SECTIONS ALONG A MAP APPLIED TO HIGHER-ORDER ...

SECTIONS ALONG A MAP APPLIED TO HIGHER-ORDER
LAGRANGIAN MECHANICS. NOETHER’S THEOREM
JOSÉ F. CARIÑENA⋆ , CARLOS LÓPEZ† , AND EDUARDO MARTÍNEZ⋆
Abstract. We show that the concept of section along a map is a fundamental
concept in the framework of the geometrical description of Classical Mechanics.
We review the higher-order Lagrangian Mechanics formulation, and simpler
redefinitions of basic objects appear in a natural way. As an application, a
Noether’s theorem for higher-order Lagrangian Mechanics admitting a converse
is developed.
1. Introduction
Geometric techniques have been applied to Physics in several ways during the
last years, and so the Geometry of the Tangent and Cotangent Bundles has been
shown to play a very relevant role in mechanics of time-independent systems from a
geometric perspective. Geometric concepts as vector fields and forms are now well
established in physics literature and the language of jets has more recently appeared
as related to the study of (partial) differential equations and variational calculus.
The theory of connections in principal bundles as geometric counterpart of gauge
theories has also been very illuminating. We feel however that a generalization of
the concept of vector field and form, or more generally, that of a section of a vector
bundle, we will deal with, is also worthy of attention. It has been very often used
in physics but without any previous geometric interpretation and we hope that a
deeper geometric study of this concept and its applications to physics may be very
fruitful for a better understanding of Classical Mechanics and Field Theory. In fact,
among other things, it is the geometric concept necessary to deal with non-point
transformations in Classical Mechanics.
The concept of vector field along a map, well-known in other fields of Mathematics
[17], has been introduced in order to describe in Lagrangian Mechanics
objects like the time evolution operator [11] or symmetries of a Lagrangian system
[5], [12], [13]. This concept has provided a new and simpler approach to some
problems, like the connection between Lagrangian and Hamiltonian formalisms or
the converse of Noether’s theorem. We aim to show in this paper that all the geometric
framework of Mechanics may be based on this concept, from the very first
definitions of the basic canonical objects, that so appear in a much more natural
way.
1991 Mathematics Subject Classification. 58F05.
Key words and phrases. Higher-order tangent bundles,Higher-order Lagrangian Mechanics,
Sections along a map, Derivations.
Eduardo Martínez thanks Ministerio de Educación y Ciencia for a grant. Partial financial
support by CICYT is also acknowledged.
1

2 JOSÉ F. CARIÑENA⋆ , CARLOS LÓPEZ† , AND EDUARDO MARTÍNEZ⋆
We propose an extensive use of the concept of section along a map in the study
of the different problems of Mechanics, not only because of the simplification so
obtained, but because of the necessity of this concept in order to enlarge the field
of available tools too.
Vector fields along a map f : N → M act on functions on M giving rise to
functions on N in much the same way as vector fields act on functions on a manifold.
On the other hand, a theory of derivations of forms along a map, similar to the
usual Frölicher and Nijenhuis’ theory, has recently been developed by Pidello and
Tulczyjew [16]. We will make an extensive use of this theory.
Along the paper we shall review from this new perspective basic objects like the
Poincaré form, total time derivative, complete lifts, etc. Once a Lagrangian function
has been introduced, objects associated to L, like the Poincaré-Cartan form, the
time evolution operator, symmetries and other related ones, are again seen from
this new perspective. In particular it is explicitely shown how the Cartan form and
the Legendre transformation are but two different features of the same object.
The paper is organized as follows: in section 2 we introduce the basic notation,
while in section 3 we give some properties of sections along a map and explain how
some of these objects appear in a canonical way. In section 4 it is shown how a
vector field along a higher-order tangent bundle projection may be prolonged, and
then we study its properties. In section 5 we study the canonical vector field along
the (first) tangent bundle projection, T (0) , its prolongations and relations with
those of other vector fields. In section 6 some properties of the variational operator
and related operators are studied. Higher-order differential equations are studied
in section 7. Finally, in section 8 we show the application of all the preceding
concepts to the study of Lagrangian Mechanics, and in particular, we give a new
version of Noether’s theorem showing that there exists a one-to-one correspondence
between the symmetries of the Lagrangian and those of the Hamiltonian dynamical
system defined by this Lagrangian, and, in consequence, between symmetries of the
Lagrangian and first integrals of the dynamics defined by it. We also point out the
difficulties that can be found when trying to generalize the theory developed by
Marmo and Mukunda for first order Lagrangians [14].
2. Notation
The basic notation of this paper is mainly that of [8]. See also [20] and [9] for a
more detailed introduction to general concepts and definitions.
Capital letters will refer to smooth manifolds and the set of smooth functions
on a manifold M will be denoted by C ∞ (M). Given a manifold M, C ∞ (R, M)
will be the set of differentiable curves over M, C ∞ (R, M) = {ρ: R → M}. The
equivalence relation defined in C ∞ (R, M) for curves having contact of first order
as follows
ρ 1 ≡ ρ 2
iff
ρ 1 (0) = ρ 2 (0) and
(2.1)
∀f ∈ C ∞ d
(M)
dt {f ◦ ρ 1} ∣ t=0
= d dt {f ◦ ρ 2} ∣ t=0
endowes the factor space with a differentiable manifold structure which is the tangent
manifold of M,
T M = C ∞ (R, M)/ ≡ . (2.2)

SECTIONS ALNG A MAP IN MECHANICS 3
The natural projection τ M : T M → M (afterwards denoted by τ), mapping each
equivalence class of curves [ρ] to the initial point m = ρ(0) ∈ M, defines a vector
bundle structure (T M, τ, M), the well known tangent bundle. The dual bundle of
T M is the cotangent bundle π : T ∗ M → M. The set of sections of τ and π, i.e.,
vector fields and 1-forms on M, respectively, will be denoted by X(M) and ∧1 (M).
Higher order tangent bundles of a manifold M are defined in a similar way by
making use of an equivalence relation for curves having a contact of higher order.
The set of equivalence classes of curves on C ∞ (R, M) with a contact of order k is the
k-th order tangent manifold of M, T k M, with natural projection τ k,0 : T k M → M.
For k > 1 the bundle structure of (T k M, τ k,0 , M) is not linear, but of polynomial
type. Elements of T k M are denoted [ρ] k , where ρ is a curve of the corresponding
equivalence class. Sometimes we write Q for an element of T k M if we do not need
to make any reference to an element of the class Q.
Given a local coordinate system of M, (U, q i ) i = 1, . . . , d = dim M, with
U ⊂ M an open subset, there are induced coordinate systems for T k M given by
(U k , q(n) i ) i = 1, . . . , d, n = 0, . . . , k
(2.3)
U k = τ −1
k (U) qi (n) ([ρ]k ) = dn {
q i
dt n ◦ ρ } ∣ t=0
.
The natural projections τ k,l : T k M → T l M, for 0 < l < k, are given by
τ k,l ([ρ] k ) = [ρ] l . A tangent vector v ∈ T (T k M) is said to be τ k,l -vertical (or
vertical of order l) if τ k,l∗ v = 0, i. e. if v is tangent to the fibres of τ k,l . The set of
τ k,l -vertical vectors will be denoted V (τ k,l ) and it is a vector subbundle of T (T k M).
A covector α ∈ T ∗ (T k M) is said to be τ k,l -semibasic (or semibasic of order l) if it
vanishes on τ k,l -vertical vectors: 〈α, v〉 = 0 ∀v ∈ V (τ k,l ). The set of τ k,l -semibasic
covectors is a vector subbundle of T ∗ (T k M). A vector field V ∈ X(T k M) is said
to be τ k,l -vertical if its image is into V (τ k,l ). Similarly, a τ k,l -semibasic 1-form α is
a 1-form α ∈ ∧1 (T k M) such that its image is made up by τ k,l -semibasic covectors.
Every function f ∈ C ∞ (M) has associated a set of functions {f(n) k ∈ C∞ (T k M)}
for n = 0, . . . , k given by
f k (n) ([ρ]k ) = dn
dt n {f ◦ ρ} ∣ ∣
t=0
(2.4)
Similarly, for every curve ρ: R → M there is a natural lifted curve ˜ρ k : R → T k M,
defined by
˜ρ k (t) = [ρ t ] k with ρ t (s) = ρ(t + s). (2.5)
In a local coordinate system (U, q), if q i (ρ(t)) = χ i (t), then q i (n) (˜ρk (t)) = χ ik (n) (t).
Natural lifts of curves allow us to define the imbeddings i k,l : T k+l M → T l (T k M)
defined by i k,l ([ρ] k+l ) = [˜ρ k ] l . These imbeddings i k,l and projections τ k,l always
define commutative diagrams:
T k+l M
i k,l
T l T k M
(2.6)
τ k+l,l
T l M
T l M
Finally, as a concept generalizing the differential of a map φ: N → M, we defined
the k-order prolongation φ k : T k N → T k M given by φ k [ρ] k = [φ ◦ ρ] k , where ρ is a
curve in N.
τ l k,0

4 JOSÉ F. CARIÑENA⋆ , CARLOS LÓPEZ† , AND EDUARDO MARTÍNEZ⋆
3. Sections along a map
Let π : E → M be a fibre bundle, f : N → M a differentiable map and denote
f ∗ E the pull-back of the bundle E by the map f [17]. By a section of E along the
map f (or over f) we mean a differentiable map σ : N → E such that π◦σ = f. The
point to be remarked here is that there exists a one-to-one canonical correspondence
between the set of sections along f and that of sections of the bundle f ∗ E over N.
Moreover, if E is a vector bundle, then both sets can be endowed with C ∞ (N)-
module structures and this correspondence is an isomorphism of C ∞ (N)-modules.
For details see e.g. [17]. The most important cases for our proposals is when the
vector bundle is either T M or (T ∗ M) ∧p or (T ∗ M) ⊗p ⊗ (T M) ⊗q , or something
similar else, and the sections along f are then said to be vector fields, p-forms
or (p, q)-type tensor fields along f, and will be denoted X(f), ∧p (f) or T p,q (f),
respectively.
The more traditional concepts of vector fields and forms in M arise here as vector
fields and forms along the identity map in the base M, i.e. X(M) = X(id M ) and
∧ p
(M) = ∧p (id M ), and similarly T p,q (M) = T p,q (id M ).
Two special instances of vector fields over f are the restriction of a vector field
X ∈ X(M) to f(N) obtained by composition of f : N → M and X : M → T M
(usually denoted X ◦f), and the image of a vector field Y ∈ X(N) under f, denoted
f ∗ ◦ Y and given by (f ∗ ◦ Y )(n) = f ∗n (Y (n)). Similarly, if α ∈ ∧p (N), then the
restriction α ◦ f of α to f(N) is a p-form along f.
More generally, if g : P → N is a differenciable map, for any X ∈ X(f) and
Y ∈ X(g), then X ◦ g and f ∗ ◦ Y are vector fields along f ◦ g. In the same way, if
α ∈ ∧p (f), then α ◦ g is a p-form along f ◦ g.
The set of vector fields over f are endowed with a C ∞ (N)-module structure:
they can be added to each other and multiplied by functions on N. In particular,
if (V, z) and (U, x) are charts in N and M, respectively, such that f(V) ⊂ U, X is
a vector field along f and n ∈ V, then the expression of X n in these coordinates is
X(n) = ξ i (z) ∂
∂x i ∣
∣f(n) , (3.1)
so that every vector field X along f can be locally written as a linear combination
X = ξ i( ∂
∂x i ◦ f) (3.2)
of the restrictions of the coordinate vector fields ∂/∂x i with coefficients ξ i in
C ∞ (N).
In the same way a p-form along f, α, has a local expression
α = α i1...i p
(dx i1 ◦ f) ∧ · · · ∧ (dx ip ◦ f), (3.3)
where α i1...i p
∈ C ∞ (N).
As another interesting example, let us consider the tangent bundle τ : T M → M
where τ plays de role of f. There exists a canonical vector field along τ, which we
will denote T (0) , given by the identity map in T M. Its coordinate expression is
T (0) = v i( ∂
∂q i ◦ τ) . (3.4)
This object will be considered in detail later on.
If f is now the projection π : T ∗ M → M of the tangent bundle, the canonical
object to be considered is the 1-form along π, ˇθ 0 , given by the identity map in

SECTIONS ALNG A MAP IN MECHANICS 5
T ∗ M. Its coordinate expression is
ˇθ 0 = p i (dq i ◦ π). (3.5)
Let us suppose that f is a submersion and denote ∧ p
0
(f) the set of p-forms α on N
satisfying the following condition:
If n ∈ N and u ∈ T n N are such that f ∗n u = 0, then i u α = 0. (3.6)
There exists a one-to-one correspondence between ∧p (f) and ∧ p
0 (f). If ω ∈ ∧p (f)
then f ∗ ◦ ω defined by
(f ∗ ◦ ω) n (u 1 , . . . , u p ) = ω n (f ∗n u 1 , . . . , f ∗n u p ) ∀u i ∈ T n N i = 1, . . . , p (3.7)
is a p-form in ∧ p
0 (f). Conversely, if α ∈ ∧ p
0 (f), n ∈ N, v 1, . . . , v p ∈ T f(n) M and
v ↑ 1 , . . . , v↑ p ∈ T n N are such that f ∗n v ↑ i = v i then ˇα defined by
is a p-form along f such that f ∗ ◦ ˇα = α.
In coordinates, this map is given by
ˇα n (v 1 , . . . , v p ) = α n (v ↑ 1 , . . . , v↑ p) (3.8)
α i1...i p
(dx i1 ◦ f) ∧ · · · ∧ (dx ip ◦ f) ↦→ α i1...i p
dx i1 ∧ · · · ∧ dx ip .
In the last example the 1-form θ 0 = π ∗ ◦ ˇθ 0 is simply the well-known canonical
1-form in T ∗ M.
Another interesting example of a (1, 1)-tensor field along τ is the vertical endomorphism.
Let T : T M → T ∗ (T M) ⊗ T M be the (1, 1)-tensor field along τ
defined as follows: T (v)X = τ ∗v X for X ∈ T v (T M). We also recall that there
exists a natural identification of the vector space T m M with its tangent space at
v ∈ T m M, T v (T m M) ≡ V v (T M), through the denoted vertical lift [7]
ξ v : T m M → T v (T m M) = V v (T M).
The vertical endomorphism S [7] can then be defined by
S(X)(v) = ξ v ◦ T (v)(X)
∀X ∈ T (T M).
Therefore, up to the vertical lift, the vertical endomorphism is but the natural (1, 1)
tensor T along τ. In local coordinates the expression of the associated tensor field
is
T (x, v) = δ j ∂
i
∂x j ⊗ dxi .
∂
∂x i → ∂
∂v i , the vertical endomorphism S be-
After the vertical lift identification
comes
S(x, v) =
∂
∂v i ⊗ dxi .
Leaving aside the canonical (1,1) tensor S on T M, S : T (T M) → T (T M), there
is another canonical map, the involution map J : T (T M) → T (T M) (see e.g. [6])
which is not a tensor field on T M, but it can be seen as a vector field along the
tangent projection τ ∗ , J ∈ X(τ ∗ ), because it has the property τ T M ◦ J = τ ∗ :
T (T M) → T M; in local coordinates (x, v, ẋ, ˙v) on T (T M), the vector field J is
given by
( ) ( )
∂
∂
J(x, v, ẋ, ˙v) = v
+ ˙v
∈ T (x,ẋ) (T M).
∂x
|(x,ẋ)
∂ẋ
|(x,ẋ)

6 JOSÉ F. CARIÑENA⋆ , CARLOS LÓPEZ† , AND EDUARDO MARTÍNEZ⋆
Its higher order generalization J k : T k (T M) → T (T k M) is also a vector field along
a map, J k ∈ X(τ k ), where τ k : T k (T M) → T k M is the k-th prolongation of the
tangent bundle map τ,
τ T k M ◦ J k = τ k .
This vector field is given in local coordinates by
J k (x, v; x 1 , v 1 ; . . . ; x k , v k ) =
k∑
l=0
vl( i ∂
∂x i ◦ τ k ).
l
The last example of this enumeration of natural sections along maps is the canonical
isomorphism Ψ : T (T ∗ M) → T ∗ (T M) (see e.g. [6]), which shows, among other
properties, the existence of a symplectic structure on the tangent manifold T (T ∗ M).
The canonical isomorphism Ψ is a one form along the tangent projection π ∗ , Ψ ∈
∧ (π∗ ), because of the property
π T M ◦ Ψ = π ∗ : T (T ∗ M) → T M.
Same as for the vector field J, Ψ has a higher order generalization, Ψ k : T k (T ∗ M) →
T ∗ (T k M) ([6]) which is a one form along the k-th prolongation π k of π. In similar
way to the case of the Poincaré 1-form θ 0 on T ∗ M, the one-form Ψ k is in
correspondence with a semibasic one form πk ∗ ◦Ψ k on T k (T ∗ M), whose exterior differential
defines a canonical symplectic structure on T k (T ∗ M). In local coordinates
(x i , p i ; x i 1, p i1 ; . . . ; x i k , p ik) on T k (T ∗ M), the one form Ψ k is given by
Ψ k (x i , p i ; . . . ; x i k, p ik ) =
k∑
( k l )p i(k−l) (dx i l ◦ π k ).
Other examples in Lagrangian Mechanics will be given in section 8.
Vector fields along f act on a function on M giving as result a function on N.
If X ∈ X(f) and n ∈ N then X(n) is a tangent vector to M at the point f(n)
which act on a function h ∈ C ∞ (M). Then, defining the function Xf ∈ C ∞ (N)
by (Xf)(n) = X(n)f, the Leibnitz rule for tangent vectors implies the property
X(hl) = f ∗ h Xl + f ∗ l Xh. A map satisfying this property is called a f ∗ -derivation
(of degree 0).
In a recent paper [16] the Frölicher and Nijenhuis’ theory has been generalized
in order to include these new derivations. We recall that a f ∗ -derivation of degree
r of scalar forms on M is a R-linear map D : ∧ (M) → ∧ (N) satisfying
l=0
D (∧p (M) ) ⊂ ∧ p+r (N)D(α ∧ β) = Dα ∧ f ∗ β + (−1) pr f ∗ α ∧ Dβ (3.9)
for β ∈ ∧ (M) and α ∈ ∧p (M).
We say that D is of type i ∗ when D vanishes on ∧0 (M) and D is of type d ∗ if
D ◦ d = (−1) r d ◦ D, where d is the exterior differentiation (in M on the left hand
side and in N on the right hand side).
If we denote V p (f) the set of p-linear maps L from T N × · · · × T N to T M
satisfying that for n ∈ N and v 1 , . . . , v p ∈ T n N, τ[L(v 1 , . . . , v p )] = f(n), then we
can associate with each element L of this set a type i ∗ derivation of degree p − 1,
denoted i L , as well as a type d ∗ derivation of degree p, denoted d L . Moreover every
derivation D of degree r can be decomposed in a unique way as a sum D = i L + d R
for some L ∈ V r+1 (f) and R ∈ V r (f).

SECTIONS ALNG A MAP IN MECHANICS 7
In particular if X ∈ V 0 (f) ≡ X(f), then d X is a derivation of degree 0 such that
its action over functions on M coincides with the the action of X over C ∞ (M) as
defined above.
Suppose that f is a submersion, P is another differentiable manifold and that
g : P → N is also a submersion. If α ∈ ∧p ∧
(f) and X ∈ X(g) then ᾱ = f ∗ ◦ α ∈
p
0 (f), and we can act with d X on ᾱ obtaining d X ᾱ ∈ ∧p (P ). It can be shown
that d X ᾱ ∈ ∧ p
0 (g) (for all α ∈ ∧p (f)) if and only if there exists Y ∈ X(f) such
that f ∗ ◦ X = Y ◦ g. In this case (d X ᾱ) ∨ is a p-form along g that we will simply
denote d X α.
Similarly, for X ∈ X(g) we can act with i X on ᾱ, and so we have a (p − 1)-form
i X ᾱ on P , that always belongs to ∧ p
0
(g). The point is that if there exists Y ∈ X(f)
such that f ∗ ◦ X = Y ◦ g, then there will be a (p − 1)-form β in ∧ p−1
0
(f) such that
f ∗ β = i X ᾱ, and we can consider β as being a (p − 1)-form along f. This form will
be denoted i X α.
4. Lifting vector fields along projections
In the following we will be concerned with cases in which f are the natural
projections τ l,k : T l M → T k M, for k and l non-negative integer numbers such that
l > k (and τ 1,0 ≡ τ).
It is well-known [8] that every function f ∈ C ∞ (M) defines functions f(n) r ∈
C ∞ (T r M) for every n ≤ r by means of
f(n)
r (
[ρ]
r ) = dn
dt n {f ◦ ρ} ∣ t=0
. (4.1)
We can extend this construction to functions F ∈ C ∞ (T k M) by making use of the
canonical immersion
i k,r : T k+r M −→ T r (T k M)
[ρ] k+r ↦−→ [˜ρ k ] r , (4.2)
where ˜ρ k is the lifting to T k M of the curve ρ, as follows: F defines F(n) r ∈
C ∞ (T r (T k M)), and then F[n] r = i ∗ k,r F (n) r is a function on T k+r M. If [ρ] k+r ∈
T k+r M then
F[n]
r (
[ρ]
k+r ) = dn {
F ◦ ˜ρ
k } ∣ dt n t=0
. (4.3)
Let now X be a vector field along τ k,0 . There exists one vector field X (r) over
τ k+r,r such that
X (r) f(n) r = (Xf)r [n] , (4.4)
for any function f ∈ C ∞ (M).
We will give an explicit construction of such vector field over τ k+r,r . Let us
choose ¯X ∈ X(T k M) such that τ k,0∗ ◦ ¯X = X. Let [ρ] k+r ∈ T k+r M be an arbitrary
but fixed point of T k+r M. Then the map χ: U ⊂ R 2 → M defined by
χ(s, t) = (τ k,0 ◦ ¯X s ◦ ˜ρ k )(t), (4.5)
where ¯X s is the flow of ¯X and U is an open neighbourhood of (0, 0) ∈ R 2 , can be
used to determine a vector ξ tangent to T r M at the point [ρ] r . More precisely,
for any real number s, the curve in M, χ s (t) = χ(s, t), defines a point in T r M.
As a function of the parameter s we obtain a curve in T r M whose tangent vector
at s = 0 is the abovementioned vector ξ at [ρ] r , and in this way we will find a
vector field X (r) along τ k+r,r . This vector field is well defined: if ¯X′ ∈ X(T k M)

8 JOSÉ F. CARIÑENA⋆ , CARLOS LÓPEZ† , AND EDUARDO MARTÍNEZ⋆
is a different vector field τ k,0∗ -related to X and ρ ′ is another curve in M defining
the same (k + r)-velocity, [ρ] k+r = [ρ ′ ] k+r , then the tangent vectors defined by
χ(s, t) = (τ k,0 ◦ ¯X s ◦ ˜ρ k )(t) and by χ ′ (s, t) = (τ k,0 ◦ ¯X s ′ ◦ ˜ρ ′k )(t) coincide. Indeed,
note that for any function f ∈ C ∞ (M) we have (f ◦ χ)(0, t) = (f ◦ ρ)(t) and
(f ◦ χ ′ )(0, t) = (f ◦ ρ ′ )(t) so that
∂ n
∂t n {f ◦ ρ} ∣ ∣
(0,0)
= ∂n
∂t n {f ◦ ρ′ } ∣ ∣
(0,0)
, (4.6)
because ρ and ρ ′ define the same r-velocity.
Moreover, note that
∂
∂s {f ◦ χ} ∣ ∣
s=0
= ∂ ∂s
{
(f ◦ τk,0 ) ( X s
(˜ρ k (t) ))} ∣ ∣
s=0
= (τ k,0∗ ¯X)˜ρk (t)f
= X˜ρk (t)f
and a similar result holds for χ ′ . Then:
and
so that
= [ (Xf) ◦ ˜ρ k] (t),
∂ n+1
∂s∂t n {f ◦ χ} ∣ ∣
(0,0)
= dn
dt n {
(Xf) ◦ ˜ρ
k } ∣ ∣
t=0
= (Xf) r [n](
[ρ]
k+r ) ,
(4.7)
(4.8)
∂ n+1
∂s∂t n {f ◦ χ′ } ∣ (0,0)
= (Xf) r (
[n] [ρ ′ ] k+r) , (4.9)
∂ n+1
∂s∂t n {f ◦ χ} ∣ (0,0)
= ∂n+1
∂s∂t n {f ◦ χ′ } ∣ (0,0)
. (4.10)
This proves that X (r)( [ρ] k+r) is well defined.
From (4.9) we see that
(Xf) r (
[n] [ρ]
k+r ) = ∂n+1
∂s∂t n {f ◦ χ} ∣ (0,0)
= d { ∂
n
ds ∂t n {f ◦ χ s} ∣ } ∣∣s=0
t=0
= d {
f r
ds
(n)(
[χs ] r)} ∣
∣s=0
(4.11)
= ( X (r) f r (n))(
[ρ]
k+r ) ,
so that X (r) satisfies the condition (4.4).
Since two tangent vectors agreeing over functions of the type f(n) r must be equal,
we can conclude that X (r) is uniquely defined.
As a consequence of (4.4) we have for every r > l ≥ 0
τ r,l∗ ◦ X (r) = X (l) ◦ τ k+r,k+l , (4.12)
(provided that X (0) = X). Indeed, for n ≤ l we have that f(n) r = τ r,l ∗ f(n) l and
F[n] r = τ k+r,k+l ∗ F[n] l , for f ∈ C∞ (M) and F ∈ C ∞ (T k M), so that from (4.4):
(τ r,l∗ ◦ X (r) )f l (n) = X(r) (τ r,l ∗ f l (n) ) = X(r) (f r (n) ) = (Xf)r [n] , (4.13)

SECTIONS ALNG A MAP IN MECHANICS 9
and
(X (l) ◦ τ k+r,k+l )f l (n) = τ k+r,k+l ∗ (X (l) f l (n) )
= τ ∗ k+r,k+l (Xf) l [n]
(4.14)
= (Xf) r [n] ,
what proves (4.12)
In the particular case of X being the restriction of a vector field Y ∈ X(M),
X = Y ◦ τ k,0 , the vector field X (r) reduces to the restriction of the complete lift
Y c,r ∈ X(T r M), X (r) = Y c,r ◦ τ k+r,r . This is the reason why we call X (r) ∈
X(τ k+r,r ) the generalized complete lift of X ∈ X(τ k,0 ).
If we consider a local chart (V, q) of M and the induced charts ( ¯V k , ¯q k ) of T k M,
then, if the coordinate expression for X is
the corresponding expression for X (r) is
r∑
X (r) = (η i ) r ( ∂
[n]
X = η i( ∂
∂q i ◦ τ) , (4.15)
n=0
∂q i (n)
5. The total time derivative operator
◦ τ k+r,r
)
. (4.16)
There exists a canonical vector field along τ k+1,k [8] given by the canonical
immersion (4.2) for r = 1, namely
i k,1 : T k+1 M −→ T (T k M)
[ρ] k+1 ↦−→ [˜ρ k ] 1 = ˙˜ρ k .
Its action on functions F ∈ C ∞ (T k M) is given by
(
dik,1 F )( [ρ] k+1) = i k,1
(
[ρ]
k+1 ) F
(5.1)
= ˙˜ρ k F
= d dt
{
F ◦ ˜ρ
k } ∣ ∣
t=0
,
(5.2)
showing that d ik,1 F = F[1] 1 .
When k = 0, the map i 0,1 is but the identity in T M, so that i 0,1 = T (0) , the
vector field along τ 1,0 described in Section 3. T (0) can be lifted to a vector field
T (k) along τ k+1,k . The resulting vector field T (k) is but i k,1 . Indeed, if F = H k−m
[n]
for H ∈ C ∞ (T m M) then
so that
(
dik,1 H k−m )(
[n] [ρ]
k+1 ) = d {H k−m
dt
[n]
◦ ˜ρ k} ∣
∣t=0
= d { d
n
dt ds n {H ◦ ˜ρm } ∣ } ∣∣t=0
s=t
d ik,1 H k−m
[n]
= dn+1
dt n+1 {H ◦ ˜ρm } ∣ t=0
= ( H k−m+1 )(
[n+1] [ρ]
k+1 ) ,
(5.3)
= H k−m+1
[n+1]
. (5.4)

10 JOSÉ F. CARIÑENA⋆ , CARLOS LÓPEZ† , AND EDUARDO MARTÍNEZ⋆
When taking m = 0 in (5.4) we have that for any function f ∈ C ∞ (M)
d ik,1 f k (n) = f k+1
(n+1) = (T (0) f) k [n] (5.5)
and taking into account that the different liftings of T (0) are characterized by this
condition, we can assure that i k,i = T (k) . The derivation d T (k) is called the total
time derivative operator in T k M.
The lifting X (r) of a vector field X ∈ X(τ k,0 ) satisfies the following ‘commutation’
rule
d X (r+1) ◦ d T (r) − d T (k+r) ◦ d X (r) = 0 (5.6)
Indeed, if f ∈ C ∞ (M), then using (5.5) and (4.5)
(
dX (r+1) ◦ d T (r))
(f
r
(n) ) = d X (r+1)(f r+1
(n+1) ) = (Xf)r+1 [n+1] , (5.7)
and ( ) )
dT (k+r) ◦ d X (r) (f
r
(n) ) = d T (k+r)(
(Xf)
r
[n] = (Xf)
r+1
[n+1] , (5.8)
what proves (5.6).
The property (5.6) is equivalent to
i X (r+1) ◦ d T (r) − d T (k+r) ◦ i X (r) = 0 (5.9)
because on C ∞ (M) both terms vanish, and on exact forms df
i X (r+1) ◦ d T (r)df − d T (k+r) ◦ i X (r)df =
= ( i X (r+1) ◦ d ◦ d T (r) − d T (k+r) ◦ i X (r) ◦ d ) f
= ( (5.10)
)
d X (r+1) ◦ d T (r) − d T (k+r) ◦ d X (r) f = 0.
Moreover, property (5.6) ( or equivalently (5.9)) characterizes prolongations of vector
fields along τ k,0 .
As a consequence of this property we have that if α ∈ ∧ (τ r,l ) then
if r ≤ k + l, and
if r > k + l.
i X (l+1)(
dT (r)α ) = d T (k+l)(
iX lα ) ∈ ∧ (τ k+l+1,l+1 ), (5.11)
i X (r+1)(
dT (r)α ) = d T (r)(
iX lα ) ∈ ∧ (τ r+1,l+1 ), (5.12)
6. The vertical endomorphism and the variational operator
There exists a canonical type (1, 1) tensor field S k on the k-th order tangent bundle
T k M called the vertical endomorphism [8]. It is defined by means of the vertical
lifting map ξ V Q : T τ k,k−1 (Q)(T k−1 M) → T Q (T k M), whose image is the set V (τ k,0 )
of the τ k,0 -vertical tangent vectors to T k M, and the differental of the projection
τ k,k−1 : T k M → T k−1 M at the point Q; for v ∈ T Q (T k M)
Its local expression in natural coordinates is
and it satisfies
(1) S k+1
k
= 0
(2) Kern(Sk n) = V (τ k,k−n)
S k (v) = ξ V Q(τ k,k−1∗Q v). (6.1)
k−1
∑
S k = (n + 1)
n=0
∂
∂q i (n+1)
⊗dq i (n) , (6.2)

SECTIONS ALNG A MAP IN MECHANICS 11
(3) Im(S n k ) = V (τ k,n−1)
(4) The Nijenhuis tensor [S k , S k ] vanishes
(5) L ∆ S k = −S k
(6) S ∗ k (df (n)) = n df (n−1)
where n = 0, . . . , k and ∆ is the Liouville vector field on T k M defined by ∆(Q) =
ξ V Q (T (k−1) (Q)).
Proposition 6.1. Let X be a vector field along τ l,0 . The operator [X (k) , S n k ] mapping
∧1 (T k M) in ∧1 (T k+l M) and defined by
for n = 1, . . . , k + l, satisfies
where F is a function on T k M.
[X (k) , S n k ] = d X (k) ◦ S ∗n
k
− S ∗n
k+l ◦ d X (k), (6.3)
[X (k) , S n k ](F α) = (τ k+l,k ∗ F )[X (k) , S n k ](α), (6.4)
Proof. Using that d X (k)
is a derivation along τ k+l,k we have
(F α) =
because τ k+l,k ∗ ◦ S ∗n
k
= d X (k)(F Sk
∗n (α)) − Sk+l(τ ∗n ∗ k+l,k F d X (k)α + d X (k)F τ ∗ k+l,k α)
= d X (k)F τ ∗ k+l,k (Sk
∗n (α)) + τ ∗ k+l,k F d X (k)(Sk
∗n (α))
− τ ∗ k+l,k F Sk+l(d ∗n
X (k)α) − d X (k)F Sk+l(τ n ∗ k+l,k (α))
= τ k+l,k ∗ F [X (k) , S n k ](α),
= S ∗n
k+l ◦ τ k+l,k ∗ .
(6.5)
□
In particular, if X = T (0) , i.e. the canonical vector field along τ 1,0 , we have [8]
[T (k) , S n k ] = −nτ k+1,k ∗ ◦ S ∗n−1
k
. (6.6)
It is easy to see from this equation (by induction, for instance) that
min{m,n}
∑
d m ◦ S ∗n
T (k) k =
r=0
(−1) r ( m
r
) n!
(n − r)! τ k+m,k+m−r ∗ ◦ S ∗n−r
k+m−r ◦ dm−r T (k) , (6.7)
(where we have written d n for d
T (m) T (m+n−1) ◦ · · · ◦ d T (m)) a relation that we will use
later in this section.
Proposition 6.2. Let X be a vector field along τ l,0 and f a function in T l M.
Then
k∑
(fX) (k) 1
=
n! (τ k+l,l+n ∗ d n f)S n T (l) k (X (k) ). (6.8)
n=0
Proof. Let us write Y k for Y k = ∑ k
n=0 1 n! (τ k+l,l+n ∗ d n T (l) f)S n k (X(k) ). First we prove
that Y k commutes with T (k) ; for α ∈ ∧1 (T k M) taking into account (5.9) and (6.6)

12 JOSÉ F. CARIÑENA⋆ , CARLOS LÓPEZ† , AND EDUARDO MARTÍNEZ⋆
we see that
(d T (k+l) ◦ i Y k − i Y k+1 ◦ d T (k))α =
( k∑
)
1
= d T (k+l)
n! (τ k+l,l+n ∗ d n f)i
T (l) X (k)Sk
∗n (α)
n=0
=
=
k+1
∑
−
n=0
n=0
1
n! (τ k+l,l+n ∗ d n f)i
T (l) X (k+1)Sk
∗n (d T (k)α)
k∑ 1
n! (τ k+l+1,l+n+1 ∗ d n+1 f)τ ∗ T (l) k+l+1,k+l i X (k)Sk
∗n (α)
k+1
∑ 1
+
n! (τ k+l+1,l+n ∗ d n f)i
T (l) X (k+1)[T (k) , Sk n ](α)
n=0
= 0.
n=0
k∑ 1
n! (τ k+l+1,l+n+1 ∗ d n+1 f)τ ∗ T (l) k+l+1,k+l i X (k)Sk
∗n (α)
k+1
∑
−
n=1
1
n! (τ k+l+1,l+n ∗ d n f)i
T (l) X (k+1)τ ∗ k+1,k S ∗n−1
k
(α)
(6.9)
This implies that Y k is the prolongation of a vector field along τ l,0 . Since τ k,0∗ ◦Y k =
fX we conclude Y k = (fX) (k) .
□
Let α be a 1-form on T k M. Then, for s = 1, . . . , k we can define operators
: ∧1 (T k M) → ∧1 (T 2k−s M) be means of
〈
〉
∑s−1
∗
τ 2k+l−s,k+l α, (fX) (k) 1
−
n! (τ k+l,l+n ∗ d n f)S n T (l) k (X (k) )
D (k)
s
= (τ 2k+l−s,s+l ∗ d s T (l) f)
n=0
〈
D (k)
s (α), X (2k−s)〉 mod d T ,
(6.10)
where mod d T means that the equality holds except for the addition of the total
time derivative of a function. A straightforward calculation shows that
k∑
D (k)
s = (−1) n+s 1 n! τ 2k−s,k+n−s ∗ ◦ d n−s ◦ S ∗n
T (k) k . (6.11)
n=s
From this expression it is easy to see that the image of D (k)
s
forms, and that the following relations holds
are τ 2k−s,k−s -semibasic
D (k)
s−1 = 1 s! τ 2k−s+1,k ∗ ◦ S ∗s−1
k
− d T (2k−s) ◦ D (k)
s (6.12)
S 2k−s ◦ D (k)
s
= sτ 2k−s,2k−s−1 ∗ ◦ D (k)
s+1 . (6.13)
Equation (6.13) implies that only D (k)
0 and D (k)
1 are relevant and they are called
the variational operator E (k) and the Cartan operator S (k) , respectively. They are
related by
E (k) = τ 2k,k ∗ − d T (2k−1) ◦ S (k) , (6.14)
as the equation (6.12) claims.

SECTIONS ALNG A MAP IN MECHANICS 13
Another usefull expression for D (k) is given by
D (k)
s =
k−s
∑
n,m=0
(−1) n 1 m
(n + s)!(
n
)
τ 2k−s,k+n ∗ ◦ S ∗n+s
k+n ◦ dn T (k) , (6.15)
as it can be easily proved using the expression (6.7). In this sum we have assumed
the convention ( m
n)
= 0 for m < n, otherwise the sum must be only for k − s ≥
m ≥ n ≥ 0.
Proposition 6.3. Let X be a vector field along τ l,0 and define a map [X (k) , S (k) ]
mapping ∧1 (T k M) in ∧1 (T 2k+2l−1 M) by means of
[X (k) , S (k) ] = τ 2k+2l−1,2k+l−1 ∗ ◦ d X (2k−1) ◦ S (k) − S (k+l) ◦ d X (k). (6.16)
Then if α is a τ k,0 -semibasic 1-form on T k M, we have
k+l−1
∑
[X (k) , S (k) ](α) = { (−1) n 1
n,m=0
(n + 1)!( m
n
)
×
τ 2k+2l−1,k+l+n ∗ ◦ [X (k+n) , S n+1
k+n ] ◦ dn T (k) }(α). (6.17)
Proof. Indeed, using the expressions (6.15) and (5.6) we have
[X (k) , S (k) ] =
=
=
k−1
∑
n,m=0
+
k+l−1
∑
n,m=0
k+l−1
∑
n,m=0
(−1) n 1 m
(n + 1)!(
n
(−1) n 1 m
(n + 1)!(
n
(−1) n 1 m
(n + 1)!(
n
− ∑ ∗(−1) n 1
(n + 1)!( m
n
)
τ ∗ 2k+2l−1,k+n+l ◦ d X (k+n) ◦ S ∗n+1
k+n
◦ dn T (k)
)
τ 2k+2l−1,k+n+l ∗ ◦ S ∗n+1
k+l+n ◦ d X (k+n) ◦ dn T (k)
)
τ 2k+2l−1,k+l+n ∗ ◦ [X (k+n) , S n+1
k+n ]∗ ◦ d n T (k)
)
τ 2k+2l−1,k+n+l ∗ ◦ d X (k+n) ◦ S ∗n+1
k+n ◦ dn T (k) (6.18)
where ∑ ∗ means sum for k − 1 ≤ n ≤ k + l − 1, 0 ≤ m ≤ k + l − 1 and k − 1 ≤ m ≤
k +l −1, 0 ≤ n ≤ k −1. But this sum vanishes on τ k,0 -semibasic forms, because if α
is τ k,0 -semibasic then d n α is τ
T (k) k+n,n -semibasic and thus S ∗n+1
k+n (dn α) = 0. □
T (k)
What it is important in this relation is that the right hand side is a sum of
‘linear’ operators acting on total time derivatives of α.
In particular if X = T (0) , then
[T (k) , S (k) ] = τ 2k+1,2k ∗ ◦ E (k) (6.19)
because S (k) ◦ d T (k−1) = τ ∗ 2k−1,k−1 and (6.14). Moreover, this implies that
[X (2k) , S (2k) ] ◦ E (k) = τ 4k+2l−1,2k+2l−1 ∗ [X (k) , S (k) ]. (6.20)

14 JOSÉ F. CARIÑENA⋆ , CARLOS LÓPEZ† , AND EDUARDO MARTÍNEZ⋆
Indeed,
[X (2k) , S (2k) ] ◦ E (k) =
= [X (2k) , S (2k) ] ◦ (τ ∗ 2k,k − d T (2k−1) ◦ S (k) )
= [X (2k) , S (2k) ] ◦ τ ∗ 2k,k + S (2k+l) ◦ d T (2k+l−1) ◦ d X (2k−1) ◦ S (k)
− τ ∗ 4k+2l−1,4k+l−1 ◦ d X (4k−1) ◦ S (2k) ◦ d T (2k−1) ◦ S (k)
= τ ∗ 4k+2l−1,2k+2l−1 ◦ [X (k) , S (k) ] + τ ∗ 4k+2l−1,2k+l−1 ◦ d X (2k−1) ◦ S (k)
− τ ∗ 4k+2l−1,4k+l−1 ◦ d X (4k−1) ◦ τ ∗ 4k−1,2k−1 ◦ S (k)
= τ ∗ 4k+2l−1,2k+2l−1 ◦ [X (k) , S (k) ].
(6.21)
7. Ordinary differential equations
The geometrical interpretation of a system of (N + 1)st-order differential equations
(hereafter shortened to (N + 1)-ode)
(
φ i q, dq
)
dt , . . . , dN+1 q
dt N+1 = 0 (7.1)
is given by a closed imbeded submanifold S of T N+1 M satisfying [19]
τ N+1,N −1 (τ N+1,N S) ≠ S. (7.2)
The solutions of this system are the curves σ : R → M whose lift to T N+1 M,
˜ρ N+1 : R → T N+1 M, takes values in S.
We are interested on (N + 1)-ode that can be presented in normal form:
d N+1 (
q
dt N+1 = f i q, dq )
dt , . . . , dN q
dt N , (7.3)
which corresponds to the image of a section γ of τ N+1,N , S = Im(γ). Then, a
solution of this (N + 1)-ode is a curve ρ in M such that ˜ρ N+1 = γ ◦ ˜ρ N . The
composition map Γ = i 1,N ◦ γ, or equivalently Γ = T (N) ◦ γ, is a vector field on
T N M that is also called a (N + 1)-ode because its integral curves are natural
liftings of solutions of γ. The associated derivations d Γ and d T (N) are related by
d Γ = γ ∗ ◦ d T (N). (7.4)
We can associate with every (N + 1)-ode a sequence of differential equations of
order greater than (N + 1).
Proposition 7.1. There exists exactly one (N + 2)-ode ˜Γ (1) in M satisfying
This ode is said to be the first ode-prolongation of Γ.
d˜Γ(1) ◦ d T (N) = d T (N) ◦ d Γ . (7.5)
Proof. In order to prove this we consider and atlas an a partition of unity subordinate
to this atlas. In a local chart the condition (7.5) imposes
˜Γ (1) q i (n) = qi (n+1)
˜Γ (1) q i (N+1) = d T (N)fi ,
n = 1, . . . , N
(7.6)

SECTIONS ALNG A MAP IN MECHANICS 15
where f i are given by f i = Γq(N) i . Since ˜Γ (1) must be a ode, its expression in this
chart is
N∑
˜Γ (1) = q i ∂
(n+1)
∂q i + d T (N)f i ∂
n=0
(n)
∂q(N+1)
i . (7.7)
Now using the partition of unity we obtain a global (N + 2)-ode.
□
The l-th ode-prolongation ˜Γ (l) of Γ is defined by recurrence as the first odeprolongation
of the (l − 1)-th ode-prolongation. The section of the projection
τ N+l+1,N+l associated to ˜Γ (l) will be denoted ˜γ (l) . We can construct from they the
sections γ (l) : T N M → T N+l+1 M of τ N+l+1,N defined by γ (l) = ˜γ (l) ◦ ˜γ (l−1) ◦· · ·◦γ.
We call γ (l) the l-th prolongation of γ because they satisfy
τ N+l+1,N+m+1 ◦ γ (l) = γ (m) , for l > m > 0, (7.8)
and a curve ρ in M is a solution of γ, ˜ρ N+1 = γ ◦ ˜ρ N , if and only if ρ is also a
solution of γ (l) : ˜ρ N+l+1 = γ (l) ◦ ˜ρ N . The equation (7.8) also holds for m = −1, 0 if
we assume the convention γ (−1) = id T N M and γ (0) = γ.
Lemma 7.2. The prolongations of a (N + 1)-ode γ satisfy
for l ≥ 0, and
for l ≥ r.
d Γ ◦ γ (l)∗ = γ (l)∗ ◦ d˜Γ(l+1), (7.9)
γ (l)∗ ◦ d l−r+1 = d l−r+1
T (N+r) Γ
◦ γ (r−1)∗ , (7.10)
Proof. We prove (7.9) by induction on l. Since the definition of ˜Γ (1) we have
d˜Γ(1) ◦ d T (N) − d T (N) ◦ d Γ = 0 and composing with γ ∗
γ ∗ ◦ (d˜Γ(1) ◦ d T (N) − d T (N) ◦ d Γ ) = γ ∗ ◦ d˜Γ(1) ◦ d T (N) − d Γ ◦ d Γ
= (γ ∗ ◦ d˜Γ(1) − d Γ ◦ γ ∗ ) ◦ d T (N),
so that γ ∗ ◦ d˜Γ(1) − d Γ ◦ γ ∗ vanishes on functions of the type f (N+1) . Moreover
(7.11)
(γ ∗ ◦ d˜Γ(1) − d Γ ◦ γ ∗ ) ◦ τ N+1,N ∗ = γ ∗ ◦ d T (N) − d Γ = 0, (7.12)
and then γ ∗ −d ◦d˜Γ(1) Γ ◦γ ∗ vanishes on functions of the type f (n) for all n ≥ 0, what
implies that it vanishes on every function. This proves (7.9) for l = 0. Note that
this implies ◦ ˜γ (l) = ˜γ (l) ◦ since ˜Γ (l+1) is the first ode-prolongation of
d˜Γ(l) d˜Γ(l+1)
˜Γ (l) . Then assuming that d Γ ◦ γ (l−1)∗ = γ (l−1)∗ ◦ ˜Γ (l) holds and taking into account
that γ (l) = ˜γ (l) ◦ γ (l−1) we have:
d Γ ◦ γ (l)∗ = d Γ ◦ γ (l−1)∗ ◦ ˜γ (l)∗
= γ (l−1)∗ ◦ d˜Γ(l) ◦ ˜γ (l)∗
= γ (l−1)∗ ◦ ˜γ (l) ◦ d˜Γ(l+1)
(7.13)
= γ (l)∗ ◦ d˜Γ(l+1).
Now, we also prove (7.10) by induction on l. For l = r since (7.4):
γ (r)∗ ◦ d T (N+r) = γ (r−1)∗ ◦ ˜γ (r)∗
= γ (r−1)∗ ◦ d˜Γ(r)
= d Γ ◦ γ (r−1)∗ .
(7.14)

16 JOSÉ F. CARIÑENA⋆ , CARLOS LÓPEZ† , AND EDUARDO MARTÍNEZ⋆
Asuming that
holds, we have
γ (l−1)∗ ◦ d T (N+l−1) ◦ · · · ◦ d T (N+r)
= d l−r
Γ
◦ γ (r−1)∗ (7.15)
γ (l)∗ ◦ d T (N+l) ◦ d T (N+l−1) ◦ · · · ◦ d T (N+r) =
= γ (l−1)∗ ◦ ˜γ (l)∗ ◦ d T (N+l) ◦ d T (N+l−1) ◦ · · · ◦ d T (N)
= γ (l−1)∗ ◦ d˜Γ(l) ◦ d T (N+l−1) ◦ · · · ◦ d T (N+r)
= d Γ ◦ γ (l−1)∗ ◦ d T (N+l−1) ◦ · · · ◦ d T (N)
= d Γ ◦ d l−r
Γ
◦ γ (r−1)∗
= d l−r+1
Γ
◦ γ (r−1)∗ .
(7.16)
□
The following proposition gives us the relation between the operators S (N) and
E (N) here defined and the operators σ Γ and E Γ defined by Crampin et. al. [8]:
σ Γ =
E Γ =
N∑
(−1) n+1 1 n! dn−1
n=1
Γ
◦ S ∗n
N ,
N∑
(−1) n 1 n! dn Γ ◦ S ∗n
n=0
N .
(7.17)
Proposition 7.3. If Γ is a (N + 1)-ode and γ is the section associated to Γ, then
(1) γ (N−2)∗ ◦ S (N) = σ Γ ,
(2) γ (N−1)∗ ◦ E (N) = E Γ .
Proof. Using (7.10), (7.8) and the definition of S (N) we have
γ (N−2)∗ ◦ S (N) =
=
=
that proves (1), and
N∑
(−1) n+1 1 n! γ(N−2)∗ ◦ τ ∗ 2N−1,N+n−1 ◦ d n−1 ◦ S ∗n
T (N) N
n=1
N∑
(−1) n+1 1 n! γ(n−2)∗ ◦ d n−1 ◦ S ∗n
T (N) N
n=1
N∑
(−1) n+1 1 n! dn−1
n=1
γ (N−1)∗ ◦ E (N) =
that proves (2).
=
=
Γ
◦ S ∗n
N ,
N∑
(−1) n+1 1 n! γ(N−1)∗ ◦ τ ∗ 2N,N+n ◦ d n ◦ S ∗n
T (N)
n=0
N∑
(−1) n+1 1 n! γ(n−1)∗ ◦ d n ◦ S ∗n
T (N)
n=0
N∑
(−1) n+1 1 n! dn Γ ◦ S ∗n
n=0
N ,
N
N
(7.18)
(7.19)
□

SECTIONS ALNG A MAP IN MECHANICS 17
Associated to a (N + 1)-ode Γ there is a set X Γ of vector fields [18] on T N M
such that its flow maps Γ into another (N + 1)-ode. Then if Y ∈ X Γ it satisfies
S N ([Γ, Y ]) = 0, or equivalently τ N,N−1∗ ◦ [Γ, Y ] = 0. The fact that all symmetries
of Γ are elements of X Γ makes this set to be very interesting. The local expression
of an element Y in X Γ is
Y = η i
∂
∂q i (0)
+
N∑
d n Γη i
n=1
∂
∂q i (n)
(7.20)
with η i functions on T N M.
Proposition 7.4. There exists a one-to-one correspondence I Γ : X(τ N,0 ) → X Γ
between the set of vector fields along τ N,0 and the set X Γ given by X ↦→ X (N) ◦
γ (N−1) . The inverse of this map is τ N,0∗
∣
∣XΓ
.
Proof. First we prove that X(γ) = X (N) ◦ γ (N−1) is an element of X Γ , i. e., it
satisfies d [Γ,X(γ)] ◦ τ N,N−1 ∗ = 0. Indeed,
d [Γ,X(γ)] ◦ τ ∗ N,N−1 = d Γ ◦ d X(γ) ◦ τ ∗ ∗
N,N−1 − d X(γ) ◦ d Γ ◦ τ N,N−1
= d Γ ◦ γ (N−1)∗ ∗
◦ d X (N) ◦ τ N,N−1
− γ (N−1)∗ ◦ d X (N) ◦ γ ∗ ∗
◦ d T (N) ◦ τ N,N−1
= d Γ ◦ γ (N−1)∗ ◦ τ ∗ 2N,2N−1 ◦ d X (N−1)
(7.21)
− γ (N−1)∗ ◦ d X (N) ◦ γ ∗ ◦ τ N+1,N ∗ ◦ d T (N−1)
= {d Γ ◦ γ (N−2)∗ − γ (N−1)∗ ◦ d T (2N−1)} ◦ d X (N−1)
= 0.
∣
In order to see that the inverse of I Γ is τ ∣XΓ N,0∗ , we must note that two vector
fields Y and Y ′ in X Γ projecting on the same vector field along τ N,0 , τ N,0∗ (Y −Y ′ ) =
0, are equal as the coordinate expression (7.20) shows. Then if Y ∈ X Γ and we
take the vector field X = τ N,0∗ ◦ Y along τ N,0 we have that X(γ) ∈ X Γ and
τ N,0∗ ◦ (Y − X(γ)) = 0 what implies X(γ) = Y .
□
8. Lagrangian Mechanics
We have seen in section 3 that the basic canonical objects on both the tangent
and cotangent bundle can be redefined as sections along maps. Once a Lagrangian
function L ∈ C ∞ (T M) is considered, new objects associated to L are defined, as
the Poincaré-Cartan 1-form θ L = S ∗ (dL), the Legendre map FL : T M → T ∗ M
or the time evolution operator [2] K L : C ∞ (T ∗ M) → C ∞ (T M), They can also be
redefined as sections along maps. For example, the 1-form θ L is semibasic, so that
it can be seen as a 1-form along τ, θ L ∈ ∧ (τ), θ L = ∂L
∂v
(dq i ◦ τ), and then, it is
i
identified with the Legendre map
FL≡θ L
T ∗ M
T M M
τ
π
(8.1)

18 JOSÉ F. CARIÑENA⋆ , CARLOS LÓPEZ† , AND EDUARDO MARTÍNEZ⋆
With L and the isomorphism Ψ : T T ∗ M → T ∗ T M we define the time evolution
operator as a vector field along FL [11], K = Ψ −1 ◦ dL
K
T T ∗ M
T M T ∗ M
FL
τ T ∗ M
(8.2)
and, as a generalization of the vertical endomorphism S, a (1,1) tensor field along
FL⊗ id can also be defined [3]
T T ∗ M
R L
T M T ∗ M⊗ M T M
FL⊗ id
π T ∗ M ⊗τ T M
(8.3)
given by R L (v) = (dq i ◦ FL)(v)⊗∂ v i. These redefinitions make some relations
trivial, for example, FL ∗ (θ o ) = θ L , taking into account that ˇθ 0 ≡ id T ∗ M and
FL ≡ ˇθ L .
In higher order Mechanics, the Lagrangian is a function on T k M so that (assuming
that L is regular) the Euler-Lagrange equations define in a unique way a
2k-th order ordinary differential equation. More precisely, the Lagrangian defines
the Cartan form θ (k)
L
= S(k) (dL) wich is a τ 2k−1,k−1 -semibasic 1-form, and the
Euler-Lagrange form δL = E (k) (dL) wich is a τ 2k,0 -semibasic 1-form. They can
(k)
be considered as forms along τ 2k−1,k−1 and τ 2k,0 , respectively, and then ˇθ
L
coincides
whith the Legendre transformation FL: T 2k−1 M → T ∗ (T k−1 M) defined
by L. Since E (k) = τ ∗ 2k,k − d T (2k−1) ◦ S (k) , the relation between θ (k)
L
and δL is
δL = τ ∗ 2k,k dL − d T (2k−1)θ (k)
L
. The Euler-Lagrange equations are the (2k)-ode γ
such that
γ ∗ δL = 0, (8.4)
or equivalently, if Γ is the vector field associated to γ,
d Γ θ (k)
L − τ 2k−1,k ∗ dL = 0. (8.5)
ˇθ
(k)
If we define ω (k)
L
= −dθ(k) L
and E L =
L
(T (k−1) ) − τ ∗ 2k−1,k L then it is easy to
prove that Γ is the Hamiltonian vector field of the Hamiltonian dynamical system
(T 2k−1 M, ω (k)
L
, E L). It is also easy to see that i T (2k−1)ω (k)
L
− τ 2k,2k−1 ∗ dE L = −δL.
Since solutions of γ are solutions of γ (l) we can deduce
γ (l)∗ δL (l) = 0 (8.6)
where δL (l) = d l T (2k) δL. Indeed, from (7.10) for r = 1 and N = 2k − 1 we have
γ (l)∗ d l T (2k) δL = d l Γ ◦ γ∗ δL = 0.
The following proposition gives us the variations of the Lagrangian and the
Cartan form under the prolongation of a vector field along τ 2k−1,0 .
Proposition 8.1. Let X be a vector field along τ 2k−1,0 , X ∈ X(τ 2k−1,0 ). Then the
following properties hold:
d X (k)L = d T (3k−2)(ˇθ L (X (k−1) )) + τ 3k−1,2k ⋆ [(δL) ∨ (X)], (8.7)

SECTIONS ALNG A MAP IN MECHANICS 19
+
3k−2
∑
n,m=0
τ 8k−3,4k−2 ∗ d X (2k−1)θ (k)
L
(−1) n 1 m
(n + 1)!(
n
= τ 8k−3,6k−3 ∗ θ (3k−1)
d X (k) L +
)
τ 8k−3,4k+n−1 ∗ [X (2k+n) , S n+1
2k+n ](δL(n) ).
(8.8)
Proof. Since δL = τ ∗ 2k,k dL−d T (2k−1)θ (k)
L
and taking into account (4.12) and (5.12)
we have
i X (2k)δL = i X (2k)[τ ∗ 2k,k dL − d T (2k−1)θ (k)
L ]
= τ ⋆ 4k−1,3k−1 d X (k)L − i X (2k)d T (2k−1)θ (k)
(8.9)
L
= τ ⋆ 4k−1,3k−1 d X (k)L − d T (4k−2)i X (2k−1)θ (k)
L .
Now, since δL is semibasic over M and θ L is semibasic over T k−1 M, we can put
and
i X (2k)δL = τ 4k−1,2k ⋆ (δL) ∨ (X) (8.10)
i X (2k−1)θ L = τ 4k−2,3k−2
⋆ ˇθL (X (k−1) ), (8.11)
so that
{ }
⋆
τ 4k−1,3k−1 d X (k)L − d T (3k−2)(ˇθ L (X (k−1) )) − τ ⋆ 3k−1,2k [(δL) ∨ (X)] = 0. (8.12)
and since τ 4k−1,3k−1 is a submersion we get the abovementioned result.
In order to prove (8.8) we apply τ 8k−3,6k−3 ∗ [X (k) , S (k) ] = [X (2k) , S (2k) ] ◦ E (k) to
dL, and so we obtain
τ 8k−3,4k−2 ∗ d X (2k−1)θ (k)
L
= τ 8k−3,6k−3 ∗ θ (3k−1)
d X (k) L + [X(2k) , S (2k) ](δL). (8.13)
Now using the expression (6.17) for [X (k) , S (k) ] we get (8.8).
Lemma 8.2. Let L be a regular Lagrangian and G a constant of the motion for
the dynamics defined by L. Then there exists a vector field X along τ 2k−1,0 such
that
d T (2k−1)G = −δL ∨ (X). (8.14)
Proof. Since L is regular, there exists a vector field Y on T 2k−1 M such that
i Y ω (k)
L
= −dG. Moreover, Y is a symmetry of the Hamiltonian dynamical system
(T 2k−1 M, ω (k)
L
, E L). Choosing a vector field Ȳ on T 2k M projecting on Y and
contracting dG + i Y ω (k)
L
= 0 with T (2k−1) we have
d T (2k−1)G + i T (2k−1)i Y ω (k)
L =
= d T (2k−1)G − i Ȳ i T (2k−1)ω (k)
L
= d T (2k−1)G − i Ȳ {i T (2k−1)ω (k)
L
− τ 2k,2k−1 ∗ dE L } − i Ȳ τ ∗ 2k,2k−1 dE L
= d T (2k−1)G + i Ȳ δL − τ ∗ 2k,2k−1 d Y E L
= d T (2k−1)G + i Ȳ δL,
□
(8.15)
where we have used d Y E L = 0. Now defining X = τ 2k−1,0∗ ◦ Y , since δL is τ 2k,0 -
semibasic we obtain d T (2k−1)G = −δL ∨ (X).
□
Definition 8.3. Let L be a Lagrangian function in T k M and X a vector field
along τ 2k−1,0 . We say that X is a symmetry of the Lagrangian [5] if there exists a
function F ∈ C ∞ (T 3k−2 M) such that d X (k)L = d T (3k−2)F .

20 JOSÉ F. CARIÑENA⋆ , CARLOS LÓPEZ† , AND EDUARDO MARTÍNEZ⋆
Now, we prove the following generalized version of Noether’s Theorem for higherorder
Lagrangians that establishes a one-to-one correspondence between symmetries
of the Lagangian function and first integrals of the dynamics defined by this Lagrangian.
Theorem 8.4. If X is a symmetry of the Lagrangian and F is the associated funtion
to X, then there exists a function G ∈ C ∞ (T 2k−1 M) such that τ ⋆ 3k−2,2k−1 G =
F − ˇθ L (X (k−1) ) and it is a constant of the motion. Conversely, given a constant
of the motion G ∈ C ∞ (T 2k−1 M) there exist a vector field X along τ 2k−1,0 and a
function F = τ ⋆ 3k−2,2k−1 G + ˇθ L (X (k−1) ) in T 3k−2 M such that X is a symmetry of
the Lagrangian with associated function F . Moreover, X(γ) is a symmetry of the
Hamiltonian dynamical system (T 2k−1 M, ω (k)
L
, E L).
Proof. Let us suppose that X ∈ X(τ 2k−1,0 ) is a symmetry of the Lagrangian, namely
Then, from (8.7) we have
d X (k)L = d T (3k−2)F. (8.16)
d T (3k−2)[F − ˇθ L (X (k−1) )] = −τ 3k−1,2k ⋆ [(δL) ∨ (X)] (8.17)
so that, there exists a funtion G ∈ C ∞ (T 2k−1 M) such that τ ⋆ 3k−2,2k−1 G = F −
ˇθ L (X (k−1) ), i. e., d T (2k−1)G = −(δL) ∨ (X), and composing it with γ ⋆ we have that
d Γ G = 0, i. e., G is a constant of the motion.
Conversely, let G ∈ C ∞ (T 2k−1 M) be a constant of the motion and X the vector
field along τ 2k−1,0 such that d T (2k−1)G = −(δL) ∨ (X). Then defining F =
ˇθ (k)
L (X(k−1) ) + τ ⋆ 3k−2,2k−1 G, the equation (8.7) reads d X (k)L = d T (3k−2)F .
Now, we prove that X(γ) is a symmetry of (T 2k−1 M, dθ (k)
L
account (7.8), (8.8) and θ (3k−1)
d T (3k−2) F = τ 6k−2,3k−2 ∗ dF we have
d X(γ) θ (k)
L = d(γ(k−2)∗ F )+
+
3k−2
∑
m,n=0
(−1) n 1 m
(n + 1)!(
n
, E L). Taking into
)
γ (2k+n−1)∗ [X (2k+n) , S n+1
2k+n ](δL(n) ), (8.18)
and since [X (2k+n) , S n−1
2k+n
] are ‘linear’ operators the last term vanishes, so that
d X(γ) θ (k)
L
= d(γ(k−2)∗ F ) and d X(γ) ω (k)
L = 0. □
Note that i X(γ) ω (k)
L
= −d(γ(k−2)∗ F − θ (k)
L
(X(γ)) and since the function F −
ˇθ L (X (k−1) ) is the pull-back τ ⋆ 3k−2,2k−1 G we obtain i X(γ) ω (k)
L
= −dG.
Note also that the equation τ ⋆ 3k−2,2k−1 G = F − ˇθ L (X (k−1) ) imposes severes
restrictions to the form of the function F . For instance F must be a polynomial in
the coordinates q(n) i for n = 2k, . . . , 3k − 2.
Finally we can see why the approach by Marmo and Mukunda [14] is not generalizable
to higher-order Mechanics. The pull-back of the equation d X (k)L = d T (3k−2)F
to T 4k−2 M is d X (2k−1)(τ ∗ 2k−1,k L) = τ 4k−2,3k−1 d T (3k−2)F . Now, if D is 2k-ode and
δ is the corresponding section we have
d X(δ) (τ 2k−1,k ∗ L) = δ (2k−2)∗ τ 4k−2,3k−1 d T (3k−2)F
= δ (k−1)∗ d T (3k−2)F
= d D (δ (k−2)∗ F ).
(8.19)

SECTIONS ALNG A MAP IN MECHANICS 21
It is easy to see that X is a symmetry of the Lagrangian if and only if we have that
d X(δ) (τ 2k−1,k ∗ L) = d D (δ (k−2)∗ F ) for every 2k-ode D. In first order theories, k=1,
this equation reduces to d X(δ) L = d D F , wich is the condition imposed by Marmo
and Mukunda. Nevertheles in higher order theories, the function that appears on
the right hand side is δ (k−2)∗ F , wich depends on the ode D.
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⋆ Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza (Spain)
† Departamento de Física Teórica II, Universidad Complutense de Madrid, 28040 Madrid
(Spain)