Coordinate-free derivation of the Euler{Lagrange equations ...andrew/papers/pdf/2005a.pdf · free derivation of the Euler{Lagrange equation is presented. Using a variational ap-proach,

Coordinate-free derivation of the Euler–Lagrange equationsand identification of global solutions via local behavior∗

Elsa K. Hansen†

2005/02/21

Abstract

Results concerning C2-minimizing curves on manifolds are presented. A coordinate-free derivation of the Euler–Lagrange equation is presented. Using a variational ap-proach, two vector fields are defined along the minimizing curve; the tangent to thecurve γ, and the infinitesimal variation δσ. The derivation presented involves completelifts of arbitrary extensions of these vector fields and it is shown that the derivation isindependent of the particular choice of extensions. Special care is also taken to ensurethat the derivation does not require any additional differentiability constraints, otherthan γ being of class C2.

Minimizing curves are also characterized in terms of their local behaviour. It isshown that if a curve is minimizing then any sub-arc of the curve is also minimizing.An important corollary of this result is that a curve, γ, on a manifold will be mini-mizing only if any collection of admissible charts which cover γ have minimizing localrepresentations.

Contents

1. Introduction 2

2. Preliminaries 32.1 Topological Definitions and Partitions of Unity. . . . . . . . . . . . . . . . . 32.2 The Flow Interpretation of the Lie Derivative. . . . . . . . . . . . . . . . . . 42.3 Lifting Vector Fields to the Tangent Bundle. . . . . . . . . . . . . . . . . . 62.4 The Exponential Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Algebraic Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3. Problem Statement Using Tangent Bundle Geometry 103.1 Classical Minimization Problem Statement. . . . . . . . . . . . . . . . . . . 103.2 Construction of Vector Fields on TM. . . . . . . . . . . . . . . . . . . . . . 13

4. Main Results 164.1 Coordinate Free Derivation of Euler–Lagrange Equations. . . . . . . . . . . 164.2 Local Minimizers from Global Minimizers. . . . . . . . . . . . . . . . . . . . 20

∗Report for project in fulfilment of requirements for MSc†Graduate student, Department of Mathematics and Statistics, Queen’s University, Kingston,ON K7L 3N6, CanadaEmail: [email protected] supported in part by a grant from the Natural Sciences and Engineering Research Council ofCanada.

1

2 E. K. Hansen

5. Conclusion and Further Work 255.1 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Further Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

List of Symbols 26

1. Introduction

It is well known that a necessary condition for a C2-curve to solve the calculus ofvariations minimization problem, is that it satisfies the Euler–Lagrange equations. Since thenature of the minimization problem is coordinate-independent, it follows that the solutionshould also be coordinate-independent. This is in fact the case and is shown directly byobserving how the Euler–Lagrange equations change when one changes coordinates [Bulloand Lewis 2004]. The coordinate-independent nature of the Euler–Lagrange equationsrecommends the existence of a coordinate-free expression. The standard approach to findingsuch an expression is to postulate its form and then prove that it agrees with the standardcoordinate expression [Crampin and Pirani 1986]. There have been a few attempts at theintellectually more satisfying approach of finding a coordinate-free derivation that parallelsthe variational method used in the standard coordinate derivation.

Two approaches towards a coordinate-free derivation which we found in the literature aregiven in [Nester 1988] and [Gamboa Saravı and Solomin 2003]. The most striking differencebetween these two approaches is that [Gamboa Saravı and Solomin 2003] postulates theexistence of a covariant derivative and then proceeds to write an intrinsic expression for theEuler–Lagrange equations in terms of that covariant derivative. In contrast, [Nester 1988]makes no use of a covariant derivative and derives the Euler–Lagrange equations using onlyvector fields defined on TM . Since the second method promises to be more general, this isthe method that is the topic of this report.

More specifically, we make an in-depth study of the derivation given by Nester in [Nester1988]. In particular, we show that the vector fields used in the derivation can be expressedas the complete lifts of the two vector fields on M which arise naturally from the variationalformulation. These two vector fields are the tangent vector field of the curve in questionand the vector field representing the infinitesimal variation. Since these vector fields onM are not necessarily defined on open subsets, it is necessary to extend them in order todefine the complete lifts. Since the choice of extension is arbitrary, this immediately bringsinto question whether or not the derivation given by Nester is independent of vector fieldextension. We first show that it is possible to extend these vector fields, and then we showthat the derivation is in fact independent of vector field extension.

Using complete lifts introduces a further wrinkle. The derivation given in [Nester 1988]takes the Lie bracket of the two vector fields defined on TM . This effectively tightens thedifferentiability constraints on the curve. We show that with a slight modification of thederivation we can remove these additional differentiability constraints. In [Nester 1988] thematter of smoothness is blithely sidestepped.

In the final section of this report we investigate the possibility of identifying a minimizingcurve on a manifold M by its local behaviour. The approach here is to require that givenany set of charts covering a minimizing curve on M , the curves coordinate representationssolve corresponding minimization problems. This approach is only valid if a minimizing

Coordinate-free derivation of the Euler–Lagrange equations 3

curve on a certain interval must also be a minimizing curve on any subinterval. We provethat this is in fact the case.

This report is organized as follows. In Chapter 2 we review some of the main conceptsand definitions which will be used. In Chapter 3 we begin by defining the concept ofa variation and giving the coordinate-free statement of the minimization problem. Nextwe show that, provided the vector fields defining the variation are defined in an openneighborhood, then the problem statement can be rephrased using complete lifts. Weconclude Chapter 3 by showing that we can always extend the tangent vector of the curveand the infinitesimal variation to a neighbourhood of the curve. In Chapter 4 we present themain results of our report. We first give the coordinate-free derivation of the Euler–Lagrangeequations and then show that if a curve is a minimizer on a certain interval, I, then it isalso a minimizing curve on any subinterval of I.

2. Preliminaries

In this chapter we review some of the major definitions and concepts that will playprominent roles in the sequel. We begin with a basic review of topology with the aim ofdefining a partition of unity. Next we define the Lie derivative in terms of flows of vectorfields. We then describe two different ways of lifting a vector field from M to TM . Followingthis, we define the exponential map and describe those properties of the exponential mapwhich will be used in our development. Lastly we give a basic review of some algebraicterminology.

2.1. Topological Definitions and Partitions of Unity. In this section only definitions andresults directly used in this work are presented. A detailed discussion of partitions of unitycan be found in [Abraham, Marsden, and Ratiu 1988]. Let (S,O) be a topological space.

2.1 Definition: The subset B ⊂ O is a basis for O if every element G ∈ O can be writtenas an arbitrary union of elements belonging to B. •

2.2 Definition: (S,O) is second countable if it has a countable basis. •

2.3 Definition: (S,O) is Hausdorff if, for any two elements x1, x2 ∈ S, there exist opensets G1,G2 ∈ O satisfying x1 ∈ G1, x2 ∈ G2 and G1 ∩ G2 = ∅. •

2.4 Definition: A cover of (S,O) is a subset {Gi}i∈I ⊂ O satisfying S = ∪i∈IGi, where Iis an arbitrary index set. •

2.5 Definition: {Gi}i∈I ⊂ O is a locally finite cover if it is a cover and if, for each x ∈ S,there exists a neighbourhood U of x for which the set { i |U ∩ Gi 6= ∅} is finite. •

2.6 Definition: A refinement of a cover {Gi}i∈I is a cover {Hj}j∈J if for each j ∈ J , thereexists an i ∈ I such that Hj ⊂ Gi. •

2.7 Definition: (S,O) is paracompact if it is Hausdorff and if every cover has a locallyfinite refinement. •

4 E. K. Hansen

2.8 Definition: A Cr-partition of unity on a manifold M is a collection of pairs{(Ui, gi)}i∈I satisfying

(i) {Ui} is a locally finite covering of M ,

(ii) gi ∈ Cr(M,R) and satisfies gi(x) ≥ 0 ∀ x ∈ Ui and gi(x) = 0 ∀ x ∈M\Ui,(iii)

∑i gi(x) = 1 for each x ∈M .

If A = {(Vj , φj)}j∈A is an atlas for M then a partition of unity {(Ui, gi)}i∈I is subordinateto A if for each i there exists a j such that Ui ⊂ Vj . If all atlases for a manifold M have asubordinate partition of unity then M admits partitions of unity . •

The proof of the following theorem can be found in [Abraham, Marsden, and Ratiu1988].

2.9 Theorem: Every second countable, Hausdorff n-dimensional manifold admits a C∞-partition of unity.

2.2. The Flow Interpretation of the Lie Derivative. In the sequel, a useful characterizationof the Lie derivative will be its flow interpretation. Before giving the flow interpretation wemust provide a few definitions.

2.10 Definition: Let f : M → N be a C1-map between manifolds M and N and letw be a k-form on N . The pull-back of w by f is defined as f∗w(x)(v1, . . . , vk) =w(f(x))(Txf(v1), . . . , Txf(vk)), where v1, . . . , vk ∈ TxM . •

That is, the pull-back of a differential form on N defines a new differential form onM by pulling back the operation of w through f . In the case where w is a function, thepull-back takes the form f∗w = w ◦ f .

2.11 Definition: Let f : M → N be a C1-diffeomorphism between manifolds M and Nand let X be a vector field on M . The push-forward of X by f is defined as f∗X(y) =Txf ◦X ◦ f−1(y). •

That is, the push-forward of a vector field on M defines a new vector field on N bypushing forward the operation of X through f . An important difference between Defini-tions 2.10 and 2.11 is that, in order for f∗X to exist, f must be a diffeomorphism so thatevery element y ∈ N can be mapped to an element x ∈M through the inverse of f .

It is also possible to define the push-forward of a k-form by a diffeomorphism:

f∗w = w(f−1(x))(Txf−1(v1), . . . , Txf

−1(vk)).

Similarly, one can define the pull-back of a vector field:

f∗X = Tf−1 ◦X ◦ f.

The push-forward of a vector field can be used to transport a vector from one point ona manifold to another. The transport construction is as follows. For t small enough, letΦXt (x) denote the one-parameter group of local diffeomorphisms which assigns to x ∈ M

the point which corresponds to starting at x and flowing along X for time t. For a fixedt and any vx ∈ TxM , the Lie transport of vx with respect to the flow of X at time t isTΦX

t (vx).With the Lie transport defined, the Lie derivative can now be interpreted as a measure

of how similar a vector field is to its Lie transport. The details are specified in the nextdefinition.


2.12 Definition: Let Y be a vector field defined along the integral curve γ of X. The Liederivative of Y with respect to X is given by,

LXY (γ(s)) =d

dt

∣∣∣∣t=0

TΦX−tY (γ(t+ s)). •

In Definition 2.12 the Lie derivative is used as a measure of how Y (γ(t+ s)) varies fromthe Lie transport of Y (γ(s)) along γ. Thus LXY (γ(s)) = 0 means that Y (γ(t+ s)) is theLie transport of Y (γ(s)) along γ.

The Lie derivative of a C1-function can also be given a flow interpretation.

2.13 Definition: The Lie derivative of a C1-function f with respect to X is givenby

LXf(γ(s)) =d

dt

∣∣∣∣t=0

ΦX∗t f(γ(s)). •

2.14 Definition: The Lie bracket of two vector fields X,Z ∈ Γr(TM), r ≥ 1, produces anew vector field [X,Z] which satisfies, for any g ∈ C2(M,R),

L[X,Z]g = LXLZg − LZLXg. •

2.15 Theorem: Let X, Z ∈ Γr(M), Y , W ∈ Γr(N), r ≥ 1, and f ∈ Ck(M,N), 2 ≤ k ≤ r.We say that X and Y are f-related , denoted by X ∼f Y , if Y ◦ f = Tf ◦X. If X ∼f Yand Z ∼f W , then [X,Z] ∼f [Y,W ].

Proof: This proof follows from one presented in [Kolar, Michor, and Slovak 1993]. Leth ∈ C2(N,R) so that h ◦ f defines a C2-function on M . Using the chain rule,

(LZ(h ◦ f))(x) = d(h ◦ f)(Z(x)) = dh(Tf(Z(x))).

Now, since Z ∼f W ,

dh(Tf(Z(x))) = dh(W ◦ f(x)) = (LWh)(f(x)).

Therefore, we have the identity,

LZ(h ◦ f)(x) = LWh(f(x)). (2.1)

Substituting (2.1) into the definition of the Lie bracket,

L[X,Z](h ◦ f)(x) = LXLZ(h ◦ f)(x)− LZLX(h ◦ f)(x)

= LXLWh(f(x))− LZLY h(f(x))

= LY LWh(f(x))− LWLY h(f(x))

= L[Y,W ]h(f(x)),

where the second equality comes from recognizing that the Lie derivative of a functionh : N → R is a new function LWh : N → R and then substituting (2.1). �

6 E. K. Hansen

2.3. Lifting Vector Fields to the Tangent Bundle. In the sequel we use two differentconstructions to lift vector fields defined on M to vector fields defined on TM . To in-troduce a discussion of these constructions, we first review a few facts about TTM . Incoordinates, an element in TTM has the form ((u, v), (e1, e2)). There are two commonlyused projections for TTM . The first is the tangent bundle projection πTM : TTM → TM ,which in coordinates looks like πTM ((u, v), (e1, e2)) = (u, v). The second is the projectionTπM : TTM → TM , which in coordinates has the form TπM ((u, v), (e1, e2)) = (u, e1).The following commutative diagram shows how these two projections commute with thestandard projection πM : TM →M :

TTMTπM

zz

πTM

$$TM

πM $$

TM

πMzzM

The vertical subbundle of TM is intrinsically defined as ker(TπM ). The first constructionwe present is the complete lift.

2.16 Definition: The complete lift of a vector field X ∈ Γr(TM), r ≥ 1, is denoted byXT ∈ Γr−1(TTM), and is defined as

XT (vx) =d

dt

∣∣∣∣t=0

TxΦXt (vx). •

Given a local chart (U, φ) on M , the coordinate representation of the principle part

of XT is given by XTφ (x, v) = (Xφ(x),

∂Xφ(x)∂x · v), where XT

φ and Xφ are the coordinate

representations of the principle parts of XT and X respectively. Comparing Definition 2.16with that of the Lie transport, we see that the complete lift can also be defined in termsof the Lie transport. That is, the complete lift of a vector field evaluated at some point vxin the tangent bundle is the tangent vector of the curve defined by the Lie transport of vxalong the flow of X.

The second lifting construction is the vertical lift.

2.17 Definition: The vertical lift of a tangent vector wx ∈ TxM is defined as

vlftvx(wx) =d

dt

∣∣∣∣t=0

(vx + twx) ∈ TvxTM. •

If X is a vector field then define the vector field vlft(X) ∈ Γ(TTM) by vlft(X)(vx) =vlftvx(X(x)).

Given a local coordinate chart (U, φ) on M , the coordinate representation of the prin-ciple part of vlftvx(X(x)) is given by (vlftvx(X(x)))φ = (0, Xφ(x)). Note that vlftvx(X(x))belongs to the vertical subbundle of TM .

A significant difference between the complete lift and the vertical lift of X is thatthe vertical lift is a pointwise construction whereas the complete lift depends on how Xis defined in a neighbourhood of the point of evaluation. This distinction is evident for


example, because XT depends on the Lie transport of vx. It is also evident by lookingat the coordinate expressions of the two different lifts; the coordinate expression for XT

contains derivatives of X.Two important constructions which use the vertical lift are the almost tangent structure

J : TTM → TTM which is defined as

J(wvx) := vlftvx(TπM (wvx)) for all wvx ∈ TTM

and the Liouville vector field V : TM → TTM , which is defined as

V (vx) := vlftvx(vx).

2.4. The Exponential Map. Before presenting properties of the exponential map, we givea few definitions.

2.18 Definition: For r ≥ 1, a Cr-affine connection on M , denoted by ∇, assigns toany two vector fields X, Z ∈ Γr(TM) a new vector field ∇XZ ∈ Γr−1(TM). For anyf ∈ Cr(M,R) let fX(x) = f(x)X(x), then the assignment satisfies the following properties:

(i) the map (X,Z) 7→ ∇XZ is R-bilinear;

(ii) ∇fXZ = f∇XZ;

(iii) ∇XfZ = f∇XZ + (LXf)Z. •

2.19 Definition: A geodesic of a Cr-connection is a differentiable curve γ : I → M thatsatisfies ∇γ(t)γ(t) = 0 for all t ∈ I. •

2.20 Definition: A Riemannian manifold , (M,G) is a manifold M with a Riemannianmetric G. •

The Riemannian metric of a Riemannian manifold uniquely determines an affine con-nection. This affine connection is called the Levi-Civita connection.

Let M be a manifold with connection ∇. For x ∈M , the exponential map exp : TxM →M is a local diffeomorphism from an open neighbourhood of the zero vector of TxM to anopen neighbourhood of x. Let βvx(t) be the geodesic, associated with the connection ∇,that satisfies the initial condition βvx(0) = vx. Then the exponential map is defined byexp(vx) = βvx(1). In order to show that the exponential map is a local diffeomorphism, wewill follow the approach in [Crampin and Pirani 1986] and show that, on an appropriate set,T exp is the identity and then apply the Inverse Function Theorem. First, for completenesswe state the Inverse Function Theorem. The following version of the Inverse FunctionTheorem is from [Bullo and Lewis 2004].

2.21 Theorem: (Inverse Function Theorem) Let f ∈ C1(M,N). If Tf is an isomor-phism at x0, then there exists a neighbourhood U of x0 for which f |U : U → f(U) is adiffeomorphism.

Let Id denote the following identification of vx ∈ TxM with w0x ∈ T0xTM ,

w0x = vlft0x(vx).

To show that T0xexp : T0x(TxM) → TxM is equal to Id, first note that, for s > 0sufficiently small, geodesics have the homogenity property that βsvx(1) = βvx(s).

8 E. K. Hansen

Applying the chain rule,

d

dt

∣∣∣∣t=0

exp((tv)x) = T0xexp

(d

dt

∣∣∣∣t=0

tv

)= T0xexp ◦ vlft0x(vx).

But by definition we know that

d

dt

∣∣∣∣t=0

exp((tv)x) =d

dt

∣∣∣∣t=0

βtv(1) =d

dt

∣∣∣∣t=0

βv(t) = vx.

Therefore we have the desired result,

T0xexp(vlft0x(v)) = vx. (2.2)

Finally, the identity is certainly non-singular so, by the Inverse Function Theorem, exp isa local diffeomorphism of a neighbourhood of 0x ∈ TxM to a neighbourhood of x ∈M .

The exponential map can be used to define a local diffeomorphism from an open neigh-bourhood of the zero section of an embedded submanifold to an open neighbourhood of thesubmanifold. This construction is defined in the following theorem and will be used laterto extend vector fields from embeddings to open neighbourhoods.

2.22 Theorem: Let M be a Riemannian manifold and N an embedded submanifold of M .Let N⊥ denote the normal bundleto TN in TM . Then exp|N⊥ defines a local diffeomor-phism from an open neighbourhood V ⊂ N⊥ to an open neighbourhood U ⊂ M , where Usatisfies N ⊂ U . U is called the tubular neighbourhood of N .

2.5. Algebraic Terminology. This section states some basic definitions and identities whichwill be used in later sections.

2.23 Definition: The permutation group, denoted by Sn, is the set of bijections σ of theset of n elements {1, . . . , n}. The operation of σ can be diagrammatically explained by(

1 . . . nσ(1) . . . σ(n)

). •

2.24 Definition: σ ∈ Sn is a transposition if(1 . . . i . . . j . . . n

σ(1) . . . j . . . i . . . σ(n)

).

That is, if σ transposes exactly two elements of {1, . . . , n} and leaves all other elementsfixed. Each σ ∈ Sn can be written as a composition of transpositions. A permutation σ iscalled even if it is a composition of an even number of transpositions and odd otherwise.It can be shown that this distinction is well-defined. The sign of σ, is 1 if σ is odd and 0if it is even. •

Let M be a manifold and let Λk denote the set of differential k-forms on M . ThenΛ = ⊕dim(M)

k=0 Λk is the graded exterior algebra of differential forms.


2.25 Definition: The exterior product , denoted by ∧, is a map ∧ : Λr(M) × Λs(M) →Λ(M)r+s defined by

(α∧β)(X1, . . . , Xr+s) =(r + s)!

r!s!

∑σ∈Sr+s

(−1)sign(σ)α(Xσ(1), . . . , Xσ(r))β(Xσ(r+1), . . . , Xσ(r+s)).

•The exterior product has the following properties:

1. α ∧ (β + γ) = α ∧ β + α ∧ γ;

2. α ∧ β = (−1)deg(α)deg(β)β ∧ α;

3. α ∧ (β ∧ γ) = (α ∧ β) ∧ γ.

2.26 Definition: Let r be an even integer. A derivation of degree r on Λ(M) is a mapD : Λ(M) → Λ(M) such that D(Λk(M)) ⊂ Λk+r(M), and which satisfies the followingproperties:

(i) D(aα+ bβ) = aD(α) + bD(β);

(ii) D(α ∧ β) = D(α) ∧ β + α ∧D(β),

for a, b ∈ R and α, β ∈ Λ(M). •

2.27 Definition: Let r be an odd integer. An antiderivation of degree r on Λ(M) is amap D : Λ(M)→ Λ(M) such that D(Λk(M)) ⊂ Λk+r(M), and which satisfies the followingproperties:

(i) D(aα+ bβ) = aD(α) + bD(β);

(ii) D(α ∧ β) = D(α) ∧ β + (−1)deg(α)α ∧D(β). •The set of all derivations and antiderivations D, is a graded Lie algebra with commuta-

tor,[D1, D2] = D1 ◦D2 − (−1)deg(D1)deg(D2)D2 ◦D1,

where, D1, D2 ∈ D.Two basic antiderivations are the interior product and the exterior derivative.

2.28 Definition: The interior product with respect to a vector V , iV : Λk(M) →Λk−1(M) is an antiderivation of degree -1 which is defined by the following two properties:

(i) iV f = 0 ∀f ∈ Λ0(M);

(ii) iV α = α(V ) ∀α ∈ Λ1(M). •

2.29 Definition: The interior product with respect to a linear endomorphism J ,iJ : Λk(M) → Λk(M) is a derivation of degree 0 which is defined by the following twoproperties:

(i) iJf = 0 ∀f ∈ Λ0(M);

(ii) 〈iJα, V 〉 = α(JV ) ∀ α ∈ Λ1(M). •

10 E. K. Hansen

2.30 Definition: The exterior derivative d : Λk(M)→ Λk+1(M) is the unique antideriva-tion of degree 1 which is defined by the following two properties:

(i) ddα = 0 ∀ α ∈ Λ(M);

(ii) df = ∂f∂xidxi ∀f ∈ Λ0(M). •

The bracket of iJ and d forms a new antiderivation, the vertical derivative:

dJ := [iJ , d] = iJd− diJ ,

which has the property, d2J = 0.The different posible brackets of the above defined antiderivations are

1. [iZ , d] = iZd+ diZ = LZ ,

2. [d, dJ ] = ddJ + dJd = 0,

3. [dJ , iV ] = dJ iV + iV dJ = iJ ,

4. [iX , iJ ] = iXiJ − iJ iX = iJX .

2.6. Notation. We will use some non-standard notation. Let σ : I × J → M be of classC1. Then σ will denote a vector field in TM defined by,

σ(s, t) =d

dtσ(s, t) ∈ Tσ(s,t)M.

Furthermore, if we define σt(s) and σs(t) to be curves parameterized by s ∈ I and t ∈ J ,respectively, then σ(s, t) = σs(t) = σt(s). That is, “ ˙ ” will always be used to denote atangent vector with respect to the variable t.

3. Problem Statement Using Tangent Bundle Geometry

In preperation for giving a coordinate free derivation of the Euler–Lagrange equations,we must first reformulate the classical version of the calculus of variations minimizationproblem in terms of vector fields on TM . This reformulation is the goal of the presentchapter. We therefore begin by giving a careful description of the minimization problem,followed by its reformulation in terms of vector fields on TM . Since our reformulation willrequire extending vector fields defined along integral curves to neighbourhoods in M , wealso show that such extensions exist.

3.1. Classical Minimization Problem Statement. In this section we give a coordinatefree statement of the minimization problem. This section parallels the discussion found in[Bullo and Lewis 2004].

Let M be a Cr-manifold, r > 1. Given a C2-function L : TM → R and a certain classof curves D, the goal is to find a curve γ0 ∈ D such that the action defined as,

AL[γ] =

∫ b

aL(γ(t))dt

is minimized with respect to all curves γ ∈ D. If the class of curves is chosen to beD(C2, xa, xb) := {γ | γ ∈ C2([a, b],M), γ(a) = xa, γ(b) = xb} then the following usefuldefinitions can be made.


3.1 Definition: Let I ⊂ R be an interval satisfying 0 ∈ int(I). A C2-variation of aC2-curve γ is a C2-map σ : I × [a, b]→M which satisfies the following properties:

(i) σ(s, a) = γ(a), s ∈ I;

(ii) σ(s, b) = γ(b), s ∈ I;

(iii) σ(0, t) = γ(t), t ∈ [a, b].

Every C2-variation σ has a corresponding C1-infinitesimal variation defined by

δσ(t) =d

ds

∣∣∣∣s=0

σ(s, t). •

From the definition of a C2-variation we see that δσ(a) = δσ(b) = 0.Let σs : [a, b] → M denote the curve satisfying σs(t) = σ(s, t) for fixed s. If γ ∈

D(C2, xa, xb), then all C2-variations of γ give rise to curves σs, s ∈ I, which also belong toD(C2, xa, xb). Also, given any two curves γ1, γ2 ∈ D(C2, xa, xb), σ(s, t) = γ1(t)(1 − s) +γ2(t)s defines a C2-variation of γ1 satisfying σ(1, t) = γ2(t). That is, given any two curvesγ1, γ2 ∈ D(C2, xa, xb) there exists a C2-variation of γ1 such that γ2 = σ1.

In order for γ0 to minimize AL, we must have

AL[γ0] ≤ AL[γ] for all γ ∈ D(C2, xa, xb). (3.1)

(3.1) will hold, if for all C2-variations of γ0, we have,

AL[γ0] ≤ AL[σs] for any fixed s ∈ I.

Fixing the particular variation, AL[σs] becomes a C2-function of s. From elementarycalculus, if γ0 = σs|s=0 minimizes the function AL[σs] : I → R then

d

ds

∣∣∣∣s=0

AL[σs] = 0. (3.2)

3.2 Remark: (3.2) is only a necessary condition for γ0 to be a minimizer. A curve satisfyingthis condition is called stationary. •

Since σs ∈ D(C2, xa, xb) for all C2-variations, if γ0 minimizes with respect to all γ ∈D(C2, xa, xb), then (3.2) must hold for all C2-variations of γ0. Expanding the left hand sideof (3.2),

d

ds

∣∣∣∣s=0

A[σ(s, t)] =d

ds

∣∣∣∣s=0

∫ b

aL(σ(s, t))dt

=

∫ b

a

d

ds

∣∣∣∣s=0

L(σ(s, t))dt

=

∫ b

a〈dL, d

ds

∣∣s=0

σ(s, t)〉dt.

The order of integration and differentiation in the second equality can be exchanged sinceL ∈ C2(TM,R).

Thus, if γ0 is stationary with respect to all γ ∈ D(C2, xa, xb), then the following equationmust hold for all C2-variations σ of γ0:∫ b

a〈dL, d

ds

∣∣s=0

σ(s, t)〉dt = 0. (3.3)

12 E. K. Hansen

3.3 Remark: (3.3) is a local condition in the sense that it depends only on the values ofσ(s, t) and d

dtσ(s, t) in a neighbourhood of s = 0. •The next step is to write (3.3) in terms of vector fields defined on TM . This is done in

the following proposition.

3.4 Proposition: Let X and Z be vector fields defined in an open neighbourhood of image(γ)which satisfy

X(σ(s, t)) =d

dtσ(s, t),

Z(σ(s, t)) =d

dsσ(s, t).

Then,∫ ba

dds

∣∣s=0

L(σ(s, t))dt = 0 if and only if∫ b

a(LZTL)γ(t)dt = 0. (3.4)

Proof: To prove Proposition 3.4 we will use the following lemma.

1 Lemma: Let X and Z be as defined in Proposition 3.4, then ΦZTs (σ(s, t)) = X(σ(s, t)).

Proof: Consider ZT (σ(s, t)):

ZT (σ(s, t)) =d

dξ

∣∣∣∣ξ=0

TΦZξ (σ(s, t))

=d

dξ

∣∣∣∣ξ=0

d

dw

∣∣∣∣w=0

ΦZξ ◦ ΦX

w (σ(s, t))

=d

dξ

∣∣∣∣ξ=0

d

dw

∣∣∣∣w=0

ΦZξ (σ(s, t+ w))

=d

dξ

∣∣∣∣ξ=0

d

dw

∣∣∣∣w=0

σ(s+ ξ, t+ w)

=d

dξ

∣∣∣∣ξ=0

d

dtσ(s+ ξ, t)

=d

dξ

∣∣∣∣ξ=0

σ(s+ ξ, t) =d

dsσ(s, t),

which impliesΦZT

s (σ(s, t)) = X(σ(s, t)). H

Now, using the definition of the Lie derivative and Lemma 1 we have,

(LZTL)(γ(t)) =d

ds

∣∣∣∣s=0

ΦZT ∗s L(γ(t)) =

d

ds

∣∣∣∣s=0

L ◦ ΦZT

s (γ(t))

= dL ◦ d

ds

∣∣∣∣s=0

X(σ(s, t)) =d

ds

∣∣∣∣s=0

L(σ(s, t)).

Integrating, ∫ b

a(LZTL)(γ(t))dt =

∫ b

a

d

ds

∣∣∣∣s=0

L(σ(s, t))dt. �


Now consider the following coordinate calculation:

((LZTL)(γ(t)))φ =∂L(γ(t), γ′(t))

∂xiZi(γ(t)) +

∂L(γ(t), γ′(t))

∂vi∂Zi(γ(t))

∂xkdγk(t)

dt

=∂L(γ(t), γ′(t))

∂xiZi(γ(t)) +

∂L(γ(t), γ′(t))

∂vidZi(γ(t))

dt.

This calculation depends only on how Z is defined along image(γ), that is, (3.4) dependsonly on how δσ is defined. Therefore, γ ∈ D(C2, xa, xb) is stationary with respect to allcurves belonging to D(C2, xa, xb) if and only if∫ b

a(LZTL)γ(t)dt = 0

for all Z satisfying that there exists some C2-variation such that Z(γ(t)) = δσ(t) for allt ∈ [a, b].

It only remains to show that we can in fact extend δσ to a neighbourhood of image(γ).

3.2. Construction of Vector Fields on TM. Given any C2-variation σ with correspondingvector fields γ and δσ ∈ Γ1(TM) defined along image(γ), we will show that there existvector fields X, Z ∈ Γ1(TM) satisfying,

1. X(γ(t)) = γ(t),

2. Z(γ(t)) = δσ(t),

3. X and Z are defined in a neighbourhood of image(γ).

3.5 Remark: To this point it is only evident that Z needs to be defined in a neighbourhoodof image(γ). In Chapter 4, however, the derivation which we discuss will also require thatthe extension of γ exists. Therefore, we will explicitly construct extensions for both δσ andγ. •

We will present two different methods for constructing the vector fields X and Z. Thefirst method we discuss uses the idea of the push-forward of a vector field. Our constructionwill use the following proposition.

3.6 Proposition: Let M be a C∞-Riemannian manifold. Let γ : [a, b] → M be a Cr-embedding, r > 1. Let X0 : image(γ) → ∪t∈[a,b]Tγ(t)M be a Ck-vector field (k ≤ r) defined

along image(γ). Then there exists a Ck-vector field X in an open neighbourhood of image(γ)such that X0(γ(t)) = X(γ(t)) for all t ∈ [a, b].

Proof: (Proof by construction) We use the push-forward with respect to the exponentialmap to construct the vector field X. Let W : ∪t∈[a,b]Tγ(t)M → ∪t∈[a,b]T (Tγ(t)M) be thevertical vector field defined for all t ∈ [a, b] given by W (vγ(t)) = vlftvγ(t)X0(γ(t)). Note

that, since the vertical lift is a pointwise construction, W is well-defined. Let image(γ)⊥

denote the normal bundle to T (image(γ)) in TM . Then for some open set V ⊂ image(γ)⊥,

14 E. K. Hansen

expγ : V → U defines a diffeomorphism from V to the tubular neighbourhood U . X is thendefined in U as,

X(x) = expγ∗W (x) = T expγ ◦W ◦ exp−1γ (x)

= T expγ ◦W (uγ(t)) = T expγ(vlftuγ(t)X0(γ(t)))

where, uγ(t) ∈ image(γ)⊥ and expγ(uγ(t)) = x.Since expγ : V → U is a C∞-diffeomorphism, both exp−1 and T exp ∈ C∞. Therefore,

since X is a composition of two C∞-maps and one Ck-map, it is itself of class Ck.From (2.2),

T0γ(t)expγ(vlft0γ(t)X0(γ(t))) = X0(γ(t)). (3.5)

(3.5), combined with the fact that exp−1γ (γ(t)) = 0γ(t) results in

X(γ(t)) = X0(γ(t)). �

Define the following maps W1 : ∪t∈[a,b]Tγ(t)M → ∪t∈[a,b]T (Tγ(t)M) and W2 :∪t∈[a,b]Tγ(t)M → ∪t∈[a,b]T (Tγ(t)M) as, W1(vγ(t)) = vlftvγ(t)(γ(t)) and W2(vγ(t)) =vlftvγ(t)(δσ(t)). Using Proposition 3.6, construct the following two vector fields on thetubular neighbourhood U ,

X(x) = expγ∗W1(x) and Z(x) = expγ∗W2(x).

To construct the appropriate vector fields on TM take the complete lifts of X and Z.The second method we discuss uses partitions of unity. Our construction will use the

following proposition.

3.7 Proposition: Let M be a C∞-manifold admitting C∞-partitions of unity, f : [a, b]→Ma Cr-embedding, r > 1. Then there exists a vector field X ∈ Γr−1(TM) on a neighbourhoodof f([a, b]) in M which satisfies X(f(t)) = Ttf · 1 for all t ∈ [a, b].

Proof: Since f([a, b]) is a submanifold of M , there exist admissible charts (Ui, φi) coveringf([a, b]) such that φi(Ui∩ image(f)) = φi(Ui)∩(R×{0}). Now, the coordinate representationof the tangent vector Ttf · 1 in the chart (TUi, Tφi) is given by ((t, 0, . . . , 0), (1, 0, . . . , 0)).This vector field agrees with the constant vector field e1 (where e1 = (1, 0, 0..., 0)) definedon the open neighbourhood φi(Ui). Since M admits a C∞-partition of unity, there existsa set of pairs {(Ui(j), gi(j))} such that Ui(j) ⊂ Ui for all j, {Ui(j)} is a locally finite coverat each x ∈ ∪iUi and the gi(j) ∈ C∞(M,R) satisfy

∑i(j) gi(j)(x) = 1. Therefore, make the

following definition:

X(x) =∑i(j)

gi(j)Txφ−1i(j) ◦ e1 ◦ φi(j)(x), (3.6)

where φi(j) is the restriction of φi to Ui(j). Now, since the sum in (3.6) is finite at any pointx ∈ ∪iUi, X is well defined in ∪iUi.

We must also show that X(f(t)) = Ttf ·1. Since f(t) = x, x ∈ ∪iUi and Tf(t)φi(Ttf ·1) =e1 we have,

Tf(t)φ−1i(j) ◦ e1 ◦ φi(j)(f(t)) = Ttf · 1


and, therefore, since∑

i(j) gi(j)(x) = 1,

X(f(t)) =∑i(j)

gi(j)(Ttf · 1) = Ttf · 1.

�

Choosing {(Ui, φi)} to be a set of submanifold charts for image(γ) and {(Ui(j), gi(j))}a partition of unity subordinate to it, direct application of Proposition 3.7 allows us toconstruct the following vector field:

X(x) =∑i(j)

gi(j)Txφ−1i(j) ◦ e1 ◦ φi(j)(x).

Using the partition of unity that was just defined, let W : ∪t∈[a,b]Tγ(t)M →∪t∈[a,b]T (Tγ(t)M) be the vertical vector field defined by,

W (vγ(t)) = vlftvγ(t)δσ(t).

Then Wφi(j) is a vector field defined at all points (t, 0, . . . , 0) ∈ φi(j)(Ui(j)), where t ∈ [a, b].Let (expγ)φi(j) be the coordinate representation of expγ . Using Proposition 3.6, define thefollowing vector field in φi(j)(Ui(j)),

Zi(j)(φi(j)(x)) = (expγ)φi(j)∗Wφi(j)(φi(j)(x)),

where x ∈ Ui(j). Now use the partition of unity {(Ui(j), gi(j))} to define the vector field Zin M . That is,

Z(x) =∑i(j)

gi(j)Txφ−1i(j) ◦ Zi(j) ◦ φi(j)(x).

To construct the appropriate vector fields on TM take the complete lifts of X and Z.Before proceeding to the coordinate-free derivation of the Euler–Lagrange equations, we

first show that the vector fields X and Z commute on the C2-variation σ. This propertywill be used in the coordinate-free derivation.

3.8 Proposition: Let X, Z ∈ Γ1(TM) be vector fields defined on open neighbourhoods ofimage(γ) satisfying,

X(σ(s, t)) =d

dtσ(s, t),

Z(σ(s, t)) =d

dsσ(s, t).

Then [X,Z](σ(s, t)) = 0.

Proof: Consider the following,

Tσ ◦ ∂

∂s(s, t) = T1σ ◦ ((s, 1), t) =

d

dsσ(s, t) = Z ◦ σ(s, t).

Similar calculations for Tσ ◦ ∂∂t(s, t) result in,

Tσ ◦ ∂∂t

(s, t) = X ◦ σ(s, t).

Therefore ∂∂t ∼σ X and ∂

∂s ∼σ Z. Theorem 2.15 implies that [ ∂∂t ,∂∂s ] ∼σ [X,Z], which in

turn implies that [X,Z]σ(s, t) = 0. �

16 E. K. Hansen

4. Main Results

In this section we give characterizing properties of a minimizing curve γ on a manifoldM . We begin by giving the coordinate free expression for the Euler–Lagrange equation andthen proceed with its coordinate-free derivation. Next we characterize a minimizing curvein terms of its local behavior. In short we prove that a curve on a manifold is a minimizerif and only if it solves a corresponding minimization problem in all of its local coordinatecharts.

4.1. Coordinate Free Derivation of Euler–Lagrange Equations.

4.1 Definition: Let L : TM → R be a C2-function and γ : I → M a curve of class C2.Then,

EL := iγdiJdL+ diV dL− dL

defines a one-form on TM . •

4.2 Theorem: Let L and γ be as defined in Definition 4.1, then γ ∈ D(C2, xa, xb) is sta-tionary with respect to all γ ∈ D(C2, xa, xb) if and only if

EL(γ(t)) = 0 for all t ∈ [a, b].

Proof: Referring to (3.2), γ is stationary if and only if for all C2-variations of γ

d

ds

∣∣∣∣s=0

AL[σs] = 0.

Therefore, by (3.4) γ is stationary if and only if∫ b

a(LZTL)(γ(t))dt = 0, (4.1)

where, Z satisfies Z(γ(t)) = δσ(t) for all t ∈ [a, b].The derivation found in [Nester 1988] uses the two identities

dω(X, Z) = LX〈ω, Z〉 − LZ〈ω, X〉 − 〈ω, [X, Z]〉 (4.2)

andLX〈ω, Z〉 = 〈LXω, Z〉+ 〈ω,LX Z〉 (4.3)

where, ω is a differentiable one-form and X and Z are vector fields on TTM . When thesubstitutions ω = iJdL, X = XT and Z = ZT are made, (4.2) and (4.3) involve the object[XT , ZT ]. This necessitates that XT and ZT are of class C1 or equivalently that γ is ofclass C3. The following two lemmas will be used to preserve our original specification thatγ is only required to be of class C2.


4.3 Lemma: Let X, Z ∈ Γ0(TTM) satisfying Tπ(X), Tπ(Z) ∈ Γ1(TM). Let L : TM → Rbe a C2-function. Then,

d(iJdL)(X, Z) =

LX〈dL, vlft(Tπ(Z))〉 − LZ〈dL, vlft(Tπ(X))〉 − 〈dL, vlft[Tπ(X), Tπ(Z)]〉. (4.4)

Furthermore, if X and Z are chosen to be the complete lifts of arbitrary extensions of γand δσ then (d(iJdL)(XT , ZT ))(γ(t)) is independent of choice of extension for all t ∈ [a, b].

Proof: In coordinates the value of the one form iJdL acting on any vector Y ∈ Γ(TTM)is ∂L

∂viY i where Y i ∂

∂xiis the coordinate representative of the projection Tπ(Y ). Therefore

the coordinate representation of iJdL is ∂L∂vidxi. Applying the definition of the exterior

derivative in coordinates we have,

(d(iJdL))φ =∂2L

∂xj∂vidxj ∧ dxi +

∂2L

∂vj∂vidvj ∧ dxi.

Letting Xi ∂∂xi

+ Si ∂∂vi

and Zi ∂∂xi

+ Ri ∂∂vi

denote the coordinate representations of X and

Z respectively,

(d(iJdL)(X, Z))φ =∂2L

∂xj∂vi(XjZi − ZjXi) +

∂2L

∂vj∂vi(SjZi −RjXi).

Now consider the coordinate expression of the first term on the right hand side of (4.4):(LX〈dL, vlft(Tπ(Z))〉

)φ

= Xj ∂

∂xj

(∂L

∂viZi)

+ Sj∂

∂vj

(∂L

∂viZi)

= Xj ∂2L

∂xj∂viZi +Xj ∂L

∂vi∂Zi

∂xj+ Sj

∂2L

∂vj∂viZi + Sj

∂L

∂vi∂Zi

∂vj.

A similar calculation for the second term results in(LZ〈dL, vlft(Tπ(X))〉

)φ

= Zj∂2L

∂xj∂viXi + Zj

∂L

∂vi∂Xi

∂xj+Rj

∂2L

∂vj∂viXi +Rj

∂L

∂vi∂Xi

∂vj

and for the final term,(〈dL, vlft[Tπ(X), Tπ(Z)]〉

)φ

=∂L

∂vi∂Zi

∂xjXj − ∂L

∂vi∂Xi

∂xjZj .

Combining terms,(LX〈dL, vlft(Tπ(Z))〉 − LZ〈dL, vlft(Tπ(X))〉 − 〈dL, vlft[Tπ(X), Tπ(Z)]〉

)φ

= Xj ∂2L

∂xj∂viZi + Sj

∂2L

∂vj∂viZis− Zj ∂2L

∂xj∂viXi −Rj ∂2L

∂vj∂viXi

=(

d(iJdL)(X, Z))φ.

To show the last part of Lemma 4.3, let X = XT and Z = ZT and evaluate (4.4) at γ(t).By the first part of Lemma 4.3 we have

d(iJdL)(XT , ZT ) = LXT 〈dL, vlft(Z)〉 − LZT 〈dL, vlft(X)〉 − 〈dL, vlft[X,Z]〉. (4.5)

18 E. K. Hansen

Writing LZT 〈dL, JXT 〉γ(t) in coordinates,

(LZT 〈dL, JXT 〉γ(t))ϕ

=∂2L(γ(t), γ′(t))

∂xj∂viXi(γ(t))Zj(γ(t)) +

∂L(γ(t), γ′(t))

∂vi∂Xi(γ(t))

∂xjZj(γ(t))

+∂2L(γ(t), γ′(t))

∂vj∂viXi(γ(t))

∂Zj(γ(t))

∂xkvk +

∂L(γ(t), γ′(t))

∂vi∂Xi(γ(t))

∂vj∂Zj(γ(t))

∂xkvk.

(4.6)

Since X is a vector field on M , the last term in (4.6) is zero. The first term is obviouslyindependent of extension off of image(γ). This leaves only the second and third term of (4.6)to consider. Writing the second term, we have,

∂L(γ(t), γ′(t))

∂vi∂Xi(γ(t))

∂xjZj(γ(t)) =

∂L(γ(t), γ′(t))

∂vi∂Zi(γ(t))

∂xjXj(γ(t))

=∂L(γ(t), γ′(t))

∂vid

dt(Zi(γ(t))), (4.7)

where the first step uses the fact that [X,Z](γ(t)) = 0 and the second step uses the factthat Xj(γ(t)) = vj . Since all extensions of Z must agree along γ, (4.7) is independent ofextension.

Similarly for the third term of (4.6),

∂2L(γ(t), γ′(t))

∂vj∂viXi(γ(t))

∂Zj(γ(t))

∂xkvk =

∂2L(γ(t), γ′(t))

∂vj∂viXi(γ(t))

d

dt(Zi(γ(t))),

which is also independent of extension.Now consider the second term in (4.5). Writing LXT 〈dL, JZT 〉γ(t) in coordinates,

(LXT 〈dL, JZT 〉γ(t))φ

=∂2L(γ(t), γ′(t))

∂xj∂viZi(γ(t))Xj(γ(t)) +

∂L(γ(t), γ′(t))

∂vi∂Zi(γ(t))

∂xjXj(γ(t))

+∂2L(γ(t), γ′(t))

∂vj∂viZi(γ(t))

∂Xj(γ(t))

∂xkvk +

∂L(γ(t), γ′(t))

∂vi∂Zi(γ(t))

∂vj∂Xj(γ(t))

∂xkvk.

Using similar arguments, we have that LXT 〈dL, JZT 〉γ(t) is also independent of vector fieldextension. The third term in (4.5), 〈dL, vlft[X,Z]〉(γ(t)), is independent of extension sinceall extensions of X and Z satisfies, [X,Z](γ(t)) = 0. Combining these results leads to thedesired conclusion that 〈iXT diJdL,Z

T 〉(γ(t)) is independent of vector field extension. �

4.4 Lemma: Let X, Z ∈ Γ1(TM). Let X ∈ Γ(TTM) be any vector field satisfyingTπ(X) = X. Then for any vx ∈ TM satisfying [X,Z](x) = 0 and X(x) = vx, we have,

(LZT 〈dL, V − JX〉)(vx) = (〈LZT dL, V − JX〉)(vx) + (〈dL, vlft[Z,X]〉)(vx). (4.8)


Proof: We begin by writing the coordinate expression for each term in (4.8). Let Xi ∂∂xi

+

Ri ∂∂vi

be the coordinate expression for ZT . Then

((LZT 〈dL, V − JX〉)(vx)

)φ

=∂2L(x, v)

∂xj∂vi(vi−Xi(x))Zj(x)+

∂2L(x, v)

∂vj∂vi(vi−Xi(x))Rj(vx)

+∂L(x, v)

∂vi∂

∂xj(vi −Xi(x))Zj(x) +

∂L(x, v)

∂vi∂

∂vj(vi −Xi(x))Rj(vx),(

(〈diZT dL, V − JX〉)(vx))φ

=∂2L(x, v)

∂xi∂vj(vj −Xj(x))Zi(x) +

∂2L(x, v)

∂vj∂vi(vj −Xj(x))Ri(vx)

+∂L(x, v)

∂xi∂Zi(x)

∂vj(vj −Xj(x)) +

∂L(x, v)

∂vi∂Ri(vx)

∂vj(vj −Xj(x)),

((〈dL, vlft[Z,X]〉)(vx))φ =∂L(x, v)

∂vi∂Xi(x)

∂xjZj(x)− ∂L(x, v)

∂vi∂Zi(x)

∂xjXj(x).

From these expressions we see that (4.8) is equivalent to requiring

∂L(x, v)

∂vi∂

∂xj(vi −Xi(x))Zj(x) +

∂L(x, v)

∂vi∂

∂vj(vi −Xi(x))Rj(vx)

=∂L(x, v)

∂xi∂Zi(x)

∂vj(vj −Xj(x)) +

∂L(x, v)

∂vi∂Ri(vx)

∂vj(vj −Xj(x))

−∂L(x, v)

∂vi∂Zi(x)

∂xjXj(x) +

∂L(x, v)

∂vi∂Xi(x)

∂xjZj(x),

which simplifies to

−∂L(x, v)

∂vi∂Xi(x)

∂xjZj(x) +

∂L(x, v)

∂viRi(vx) =

∂L(x, v)

∂xi∂Zi(x)

∂vj(vj −Xj(x))

+∂L(x, v)

∂vi∂Ri(vx)

∂vj(vj −Xj(x))− ∂L(x, v)

∂vi∂Zi(vx)

∂xjXj(vx) +

∂L(x, v)

∂vi∂Xi(x)

∂xjZj(x).

Therefore, by the definition of complete lift we have

−∂L(x, v)

∂vi∂Xi(x)

∂xjZj(x) +

∂L(x, v)

∂vi∂Zi(x)

∂xkvk =

∂L(x, v)

∂xi∂Zi(x)

∂vj(vj −Xj(x))

+∂L(x, v)

∂vi∂Ri(vx)

∂vj(vj −Xj(x))− ∂L(x, v)

∂vi∂Zi(x)

∂xjXj(x) +

∂L(x, v)

∂vi∂Xi(x)

∂xjZj(x).

Now, since vx = X(x),

0 = 2∂L(x, v)

∂vi

(−∂Z

i(x)

∂xjXj(x) +

∂Xi(x)

∂xjZj(x)

),

which holds since [X,Z](x) = 0. �

By Lemma 4.4 we have,

(LZT 〈dL, V −J(XT )〉)(γ(t)) = 〈(LZT dL, V −J(XT )〉)(γ(t))+(〈dL, vlft[Z,X]〉)(γ(t)). (4.9)

20 E. K. Hansen

Now, using (4.5) and (4.9), the integrand in (4.1) can be written as

(LZTL)(γ(t)) = (LZT 〈dL, V 〉)(γ(t))− (〈diV dL− dL,ZT 〉)(γ(t))

= (LZT 〈dL, V 〉)(γ(t))− (〈EL, ZT 〉)(γ(t)) + (〈iXT diJdL,ZT 〉)(γ(t))

= (LZT 〈dL, V 〉)(γ(t))− (〈EL, ZT 〉)(γ(t)) + (LXT 〈dL, vlft(Z)〉)(γ(t))

−(LZT 〈dL, vlft(X)〉)(γ(t))− (〈dL, vlft([X,Z])〉)(γ(t))

= −(〈EL, ZT 〉)(γ(t)) + (LXT 〈dL, vlft(Z)〉)(γ(t))

+(LZT 〈dL, V − vlft(X)〉)(γ(t))− (〈dL, vlft([X,Z])〉)(γ(t))

= −(〈EL, ZT 〉)(γ(t)) + (LXT 〈dL, vlft(Z)〉)(γ(t))

+(〈LZT dL, V − vlft(X)〉)(γ(t)) + (〈dL, vlft([Z,X])〉)(γ(t))

−(〈dL, vlft([X,Z])〉)(γ(t))

= −(〈EL, ZT 〉)(γ(t)) + (LXT 〈dL, vlft(Z)〉)(γ(t)). (4.10)

Using the flow interpretation of the Lie derivative the second term in (4.10) becomes

(LXT 〈dL, JZT 〉)(γ(t)) = (d

dt

∣∣∣∣t=0

ΦXT ∗t 〈dL, JZT 〉)(γ(t)),

which, due to the condition JZT∣∣σ(a,s)

= JZT∣∣σ(a,s)

= 0, integrates to zero. This leads to

the equation ∫ b

a(LZTL)(γ(t))dt = −

∫ b

a〈EL, ZT 〉(γ(t))dt.

Therefore EL(γ(t)) = 0 is a sufficient condition for γ to be stationary. It only remainsto show that EL(γ(t)) = 0 is also a necessary condition. This is done in [Nester 1988] byshowing that, for all t ∈ [a, b], the value of EL(γ(t)) acting on any vertical vector is zero:

4.5 Lemma: iJEL(γ(t)) = 0.

Proof: First note that iJdL = dJL and EL = iXT diJdL+diV dL−dL. Therefore, using therelations from Section 2.5, we have,

iJ iXT diJdL = [iJ , iX ]diJdL+ iXT iJdiJdL

= −iJXT ddJL+ iXT d2JL+ iXT diJdJL = −iJXT diJdL

and

iJdiV dL− iJdL = dJ iV dL− dJL = [dJ , iV ]dL− dJL− iV dJdL = −iV dJdL = iV ddJL.

Therefore, iJEL = iV−JXT diJdL and since V (γ(t)) = JXT (γ(t)) for all t ∈ [a, b] we have,

iJEL(γ(t)) = 0

�

Therefore, EL(γ(t)) = 0 for all t ∈ [a, b] is a necessary and sufficient condition for γ to bea stationary curve. �

4.2. Local Minimizers from Global Minimizers. The object of this section is to provethat, if γ0 is a solution to the minimization problem on some time interval [a, b], then it alsosolves a minimization problem on any subinterval of [a, b]. What follows uses techniquesfound in [Bullo and Lewis 2004] and [Troutman 1996].


4.6 Theorem: Let L : TM → R be a C0-function. Suppose a < a < b < b. If γ0 ∈D(C2, xa, xb) minimizes AL[γ] with respect to all γ ∈ D(C2, xa, xb), then γ0 minimizesAL[γ] with respect to all γ ∈ D(C2, γ0(a), γ0(b)). Where AL[γ] is the restriction of AL[γ] tothe interval [a, b] and D(C2, γ0(a), γ0(b)) = {γ ∈ C2([a, b]) | γ(a) = γ0(a), γ(b) = γ0(b)}.

Proof: We prove the contra-positive. Suppose that γ0 does not minimize AL[γ] with respectto all γ ∈ D(C2, γ0(a), γ0(b)). Then there exists a curve γ ∈ D(C2, γ0(a), γ0(b)) satisfying,

AL[γ0]− AL[γ] = δ (4.11)

for some δ > 0.Now, if the curve

γ∗(t) =

γ0(t) for all t ∈ [a, a]

γ(t) for all t ∈ (a, b)

γ0(t) for all t ∈ [b, b]

(4.12)

is of class C2, then AL[γ∗] < AL[γ0] and then certainly γ0 is not a minimizer. Unfortunately,it is very likely that γ∗ is not of class C2. In the next lemma we show that there exists acurve γ satisfying ∣∣AL[γ]− AL[γ]

∣∣ =1

2δ

and whose concatenation

γ0(t) =

γ0(t) for all t ∈ [a, a]

γ(t) for all t ∈ (a, b)

γ0(t) for all t ∈ [b, b]

(4.13)

is of class C2.

4.7 Lemma: For any ε > 0, sufficiently small, there exists a curve γ ∈ D(C2, γ0(a), γ0(b))such that γ(t) = γ(t) for all t ∈ (a+ε, b−ε), | AL[γ]−AL[γ] |= 1

2δ (where δ satisfies (4.11))and the concatenated curve in (4.13) is of class C2.

Proof: By hypothesis, γ(t) is specified for all t except for t ∈ [a, a+ε] and t ∈ [b−ε, b]. Theconstruction of γ(t) is similar on both time intervals so we will only show the constructionfor t ∈ [a, a + ε]. Let (U1, φ1) be an admissible chart with γ(a) ∈ U1. Choose ε > 0 suchthat γ(a + ε) ∈ U1. We will construct γ(t) such that, in the local coordinate chart, thefollowing properties are satisfied:

1. γφ1(a) = γφ1(a);

2. ˙γφ1(a) = γφ1(a);

3. ¨γφ1(a) = γφ1(a);

4. γφ1(a+ ε) = γφ1(a+ ε);

5. ˙γφ1(a+ ε) = ˙γφ1(a+ ε);

6. ¨γφ1(a+ ε) = ¨γφ1(a+ ε).

22 E. K. Hansen

t

¨γj(t)

a a + ε

¨γj(a)

¨γj(a + ε)

hj

kj

Figure 1. Proposed choice for a continuous ¨γ on the time intervalt ∈ [a, a+ ε].

Let γj(t) denote the jth coordinate of γφ1(t). We begin by defining ¨γj(t) for t ∈ [a, a+ ε].

Choose ¨γj(t) to be the continuous sawtooth function depicted in Figure 1. As shown in

Figure 1, ¨γj(a) and ¨γj(a+ ε) are chosen to satisfy Properties 3 and 4 and kj and hj are to

be determined. The coordinate expression for ¨γj(t) is given by,

¨γj(t) =

3(kj−γj(a))

ε t+ γj(a)− 3(kj−γj(a))aε , t ∈ [a, a+ 1

3ε),3(hj−kj)

ε t+ 2kj − hj + 3aε (kj − hj), t ∈ [a+ 1

3ε, a+ 23ε],

3(−hj+¨γj(a+ε))ε t+ ¨γj(a+ ε)− 3(−hj+¨γj(a+ε))(a+ε)

ε , t ∈ (a+ 23ε, a+ ε].

(4.14)

To find the coordinate expressions for ˙γj(t), and γj(t) we integrate (4.14). This yields thefollowing equations,

˙γj(t) =

3(kj−γj(a))

2ε t2 + γj(a)t− 3(kj−γj(a))aε t+Bj , t ∈ [a, a+ 1

3ε),3(hj−kj)

2ε t2 + 2kjt− hjt+ 3aε (kj − hj)t+Bj , t ∈ [a+ 1

3ε, a+ 23ε],

3(−hj+¨γj(a+ε))2ε t2 + ¨γj(a+ ε)t− 3(−hj+¨γj(a+ε))(a+ε)

ε t+Bj , t ∈ (a+ 23ε, a+ ε],

γj(t) =

(kj−γj(a))

2ε t3 + 12 γ

j(a)t2 − 3(kj−γj(a))a2ε t2 +Bjt+ Cj , t ∈ [a, a+ 1

3ε),(hj−kj)

2ε t3 + kjt2 − 12h

jt2 + 3a2ε (kj − hj)t2 +Bjt+ Cj , t ∈ [a+ 1

3ε, a+ 23ε],

(−hj+¨γj(a+ε))2ε t3 + 1

2¨γj(a+ ε)t2 − 3(−hj+¨γj(a+ε))(a+ε)

2ε t2 +Bjt+ Cj , t ∈ (a+ 23ε, a+ ε].

Bj , Cj , hj and kj are chosen to satisfy Properties 1–5. A tedious computation gives

hj =1

3(a2 + 2aε+ ε2)(γj(a)a2 + 4 ˙γj(a+ ε)a+ ¨γj(a+ ε)a2 + ¨γj(a+ ε)aε+ 6γ(a)j

− 6γj(a+ ε)− 4γj(a)a+ 6 ˙γj(a+ ε)ε)


and

kj =1

3a2(2γj(a)a2 + 2γj(a)aε− 4 ˙γj(a+ ε)a+ ¨γj(a+ ε)a2 + 2¨γj(a+ ε)εa− 6γ(a)j

+ 6γj(a+ ε) + 4γj(a)a− 4 ˙γj(a+ ε)ε+ ¨γj(a+ ε)ε2 − 2γj(a)ε).

Provided 0 < ε < 1 and letting

mj = max{∣∣γj(t)∣∣ , ∣∣γj(t)∣∣ , ∣∣γj(t)∣∣ , ∣∣γj(t)∣∣ , ∣∣ ˙γj(t)∣∣ , ∣∣¨γj(t)∣∣} (4.15)

we have ∣∣hj∣∣ ≤ 1

3a2(3mj a2 + 9mj |a|+ 18mj)

and ∣∣kj∣∣ ≤ 1

3a2(3mj a2 + 12mj |a|+ 19mj).

Therefore, there exists a positive constant µj such that∣∣hj∣∣ , ∣∣kj∣∣ ≤ µj (4.16)

Referring to Figure 1 we see that (4.15) and (4.16) imply∣∣¨γj∣∣ ≤ 2mj + 2µj ,

and therefore,

∣∣ ˙γj(t)∣∣ =

∣∣∣∣∫ t

a

¨γj(τ)dτ + γj(a)

∣∣∣∣ ≤ ∫ t

a

∣∣¨γj(τ)∣∣dτ +mj

≤ 2(mj + µj)(t− a) +mj ≤ 2(mj + µj)(ε) +mj ≤ 3mj + 2µj .

Also,

∣∣γj(t)− γj(t)∣∣ =

∣∣∣∣∫ t

a( ˙γj(τ)− ˙γj(τ))dτ + γj(a)− γj(a)

∣∣∣∣ ≤ ∫ t

a

∣∣ ˙γj(τ)− ˙γj(τ)∣∣dτ

≤∫ t

a

∣∣ ˙γj(τ)∣∣+∣∣ ˙γj(τ)

∣∣dτ ≤ ∫ a+ε

a

∣∣ ˙γj(τ)∣∣+∣∣ ˙γj(τ)

∣∣dτ ≤ (4mj + 2µj)ε.

Therefore, by appropriate choice of ε, we can construct a curve γφ(t) which is arbitrarilyclose to γφ(t) for t ∈ (a, a+ ε) and satisfies Properties 1–6. Applying φ−11 to γφ(t) gives thedesired curve in M .

In order to show that | AL[γ] − AL[γ] |≤ 12δ, we first show that there exists a βa > 0,

independent of ε, satisfying ∣∣L( ˙γ(t))∣∣ < βa ∀ t ∈ [a, a+ ε].

To do this, first choose 0 < εa < 1 such that if γj(t) ∈ [γj(t) − (4mj + 2µj)εa, γj(t) +

(4mj+2µj)εa] for all t ∈ [a, a+εa] then γ(t) ∈ φ1(U1) for all t ∈ [a, a+εa]. For any 0 < ε < εa,γj(t) ∈ [γj(t)−(4mj+2µj)εa, γ

j(t)+(4mj+2µj)εa], and ˙γj(t) ∈ [−(3mj+2µj), (3mj+2µj)].

Since these intervals are compact there exists a constant βa such that∣∣∣L( ˙γ(t))

∣∣∣ ≤ βa for all

24 E. K. Hansen

t ∈ [a, a+ εa]. Using similar arguments we can show that there exist analogous constants,εb and βb, for the time interval [b− ε, b]. Let β = max{βa, βb} and choose ε < max{εa, εb} .

With the above constructions we have∣∣AL[γ(t)]− AL[γ(t)]∣∣ =

∣∣ ∫ b

aL( ˙γ(t))dt−

∫ b

aL( ˙γ(t))dt

∣∣=

∣∣ ∫ b

a(L( ˙γ(t))− L( ˙γ(t)))dt

∣∣=

∣∣ ∫ a+ε

a(L( ˙γ(t))− L( ˙γ(t)))dt+

∫ b

b−ε(L( ˙γ(t))− L( ˙γ(t)))dt

∣∣.Now since L : TM → R is a continuous function with respect to elements in TM and γ

∈ C2([a, b],M) the composition L◦ ˙γ belongs to C([a, b],R). Since [a, b] is a compact interval,this implies that L ◦ ˙γ achieves its maximum and minimum values. Let M = max

∣∣L ◦ ˙γ∣∣

for all t ∈ [a, b].Therefore, ∣∣∣AL[γ(t)]− AL[γ(t)]

∣∣∣ ≤ ∫ a+ε

a(β +M)dt+

∫ b

b−ε(β +M)dt

≤ 2(β +M)ε. (4.17)

Choosing ε < 12

δ2(β+M) (where δ satisfies (4.11)), (4.17) results in,∣∣∣AL[γ(t)]− AL[γ(t)]

∣∣∣ < δ

2.

Now, since γ0 ∈ C2([a, b],M) and γ ∈ C2([a, b],M) satisfies Properties (i) and (vi), thecurve γ0 defined in (4.13) certainly belongs to D(C2, xa, xb). �

Since γ is a minimizer of AL we have,∫ b

aL(γ0(t))dt =

∫ a

aL(γ0(t))dt+

∫ b

aL(γ0)dt+

∫ b

bL(γ0)dt

=

∫ a

aL(γ0(t))dt+

∫ b

aL( ˙γ(t))dt+ δ +

∫ b

bL(γ0(t))dt

≥∫ a

aL(γ0)dt+

∫ b

aL( ˙γ(t))dt+

1

2δ +

∫ b

bL(γ0(t))dt

≥∫ a

aL(γ0)dt+

∫ b

aL( ˙γ(t))dt+

∫ b

bL(γ0(t))dt

=

∫ b

aL( ˙γ0(t))dt.

Therefore, γ0(t) cannot minimize AL. �

An important consequence of Theorem 4.6 is that to check that a curve γ on a manifoldM minimizes an action AL, it is equivalent to check that for any set of charts {(φi, Ui)}covering γ, all of the coordinate representations γφi are solutions to corresponding localminimization problems.


5. Conclusion and Further Work

5.1. Conclusion. In this report we have elaborated on the previous work of Nester [Nester1988], in order to provide a thorough coordinate-free derivation of the Euler–Lagrangeequations. We began by using a variational approach to define two vector fields; γ, thetangent vector field to the curve, and δσ, the vector field which defines the infinitesimalvariation. We showed that the condition for stationarity can be expressed in terms of theLie derivative of the Lagrangian with respect to the complete lift of an arbitrary extensionof δσ.

After proving two identities which allowed us to preserve our original requirement thatγ(t) be only of class C2, we presented a coordinate-free derivation of the Euler–Lagrangeequations. Most importantly, we addressed the fact that using complete lifts in the deriva-tion necessitated first extending γ(t) and δσ to neighbourhoods of image(γ). In line withthis we showed, using coordinate calculations, that the derivation was in fact independentof our choice of extension.

Lastly, we showed that, if a curve is a C2-minimizer on a certain interval then it must alsosolve a corresponding minimization problem on any subinterval. A corollary of this resultis that a C2-curve on a manifold solves a minimization problem if and only if its coordinateexpression solves a corresponding minimization problem in a set of charts covering thecurve.

A summary of the results given in this report follows:

• An equivalent condition for stationarity is∫ b

a(LZTL)γ(t)dt = 0,

where ZT is the complete lift of an arbitrary extension of δσ. Moreover this equationis independent of how δσ is extended.

• If γ and δσ are defined on a Riemannian manifold, then they can be extended tocover the tubular neighbourhood defined by the exponential map. This extension isconstructed using the push-forward of certain vertical vector fields with respect to theexponential map.

• If γ and δσ are defined on a manifold which admits partitions of unity, then they canbe extended to cover the open neighbourhoods Ui of the submanifold charts {(Ui, φi)}.This extension is constructed by extending the local vectors in the coordinate neigh-bourhoods given by the submanifold charts, and then patching together the resultingvector fields on M using a partition of unity.

• Given that X, Z ∈ Γ0(TTM) satisfy Tπ(X), Tπ(Z) ∈ Γ1(TM), then

d(iJdL)(X, Z) =

LX〈dL, vlft(Tπ(Z))〉 − LZ〈dL, vlft(Tπ(X))〉 − 〈dL, vlft[Tπ(X), Tπ(Z)]〉.

• Let X, Z ∈ Γ1(TM). At points vx ∈ TM satisfying [X,Z](x) = 0 and X(x) = vx wehave

(LZT 〈dL, V − JX〉)(vx) = 〈LZT dL, V − JX〉(vx) + 〈dL, vlft([Z,X])〉(vx).

26 E. K. Hansen

• The coordinate-free derivation (4.10) of the Euler–Lagrange equation is independentof vector field extension.

• If a C2-curve solves the minimization problem on a certain interval, then it solves acorresponding minimization problem on any subinterval.

5.2. Further Work. The results in this report admit further generalization and suggestfurther study.

• Time-dependent Lagrangian: All results in this report have been for a time inde-pendent Lagrangian. For a more general theory the results should be generalized toinclude time-dependent Lagrangians. This is straightforward.

• Coordinate-free derivations of Major Theorems of Calculus of Variations: In partic-ular, it would be useful to give the coordinate-free derivations of Hilbert’s criterion,Lagrange’s Lemma and the Weierstrass–Erdmann Corner conditions. These wouldserve to further elucidate the issues involving differentiability constraints.

• Minimization over a more general class of curves: Armed with a coordinate-free ver-sions of the major theorems of Calculus of Variations, the next step would be toextend the theory to a more general class of curves (C1- or piecewise C1-curves).

• Geometric study of identities from Section 4.1: The identities used to relax the differ-entiability constraints on XT and ZT were proven using coordinate calculations. Itwould be informative to both determine the geometric significance of these identitiesand to find their coordinate-free proofs.

Acknowledgment. Foremost I would like to thank my supervisor Andrew Lewis for alwaysbeing available for invaluable discussion and for taking a vested interest in my educationand work. Thanks are also due to John Chapman, David Tyner and Ajit Bhand; John foralways having interesting answers to my questions, Dave for directing me towards numerousindispensable resources and Ajit for always being willing to discuss differential geometryover a cup of coffee.

List of Symbols

Below is a roughly alphabetical list of symbols that reoccur often throughout the text.

AL the action of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

AL restriction of AL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

d the exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10dJ the vertical derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10D(C2, γ0(a), γ0(b)) a certain class of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21D(C2, xa, xb) a certain class of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10δσ an infinitesimal variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11EL the one-form used in expression for Euler–Lagrange equation . . 16exp the exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7


f∗ω the pull-back of ω by f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4f∗X the push-forward of X by f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4γ∗ a concatenation of two curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21γ0 a global minimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21γ a local minimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21γ an approximation to γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

γ0 a concatenation of two curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21γφ1 a local coordinate representation of γ . . . . . . . . . . . . . . . . . . . . . . . . . . 21

γj the jth coordinate of γφ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

i the interior product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

J the almost tangent structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

L the Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

N⊥ the normal bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8∇ a connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7πTM the canonical projection from TTM to TM . . . . . . . . . . . . . . . . . . . . . 6σ a variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11σ(s, t), σs(t), σt(s) a vector field in TM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10{(Ui, gi)}i∈I a partition of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

V the Liouville vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7vlft the vertical lift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6∧ the exterior product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

XT the complete lift of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

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Arnol′d, V. I. [1978] Mathematical Methods of Classical Mechanics, translated by K. Vogt-mann and A. Weinstein, number 60 in Graduate Texts in Mathematics, Springer-Verlag:New York/Heidelberg/Berlin, isbn: 0-387-96890-3, New edition: [Arnol′d 1989].

— [1989] Mathematical Methods of Classical Mechanics, translated by K. Vogtmann and A.Weinstein, 2nd edition, number 60 in Graduate Texts in Mathematics, Springer-Verlag:New York/Heidelberg/Berlin, isbn: 978-0-387-96890-2, First edition: [Arnol′d 1978].

Bullo, F. and Lewis, A. D. [2004] Geometric Control of Mechanical Systems, Modeling,Analysis, and Design for Simple Mechanical Systems, number 49 in Texts in AppliedMathematics, Springer-Verlag: New York/Heidelberg/Berlin, isbn: 978-0-387-22195-3.

Crampin, M. and Pirani, F. A. E. [1986] Applicable Differential Geometry, number 59 inLondon Mathematical Society Lecture Note Series, Cambridge University Press: NewYork/Port Chester/Melbourne/Sydney, isbn: 978-0-521-23190-9.

Edelen, D. G. B. [1985] Applied Exterior Calculus, John Wiley and Sons: NewYork, NY,isbn: 0-471-80773-7, Reprint: [Edelen 2005].

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Coordinate-free derivation of the Euler{Lagrange equations ...andrew/papers/pdf/2005a.pdf · free derivation of the Euler{Lagrange equation is presented. Using a variational ap-proach,