In
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, the jet bundle is a certain construction that makes a new
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
out of a given smooth fiber bundle. It makes it possible to write
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s on
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
s of a fiber bundle in an invariant form.
Jets may also be seen as the coordinate free versions of
Taylor expansions.
Historically, jet bundles are attributed to
Charles Ehresmann, and were an advance on the method (
prolongation
In music theory, prolongation is the process in tonal music through which a pitch, interval, or consonant triad is considered to govern spans of music when not physically sounding. It is a central principle in the music-analytic methodology of ...
) of
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
, of dealing ''geometrically'' with
higher derivatives, by imposing
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
conditions on newly introduced formal variables. Jet bundles are sometimes called sprays, although
sprays usually refer more specifically to the associated
vector field induced on the corresponding bundle (e.g., the
geodesic spray
In geometry, a geodesic () is a curve representing in some sense the shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in ...
on
Finsler manifold
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth c ...
s.)
Since the early 1980s, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. Consequently, the jet bundle is now recognized as the correct domain for a
geometrical covariant field theory and much work is done in
general relativistic formulations of fields using this approach.
Jets
Suppose ''M'' is an ''m''-dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
and that (''E'', π, ''M'') is a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
. For ''p'' ∈ ''M'', let Γ(p) denote the set of all local sections whose domain contains ''p''. Let be a
multi-index
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
(an ''m''-tuple of non-negative integers, not necessarily in ascending order), then define:
:
Define the local sections σ, η ∈ Γ(p) to have the same ''r''-jet at ''p'' if
:
The relation that two maps have the same ''r''-jet is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
. An ''r''-jet is an
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
under this relation, and the ''r''-jet with representative σ is denoted
. The integer ''r'' is also called the order of the jet, ''p'' is its source and σ(''p'') is its target.
Jet manifolds
The ''r''-th jet manifold of π is the set
:
We may define projections ''π
r'' and ''π''
''r'',0 called the source and target projections respectively, by
:
If 1 ≤ ''k'' ≤ ''r'', then the ''k''-jet projection is the function ''π
r,k'' defined by
:
From this definition, it is clear that ''π
r'' = ''π''
o ''π''
''r'',0 and that if 0 ≤ ''m'' ≤ ''k'', then ''π
r,m'' = ''π
k,m''
o ''π
r,k''. It is conventional to regard ''π
r,r'' as the
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on ''J
r''(''π'') and to identify ''J''
0(''π'') with ''E''.
The functions ''π
r,k'', ''π''
''r'',0 and ''π
r'' are
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
submersions.
![Jet Bundle Image FbN](https://upload.wikimedia.org/wikipedia/commons/7/7a/Jet_Bundle_Image_FbN.png)
A
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
on ''E'' will generate a coordinate system on ''J
r''(''π''). Let (''U'', ''u'') be an adapted
coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
on ''E'', where ''u'' = (''x
i'', ''u
α''). The induced coordinate chart (''U
r'', ''u
r'') on ''J
r''(''π'') is defined by
:
where
:
and the
functions known as the derivative coordinates:
:
Given an atlas of adapted charts (''U'', ''u'') on ''E'', the corresponding collection of charts (''U
r'', ''u
r'') is a
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
''C''
∞ atlas on ''J
r''(''π'').
Jet bundles
Since the atlas on each
defines a manifold, the triples ''
'', ''
'' and ''
'' all define fibered manifolds. In particular, if ''
''is a fiber bundle, the triple ''
'' defines the ''r''-th jet bundle of π.
If ''W'' ⊂ ''M'' is an open submanifold, then
:
If ''p'' ∈ ''M'', then the fiber
is denoted
.
Let σ be a local section of π with domain ''W'' ⊂ ''M''. The ''r''-th jet prolongation of σ is the map
defined by
:
Note that
, so
really is a section. In local coordinates,
is given by
:
We identify ''
'' with
.
Algebraic-geometric perspective
An independently motivated construction of the sheaf of sections
'' is given''.''
Consider a diagonal map
, where the smooth manifold
is a
locally ringed space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
by
for each open
. Let
be the
ideal sheaf In algebraic geometry and other areas of mathematics, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal in a ring. The ideal sheaves on a geometric object are closely connected to its subspaces.
Definition
Let ''X'' be a t ...
of
, equivalently let
be the
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
of smooth
germs which vanish on
for all
. The
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
of the
quotient sheaf from
to
by
is the sheaf of k-jets.
The
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
of the sequence of injections given by the canonical inclusions
of sheaves, gives rise to the infinite jet sheaf
. Observe that by the direct limit construction it is a filtered ring.
Example
If π is the
trivial bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
(''M'' × R, pr
1, ''M''), then there is a canonical
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
between the first jet bundle
and ''T*M'' × R. To construct this diffeomorphism, for each σ in
write
.
Then, whenever ''p'' ∈ ''M''
:
Consequently, the mapping
:
is well-defined and is clearly
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. Writing it out in coordinates shows that it is a diffeomorphism, because if ''(x
i, u)'' are coordinates on ''M'' × R, where ''u'' = id
R is the identity coordinate, then the derivative coordinates ''u
i'' on ''J
1(π)'' correspond to the coordinates ∂
''i'' on ''T*M''.
Likewise, if π is the trivial bundle (R × ''M'', pr
1, R), then there exists a canonical diffeomorphism between
and R × ''TM''.
Contact structure
The space ''J
r''(π) carries a natural
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
, that is, a sub-bundle of the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
''TJ
r''(π)), called the ''Cartan distribution''. The Cartan distribution is spanned by all tangent planes to graphs of holonomic sections; that is, sections of the form ''j
rφ'' for ''φ'' a section of π.
The annihilator of the Cartan distribution is a space of
differential one-forms called
contact form
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution m ...
s, on ''J
r''(π). The space of differential one-forms on ''J
r''(π) is denoted by
and the space of contact forms is denoted by
. A one form is a contact form provided its
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
along every prolongation is zero. In other words,
is a contact form if and only if
:
for all local sections σ of π over ''M''.
The Cartan distribution is the main geometrical structure on jet spaces and plays an important role in the geometric theory of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s. The Cartan distributions are completely non-integrable. In particular, they are not
involutive. The dimension of the Cartan distribution grows with the order of the jet space. However, on the space of infinite jets ''J
∞'' the Cartan distribution becomes involutive and finite-dimensional: its dimension coincides with the dimension of the base manifold ''M''.
Example
Consider the case ''(E, π, M)'', where ''E'' ≃ R
2 and ''M'' ≃ R. Then, ''(J
1(π), π, M)'' defines the first jet bundle, and may be coordinated by ''(x, u, u
1)'', where
:
for all ''p'' ∈ ''M'' and σ in Γ
''p''(π). A general 1-form on ''J
1(π)'' takes the form
:
A section σ in Γ
''p''(π) has first prolongation
:
Hence, ''(j
1σ)*θ'' can be calculated as
:
This will vanish for all sections σ if and only if ''c'' = 0 and ''a'' = −''bσ′(x)''. Hence, θ = ''b(x, u, u
1)θ
0'' must necessarily be a multiple of the basic contact form θ
0 = ''du'' − ''u
1dx''. Proceeding to the second jet space ''J
2(π)'' with additional coordinate ''u
2'', such that
:
a general 1-form has the construction
:
This is a contact form if and only if
:
which implies that ''e'' = 0 and ''a'' = −''bσ′(x)'' − ''cσ′′(x)''. Therefore, θ is a contact form if and only if
:
where θ
1 = ''du''
1 − ''u''
2''dx'' is the next basic contact form (Note that here we are identifying the form θ
0 with its pull-back
to ''J
2(π)'').
In general, providing ''x, u'' ∈ R, a contact form on ''J
r+1(π)'' can be written as a
linear combination of the basic contact forms
:
where
:
Similar arguments lead to a complete characterization of all contact forms.
In local coordinates, every contact one-form on ''J
r+1(π)'' can be written as a linear combination
:
with smooth coefficients
of the basic contact forms
:
'', I, '' is known as the order of the contact form
. Note that contact forms on ''J
r+1(π)'' have orders at most ''r''. Contact forms provide a characterization of those local sections of ''π
r+1'' which are prolongations of sections of π.
Let ψ ∈ Γ
''W''(''π
r+1''), then ''ψ'' = ''j
r+1''σ where σ ∈ Γ
''W''(π) if and only if
Vector fields
A general
vector field on the total space ''E'', coordinated by
, is
:
A vector field is called horizontal, meaning that all the vertical coefficients vanish, if
= 0.
A vector field is called vertical, meaning that all the horizontal coefficients vanish, if ''ρ
i'' = 0.
For fixed ''(x, u)'', we identify
:
having coordinates ''(x, u, ρ
i, φ
α)'', with an element in the fiber ''T
xuE'' of ''TE'' over ''(x, u)'' in ''E'', called a
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
in ''TE''. A section
:
is called a vector field on ''E'' with
:
and ψ in ''Γ(TE)''.
The jet bundle ''J
r(π)'' is coordinated by
. For fixed ''(x, u, w)'', identify
:
having coordinates
:
with an element in the fiber
of ''TJ
r(π)'' over ''(x, u, w)'' ∈ ''J
r(π)'', called a tangent vector in ''TJ
r(π)''. Here,
:
are real-valued functions on ''J
r(π)''. A section
:
is a vector field on ''J
r(π)'', and we say
Partial differential equations
Let ''(E, π, M)'' be a fiber bundle. An ''r''-th order
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
on π is a
closed embedded submanifold ''S'' of the jet manifold ''J
r(π)''. A solution is a local section σ ∈ Γ
''W''(π) satisfying
, for all ''p'' in ''M''.
Consider an example of a first order partial differential equation.
Example
Let π be the trivial bundle (R
2 × R, pr
1, R
2) with global coordinates (''x''
1, ''x''
2, ''u''
1). Then the map ''F'' : ''J''
1(π) → R defined by
:
gives rise to the differential equation
:
which can be written
:
The particular
:
has first prolongation given by
:
and is a solution of this differential equation, because
:
and so
for ''every'' ''p'' ∈ R
2.
Jet prolongation
A local diffeomorphism ''ψ'' : ''J
r''(''π'') → ''J
r''(''π'') defines a contact transformation of order ''r'' if it preserves the contact ideal, meaning that if θ is any contact form on ''J
r''(''π''), then ''ψ*θ'' is also a contact form.
The flow generated by a vector field ''V
r'' on the jet space ''J
r(π)'' forms a one-parameter group of contact transformations if and only if the
Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
of any contact form θ preserves the contact ideal.
Let us begin with the first order case. Consider a general vector field ''V''
1 on ''J''
1(''π''), given by
:
We now apply
to the basic contact forms
and expand the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
of the functions in terms of their coordinates to obtain:
:
Therefore, ''V
1'' determines a contact transformation if and only if the coefficients of ''dx
i'' and
in the formula vanish. The latter requirements imply the contact conditions
:
The former requirements provide explicit formulae for the coefficients of the first derivative terms in ''V
1'':
:
where
:
denotes the zeroth order truncation of the total derivative ''D
i''.
Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if
satisfies these equations, ''V
r'' is called the ''r''-th prolongation of ''V'' to a vector field on ''J
r(π)''.
These results are best understood when applied to a particular example. Hence, let us examine the following.
Example
Consider the case ''(E, π, M)'', where ''E'' ≅ R
2 and ''M'' ≃ R. Then, ''(J
1(π), π, E)'' defines the first jet bundle, and may be coordinated by ''(x, u, u
1)'', where
:
for all ''p'' ∈ ''M'' and ''σ'' in Γ
''p''(''π''). A contact form on ''J
1(π)'' has the form
:
Consider a vector ''V'' on ''E'', having the form
:
Then, the first prolongation of this vector field to ''J
1(π)'' is
:
If we now take the Lie derivative of the contact form with respect to this prolonged vector field,
we obtain
:
Hence, for preservation of the contact ideal, we require
:
And so the first prolongation of ''V'' to a vector field on ''J
1(π)'' is
:
Let us also calculate the second prolongation of ''V'' to a vector field on ''J
2(π)''. We have
as coordinates on ''J
2(π)''. Hence, the prolonged vector has the form
:
The contact forms are
:
To preserve the contact ideal, we require
:
Now, ''θ'' has no ''u''
2 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
1''. Therefore, the problem is analogous to prolonging the vector field ''V
1'' to ''J''
2(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
:
and so
:
Therefore, the Lie derivative of the second contact form with respect to ''V
2'' is
:
Hence, for
to preserve the contact ideal, we require
:
And so the second prolongation of ''V'' to a vector field on ''J''
2(π) is
:
Note that the first prolongation of ''V'' can be recovered by omitting the second derivative terms in ''V
2'', or by projecting back to ''J
1(π)''.
Infinite jet spaces
The
inverse limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
of the sequence of projections
gives rise to the infinite jet space ''J
∞(π)''. A point
is the equivalence class of sections of π that have the same ''k''-jet in ''p'' as σ for all values of ''k''. The natural projection π
∞ maps
into ''p''.
Just by thinking in terms of coordinates, ''J
∞(π)'' appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure on ''J
∞(π)'', not relying on differentiable charts, is given by the
differential calculus over commutative algebras In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this ...
. Dual to the sequence of projections
of manifolds is the sequence of injections
of commutative algebras. Let's denote
simply by
. Take now the
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
of the
's. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric object ''J
∞(π)''. Observe that
, being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.
Roughly speaking, a concrete element
will always belong to some
, so it is a smooth function on the finite-dimensional manifold ''J
k''(π) in the usual sense.
Infinitely prolonged PDEs
Given a ''k''-th order system of PDEs ''E'' ⊆ ''J
k(π)'', the collection ''I(E)'' of vanishing on ''E'' smooth functions on ''J
∞(π)'' is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
in the algebra
, and hence in the direct limit
too.
Enhance ''I(E)'' by adding all the possible compositions of
total derivative
In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with resp ...
s applied to all its elements. This way we get a new ideal ''I'' of
which is now closed under the operation of taking total derivative. The submanifold ''E''
(∞) of ''J''
∞(π) cut out by ''I'' is called the infinite prolongation of ''E''.
Geometrically, ''E''
(∞) is the manifold of formal solutions of ''E''. A point
of ''E''
(∞) can be easily seen to be represented by a section σ whose ''k''-jet's graph is tangent to ''E'' at the point
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
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
of
at the point ''p''.
Most importantly, the closure properties of ''I'' imply that ''E''
(∞) is tangent to the infinite-order contact structure
on ''J
∞(π)'', so that by restricting
to ''E''
(∞) one gets the
diffiety
In mathematics, a diffiety () is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more co ...
, and can study the associated
Vinogradov (C-spectral) sequence.
Remark
This article has defined jets of local sections of a bundle, but it is possible to define jets of functions ''f: M'' → ''N'', where ''M'' and ''N'' are manifolds; the jet of ''f'' then just corresponds to the jet of the section
:''gr
f: M'' → ''M'' × ''N''
:''gr
f(p)'' = ''(p, f(p))''
(''gr
f'' is known as the graph of the function ''f'') of the trivial bundle (''M'' × ''N'', π
1, ''M''). However, this restriction does not simplify the theory, as the global triviality of π does not imply the global triviality of π
1.
See also
*
Jet group In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coor ...
*
Jet (mathematics) In mathematics, the jet is an operation that takes a differentiable function ''f'' and produces a polynomial, the truncated Taylor polynomial of ''f'', at each point of its domain. Although this is the definition of a jet, the theory of jets regards ...
*
Lagrangian system
In mathematics, a Lagrangian system is a pair , consisting of a smooth fiber bundle and a Lagrangian density , which yields the Euler–Lagrange differential operator acting on sections of .
In classical mechanics, many dynamical systems are Lagr ...
*
Variational bicomplex
In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber b ...
References
Further reading
* Ehresmann, C., "Introduction à la théorie des structures infinitésimales et des pseudo-groupes de Lie." ''Geometrie Differentielle,'' Colloq. Inter. du Centre Nat. de la Recherche Scientifique, Strasbourg, 1953, 97-127.
* Kolář, I., Michor, P., Slovák, J.,
Natural operations in differential geometry.' Springer-Verlag: Berlin Heidelberg, 1993. , .
* Saunders, D. J., "The Geometry of Jet Bundles", Cambridge University Press, 1989,
* Krasil'shchik, I. S., Vinogradov, A. M.,
t al.
T, or t, is the twentieth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''tee'' (pronounced ), plural ''tees''. It is deri ...
"Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, .
*
Olver, P. J., "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995,
{{Authority control
Differential topology
Differential equations
Fiber bundles