(\pi)\,</math>. A general 1-form on <math>J^{1}\pi\,</math> 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 <math>\sigma \in \Gamma_{p}(\pi)\,</math> has first prolongation <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, <math>(j^{1}\sigma)^{*} \theta\,</math> 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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| |}
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| This will vanish for all sections <math>\sigma\,</math> if and only if <math>c=0\,</math> and <math>a = -b\sigma'(x)\,</math>. Hence, <math>\theta=b(x, u, u_{1})\theta_{0}\,</math> must necessarily be a multiple of the basic contact form <math>\theta_{0}=du-u_{1}dx\,</math>.
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| Proceeding to the second jet space <math>J^{2}\pi\,</math> with additional coordinate <math>u_{2}\,</math>, 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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| which implies that <math>e=0\,</math> and <math>a=-b\sigma'(x)-c\sigma''(x)\,</math>. Therefore, <math>\theta\,</math> 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 <math>\theta_{1} = du_{1} - u_{2}dx\,</math> is the next basic contact form
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| (Note that here we are identifying the form <math>\theta_{0}\,</math> with its pull-back <math>(\pi_{2,1})^{*}\theta_{0}\,</math> to <math>J^{2}\pi\,</math>).
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| In general, providing <math>x,u, \in \mathbb{R}\,</math>, a contact form on <math>J^{r+1}\pi\,</math> 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 <math>J^{r+1}\pi\,</math> 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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| <math>|I|\,</math> is known as the '''order''' of the contact form <math>\theta_{I}^{\alpha}</math>. Note that contact forms on <math>J^{r+1}\pi\,</math> have orders at most <math>r\,</math>.
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| Contact forms provide a characterization of those local sections of <math>\pi_{r+1}\,</math> which are prolongations of sections of <math>\pi\,</math>.
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| Let <math>\psi \in \Gamma_{W}(\pi_{r+1})\,</math>, then <math>\psi = j^{r+1}\sigma\,</math> where <math>\sigma \in \Gamma_{W}(\pi)\,</math> 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 <math>\mathcal{E}</math>, 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 all the vertical coefficients vanish, if <math>\phi^{\alpha}=0\,</math>.
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| A vector field is called '''vertical''', meaning all the horizontal coefficients vanish, if <math>\rho^{i}=0\,</math>.
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| For fixed <math>(x,u)\,</math>, 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 <math>(x,u,\rho^{i},\phi^{\alpha})\,</math>, with an element in the fiber <math>T_{xu}\mathcal{E}</math> of <math>T\mathcal{E}</math> over <math>(x,u) \in \mathcal{E}</math>, called '''a [[tangent vector]] in <math>T\mathcal{E}</math>'''. A section
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| :{|
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| |<math>\psi : \mathcal{E} \,</math>
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| |<math>\longrightarrow T\mathcal{E} </math>
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| |-
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| |align=right|<math>(x,u) \,</math>
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| |<math>\longmapsto \psi(x,u) = V\,</math>
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| |-
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| |}
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| is called '''a vector field on <math>\mathcal{E}</math>''' with <math> V = \rho^{i}(x,u) \frac{\partial}{\partial x^{i}} + \phi^{\alpha}(x,u) \frac{\partial}{\partial u^{\alpha}}\,</math> and <math>\psi \in \Gamma(T\mathcal{E})\,</math>.
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| The jet bundle <math>J^{r}\pi\,</math> is coordinated by <math>(x,u,w) \ \stackrel{\mathrm{def}}{=}\ (x^{i},u^{\alpha},w_{i}^{\alpha})\,</math>. For fixed <math>(x,u,w)\,</math>, 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>
| |
| |-
| |
| |}
| |
| | |
| 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 <math>T(J^{r}\pi)\,</math> over <math>(x,u,w) \in J^{r}\pi\,</math>, called '''a tangent vector in <math>T(J^{r}\pi)\,</math>'''.
| |
| Here, <math>v^{\alpha}_{i}, v^{\alpha}_{i_{1}i_{2}},\ldots,v^{\alpha}_{i_{1}i_{2} \cdots i_{r}}\,</math> are real-valued functions on <math>J^{r}\pi\,</math>. A section
| |
| | |
| :{|
| |
| |-
| |
| |<math>\Psi : J^{r}\pi \,</math>
| |
| |<math>\longrightarrow T(J^{r}\pi) \,</math>
| |
| |-
| |
| |align=right|<math>(x,u,w) \, </math>
| |
| |<math>\longmapsto \Psi(u,w) = V \, </math>
| |
| |-
| |
| |}
| |
| | |
| is '''a vector field on <math>J^{r}\pi\,</math>''', and we say <math>\Psi \in \Gamma(T(J^{r}\pi))\,</math>.
| |
| | |
| ==Partial differential equations==
| |
| | |
| Let <math>(\mathcal{E},\pi,\mathcal{M})</math> be a fiber bundle. An '''<math>r^{th}\,</math> order [[partial differential equation]]''' on <math>\pi\,</math> is a [[closed]]{{dn|date=July 2012}} [[embedding|embedded]] submanifold <math>\mathcal{S}</math> of the jet manifold <math>J^{r}\pi\,</math>.
| |
| A solution is a local section <math>\sigma \in \Gamma_{W}(\pi)\,</math> satisfying <math>j^{r}_{p}\sigma \in \mathcal{S}, \forall p \in \mathcal{M}</math>.
| |
| | |
| Let us consider an example of a first order partial differential equation.
| |
| | |
| ===Example===
| |
| Let <math>\pi\,</math> be the trivial bundle <math>(\mathbb{R}^{2} \times \mathbb{R}, pr_{1}, \mathbb{R}^{2})\,</math> with global coordinates <math>(x^{1}, x^{2}, u^{1})\,</math>.
| |
| Then the map <math>F:J^{1}\pi \longrightarrow \mathbb{R}\,</math> defined by
| |
| | |
| :<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 <math>\sigma:\mathbb{R}^{2} \longrightarrow \mathbb{R}^{2} \times \mathbb{R}\,</math> 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 \mathcal{S}\,</math> for ''every'' <math>p \in \mathbb{R}^{2}\,</math>.
| |
| | |
| ==Jet Prolongation==
| |
| A local diffeomorphism <math>\psi:J^{r}\pi \longrightarrow J^{r}\pi\,</math> defines a contact transformation of order <math>r\,</math> if it preserves the contact ideal, meaning that if <math>\theta\,</math> is any contact form on <math>J^{r}\pi\,</math>, then <math>\psi^{*}\theta\,</math> is also a contact form.
| |
| | |
| The flow generated by a vector field <math>V^{r}\,</math> on the jet space <math>J^{r}\,</math> 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 <math>\theta\,</math> preserves the contact ideal.
| |
| | |
| Let us begin with the first order case. Consider a general vector field <math>V^{1}\,</math> on <math>J^{1}\pi\,</math>, 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, <math>V^{1}\,</math> determines a contact transformation if and only if the coefficients of <math>dx^{i}\,</math> 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 <math>V^{1}\,</math>:
| |
| | |
| :<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 <math>D_{i}\,</math>.
| |
| | |
| 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, <math>V^{r}\,</math> is called the '''<math>r^{th}\,</math> prolongation of <math>V\,</math> to a vector field on <math>J^{r}\pi\,</math>'''.
| |
| | |
| These results are best understood when applied to a particular example. Hence, let us examine the following.
| |
| | |
| ===Example===
| |
| | |
| Let us consider the case <math>(\mathcal{E},\pi,\mathcal{M})</math>, where <math>\mathcal{E} \simeq \mathbb{R}^{2}</math> and <math>\mathcal{M} \simeq \mathbb{R}</math>.
| |
| Then, <math>(J^{1}\pi, \pi, \mathcal{E})</math> defines the first jet bundle, and may be coordinated by <math>(x,u,u_{1})\,</math>, 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 <math>p \in \mathcal{M}</math> and <math>\sigma \in \Gamma_{p}(\pi)\,</math>. A contact form on <math>J^{1}\pi\,</math> has the form
| |
| | |
| :<math>\theta = du - u_{1}dx \,</math>
| |
| | |
| Let us consider a vector <math>V\,</math> on <math>\mathcal{E}</math>, 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 <math>J^{1}\pi\,</math> 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 <math>du = \theta + u_{1}dx\,</math>. 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 <math>V\,</math> to a vector field on <math>J^{1}\pi\,</math> 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 <math>V\,</math> to a vector field on <math>J^{2}\pi\,</math>.
| |
| We have <math>\{x,u,u_{1}, y_{2}\}\,</math> as coordinates on <math>J^{2}\pi\,</math>. 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, <math>\theta\,</math> has no <math>u_{2}\,</math> dependency. Hence, from this equation we will pick up the formula for <math>\rho\,</math>, which will necessarily be the same result as we found for <math>V^{1}\,</math>. Therefore, the problem is analogous to prolonging the vector field <math>V^{1}\,</math> to <math>J^{2}\pi\,</math>.
| |
| That is to say, we may generate the <math>r^{th}\,</math>-prolongation of a vector field by recursively applying the Lie derivative of the contact forms with respect to the prolonged vector fields, <math>r\,</math> 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 <math>V^{2}\,</math> 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 <math>du=\theta + u_{1}dx \,</math> and <math>du_{1}=\theta_{1} + u_{2}dx \,</math>. 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 <math>V\,</math> to a vector field on <math>J^{2}\pi\,</math> 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 <math>V\,</math> can be recovered by omitting the second derivative terms in <math>V^{2}\,</math>, or by projecting back to <math>J^{1}\pi\,</math>.
| |
| | |
| ==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''' <math>J^\infty(\pi)</math>. A point <math>j_p^\infty(\sigma)</math> is the equivalence class of sections of <math>\pi</math> that have the same <math>k</math>-jet in <math>p</math> as <math>\sigma</math> for all values of <math>k</math>. The natural projection <math>\pi_\infty</math> maps <math>j_p^\infty(\sigma)</math> into <math>p</math>.
| |
| | |
| Just by thinking in terms of coordinates, <math>J^\infty(\pi)</math> appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure on <math>J^\infty(\pi)</math>, 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 <math>J^\infty(\pi)</math>. 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 <math>J^k(\pi)</math> in the usual sense.
| |
| | |
| ===Infinitely prolonged PDE's===
| |
| Given a <math>k</math>-th order system of PDE's <math>\mathcal{E}\subseteq J^k(\pi)</math>, the collection <math>I(\mathcal{E})</math> of vanishing on <math>\mathcal{E}</math> smooth functions on <math>J^\infty(\pi)</math> is an [[ideal]] in the algebra <math>\mathcal{F}_k(\pi)</math>, and hence in the direct limit <math>\mathcal{F}(\pi)</math> too.
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| Enhance <math>I(\mathcal{E})</math> by adding all the possible compositions of [[total derivative]]s applied to all its elements. This way we get a new ideal <math>I</math> of <math>\mathcal{F}(\pi)</math> which is now closed under the operation of taking total derivative. The submanifold <math>\mathcal{E}_{(\infty)}</math> of <math>J^\infty(\pi)</math> cut out by <math>I</math> is called the '''infinite prolongation''' of <math>\mathcal{E}</math>.
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| Geometrically, <math>\mathcal{E}_{(\infty)}</math> is the manifold of '''formal solutions''' of <math>\mathcal{E}</math>. A point <math>j_p^\infty(\sigma)</math> of <math>\mathcal{E}_{(\infty)}</math> can be easily seen to be represented by a section <math>\sigma</math> whose <math>k</math>-jet's graph is tangent to <math>\mathcal{E}</math> at the point <math>j_p^k(\sigma)</math> with arbitrarily high order of tangency.
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| Analytically, if <math>\mathcal{E}</math> is given by <math>\varphi=0</math>, a formal solution can be understood as the set of Taylor coefficients of a section <math>\sigma</math> in a point <math>p</math> that make vanish the [[Taylor series]] of <math>\varphi\circ j^k(\sigma)</math> at the point <math>p</math>.
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| Most importantly, the closure properties of <math>I</math> imply that <math>\mathcal{E}_{(\infty)}</math> is tangent to the '''infinite-order contact structure''' <math>\mathcal{C}</math> on <math>J^\infty(\pi)</math>, so that by restricting <math>\mathcal{C}</math> to <math>\mathcal{E}_{(\infty)}</math> one gets the [[diffiety]] <math>(\mathcal{E}_{(\infty)},\mathcal{C}|_{\mathcal{E}_{(\infty)}})</math>, and can study the associated [[C-spectral sequence]].
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| ==Remark==
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| This article has defined jets of local sections of a bundle, but it is possible to define jets of functions <math>f:\mathcal{M} \longrightarrow \mathcal{N}\,</math>, where <math>\mathcal{M}</math> and <math>\mathcal{N}</math> are manifolds; the jet of <math>f\,</math> then just corresponds to the jet of the section
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| :{|
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| |-
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| |<math>gr_{f}:\mathcal{M} \,</math>
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| |<math>\longrightarrow \mathcal{M} \times \mathcal{N} \,</math>
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| |-
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| |align=right|<math>p \,</math>
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| |<math>\longmapsto gr_{f}(p) = (p, f(p) )\,</math>
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| |-
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| |}
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| (<math>gr_{f}\,</math> is known as the '''graph of the function <math>f\,</math>''') of the trivial bundle <math>(\mathcal{M} \times \mathcal{N}, \pi_{1}, \mathcal{M})</math>. However, this restriction does not simplify the theory, as the global triviality of <math>\pi\,</math> does not imply the global triviality of <math>\pi_{1}\,</math>.
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| == See also ==
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| * [[Jet group]]
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| * [[Jet (mathematics)]]
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| ==References==
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| * 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.
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| * 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.
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| * Saunders, D. J., "The Geometry of Jet Bundles", Cambridge University Press, 1989, ISBN 0-521-36948-7
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| * 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.
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| * 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.]], Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians, [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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| [[zh:节丛]] | |