Bundle-theoretic methods for higher-order variational calculus
aa r X i v : . [ m a t h . DG ] J un Bundle-theoretic methods for higher-order variational calculus ∗ Michał Jó´zwikowski † , Mikołaj Rotkiewicz ‡ October 16, 2018
Abstract
We present a geometric interpretation of the integration-by-parts formula on an arbitrary vec-tor bundle. As an application we give a new geometric formulation of higher-order variationalcalculus.
Keywords: higher tangent bundles, variational calculus, higher-order Euler-Lagrange equations,graded bundles, integration by parts, geometric mechanics
MSC:
Our results.
The main motivation of this paper is to clarify the geometry of higher-order variationalcalculus. In order to study this topic we had to introduce new geometric tools and prove some resultswhich we quickly sketch below.First, we observe that, given a vector bundle σ : E → M , it is possible to characterize two canon-ical morphisms Υ k,σ : e T k,k σ → σ and υ k,σ : e T k,k σ → T k σ , where the bundle of semi-holonomicvectors e T k,k σ consists of all elements of T k T k E which project to the elements of T k M ⊂ T k T k M (i.e., to holonomic vectors). Morphisms Υ k,σ and υ k,σ provide an elegant geometric description ofthe k th order integration-by-parts formula on the bundle σ . A more precise formulation is providedby Theorem 3.3 below.Then, we use this result to obtain a comprehensive and, to certain extent, simpler geometric in-terpretation of the standard procedure of deriving both the Euler-Lagrange equations (forces) and theboundary terms (generalized momenta) for a k th order variational problem. Our description is “natu-ral” in the sense that it mimics the standard way of deriving the Euler-Lagrange equations. We startfrom a homotopy γ ( t, s ) in M and then transform the variation of the action t s =0 R t t L ( t kt γ ( t, s )) d t to extract the integral and the boundary terms. Here t kt γ stands for the k th jet of a curve γ = γ ( t ) ata point t ∈ R . In particular, t t = t γ ( t ) is the tangent vector to the curve γ at t . On the computationallevel this procedure consists basically of two steps: • reversing the order of differentiation to get t kt t s γ ( t, s ) from t s t kt γ ( t, s ) , • performing the k th order integration by parts to extract t s =0 γ ( t, s ) from t kt t s =0 γ ( t, s ) . ∗ This research was supported by Polish National Science Center grant under the contract number DEC-2012/06/A/ST1/00256. † Institute of Mathematics. Polish Academy of Sciences (email: [email protected] ) ‡ Institute of Mathematics. Polish Academy of Sciences and
Faculty of Mathematics, Informatics and Mechanics. Uni-versity of Warsaw (email: [email protected] ) t s t kt γ ( t, s ) is the image of t kt t s γ ( t, s ) under the canonical flip κ k : T k T M → TT k M , hence reversing this operation requiresapplying the dual map ε k : T ∗ T k M → T k T ∗ M to the differential of the Lagrangian [2]. Concerningthe second, we can apply the maps Υ k,τ ∗ and υ k − ,τ ∗ introduced by us (here τ ∗ : T ∗ M → M standsfor the cotangent fibration). In this way we obtain the geometric formula Υ k,τ ∗ (cid:0) t k ε k (cid:0) d L ( t k γ ( t )) (cid:1)(cid:1) describing the force (integral term) along the trajectory γ ( t ) = γ ( t, . Thus the Euler-Lagrangeequations along γ ( t ) read as Υ k,τ ∗ (cid:16) t kt ε k (cid:16) d L ( t k γ ( t )) (cid:17)(cid:17) = 0 . The geometric formula for the momentum (boundary term) is obtained in a similar way using the map υ k − ,τ ∗ . The precise description is provided by Theorems 4.2 and 4.3. State of research, applications.
In the theory of jet bundles there exists a well established notionof semi-holonomic jets [19]. These objects share some similarities with our semi-holonomic vectors ,however, in principle, are different, so that the similarity of names shall not be confusing. Morphisms Υ k,σ and υ k,σ were, to our best knowledge, so far not present in the literature. On the contrary, theproblem of geometric formulation of higher-order variational calculus has gained a lot of interestin the mathematical community and has many solutions (e.g., [3, 14, 24, 25]), briefly reviewed inSection 5.Since the geometry of higher-order variational calculus is a well-established topic, one should askabout other applications of morphisms Υ k,σ and υ k,σ . We believe that our result makes it easier togeneralize higher-order variational calculus and mechanics to the framework of algebroids. Apartfrom some very special cases (see [5]), we are not aware of any such generalization in the spirit of theclassical papers quoted above. Our preprint [10] contains some results in this direction, which makeuse of the tools introduced here. We present them briefly in Section 5. Outline of the paper.
In the preliminary Section 2 we revise basic information on higher tangentbundles, vector bundles and their lifts, as well as canonical pairings between such objects. We alsorecall the notion of the canonical flip κ k : T k T M −→ TT k M and its dual ε k : T ∗ T k M → T k T ∗ M .Section 3, with its central Theorem 3.3, contains our main results. We begin by introducing thenotion of the bundle of semi-holonomic vectors e T n ,...,n k σ and the canonical projection P k : e T ( k ) σ := e T ,..., σ → T k σ . In Theorem 3.3 we provide a geometric construction of the canonical maps Υ k,σ and υ k,σ which give a comprehensive geometric interpretation of the integration-by-parts procedureon a vector bundle σ . We also state Lemma 3.4 about the universality of the map Υ k,σ , whose proofis postponed to the Appendix.In Section 4 we show how to apply our results to higher-order variational problems (Problem 4.1).In particular, in Theorem 4.2, we obtain the general formula for the variation of the action includingthe forces and the generalized momenta along the trajectory. As a corollary we prove Theorem 4.3,which characterizes the extremals of Problem 4.1 in terms of higher-order Euler-Lagrange equationsand general transversality conditions. We also give a geometric and local description of the Euler-Lagrange equations and generalized momenta.Finally, in Section 5, we briefly discuss different approaches to the geometry of higher-ordervariational calculus and relate our work in this direction with the papers of Tulczyjew [20, 24]. Laterwe sketch some result from [10], which show an application of our results to higher-order variationalproblems on algebroids. We end with an example of Riemannian cubic polynomials.undle-theoretic methods for higher-order variational calculus 3 Higher tangent bundles.
Throughout the paper we shall work with higher tangent bundles and usethe standard notation T k M for the k th tangent bundle of the manifold M . Points in the total spaceof this bundle will be called k –velocities . An element represented by a curve γ : [ t , t ] → M at t ∈ [ t , t ] will be denoted by t k γ ( t ) or t kt γ ( t ) .The k + 1 st tangent bundle is canonically included in the tangent space of the k th tangent bundle(see, e.g., [21]): ι ,k : T k +1 M ⊂ TT k M, t k +1 t =0 γ ( t ) t t =0 t ks =0 γ ( t + s ) . The composition of this injection with the canonical projection τ T k M : TT k M → T k M defines thestructure of the tower of higher tangent bundles : T k M −→ T k − M −→ T k − M −→ . . . −→ T M −→ M. The canonical projections from higher to lower-order tangent bundles will be denoted by τ ks : T k M → T s M (for k ≥ s ). Instead of τ k : T k M → M we simply write τ k and, instead of τ = τ : T M → M , just use the standard symbol τ . The cotangent fibration is denoted by τ ∗ : T ∗ M → M .Another important constructions are the iterated tangent bundles T ( k ) M := T . . . T M and the iterated higher tangent bundles T n . . . T n r M . Elements of the latter will be called ( n , . . . , n r ) –velocities . For notational simplicity we shall also denote the iterated higher tangent functor T n T n . . . T n r by T n ,...,n r .Of our particular interest will be the bundles T k T l M = T k,l M . These bundles admit naturalprojections to lower-order tangent bundles which will be denoted by τ ( k,l )( k ′ ,l ′ ) : T k T l M → T k ′ T l ′ M (for k ≥ k ′ and l ≥ l ′ ). (Iterated higher) tangent bundles are subject to a number of natural inclusionssuch as already mentioned ι ,k : T k +1 M ⊂ TT k M , ι l,k : T k + l M ⊂ T l T k M , ι k : T k M ⊂ T ( k ) M ,etc., which will be used extensively.Occasionally, when dealing with manifolds other than M , or when it can lead to confusions, wewill add a suffix to the maps ι ...... , τ ...... , etc., to emphasize which manifold we are working with, e.g., τ kl = τ kl,M , etc.Given a smooth function f on a manifold M one can construct functions f ( α ) on T k M , with ≤ α ≤ k , the so called ( α ) -lifts of f (see [17]), defined by(2.1) f ( α ) ( t k γ ( t )) := d α d t α (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( γ ( t )) . The functions f ( k ) : T k M → R and f (1) : T M → R are called the complete lift and the tangent lift of f , respectively. By iterating this construction we obtain functions f ( α,β ) := ( f ( β ) ) ( α ) on T k T l M for ≤ α ≤ k , ≤ β ≤ l , and, generally, functions f ( ǫ ,...,ǫ r ) on T n . . . T n r M for ≤ ǫ j ≤ n j , ≤ j ≤ r . A coordinate system ( x a ) on M gives rise to the so-called adapted coordinate systems ( x a, ( α ) ) ≤ α ≤ k on T k M and ( x a, ( ǫ ) ) ǫ on T n . . . T n r M where the multi-index ǫ = ( ǫ , . . . ǫ r ) is asbefore, and x a, ( α ) , x a, ( ǫ ) are obtained from x a by the above lifting procedure. Within this notation weeasily find that the canonical inclusion ι k : T k M → T ( k ) M is given by(2.2) ( ι k ) ∗ ( x a, ( ǫ ) ) = x a, ( ǫ + ... + ǫ k ) , where ǫ ∈ { , } k .Let us remark that in the definition of f ( α ) we follow the convention of [5, 20], whereas theoriginal convention of [17] is slightly different, since it contains a normalizing factor α ! in front ofthe derivative. Tangent lifts of vector bundles and canonical pairings.
Let us pass to another important tool forour analysis, namely the (iterated) higher tangent bundles of vector bundles. Let σ : E → M bea vector bundle. It is clear that σ may be lifted to vector bundles T k σ : T k E → T k M , T ( k ) σ :T ( k ) E → T ( k ) M , T k T l σ : T k T l E → T k T l M , etc. Let σ ∗ : E ∗ → M be the bundle dual to σ .Throughout the paper we denote by ( x a ) the coordinates on the base M , by ( y i ) the linear co-ordinates on the fibers of σ , and by ( ξ i ) the linear coordinates on the fibers of the dual bundle σ ∗ : E ∗ → M . Natural weighted coordinates on (iterated higher) tangent lifts of σ and σ ∗ areconstructed from x a , y i , ξ j by the lifting procedure mentioned above. They are denoted by addingthe proper degree to a coordinate symbol. Degrees will be denoted by bracketed small Greek letters: ( ǫ ) , ( α ) , ( β ) , etc.The natural pairing h· , ·i σ between E and E ∗ induces a non-degenerated pairing between T E and T E ∗ over T M : h· , ·i T σ := h· , ·i (1) σ : T( E ∗ × M E ) ≃ T E ∗ × T M T E → R , i.e., h· , ·i T σ is the tangent lift of h· , ·i σ . In a similar way, h· , ·i σ can be lifted to non-degenerate pairingson higher tangent prolongations of E and E ∗ . Proposition 2.1.
Let h· , ·i σ : E ∗ × M E → R be the natural pairing. Let us define inductively h· , ·i T ( k ) σ : T ( k ) E ∗ × T ( k ) M T ( k ) E → R as the tangent lift of h· , ·i T ( k − σ (for k ≥ ) and let h· , ·i T k σ : T k E ∗ × T k M T k E → R be the restriction of h· , ·i T ( k ) σ to the product of the subbundles T k σ ∗ ⊂ T ( k ) σ ∗ and T k σ ⊂ T ( k ) σ .Then(a) in the local coordinates denoted by ( x a, ( ǫ ) , y i, ( ǫ ) ) and ( x a, ( ǫ ) , ξ ( ǫ ) i ) , with ǫ ∈ { , } k (resp. ( x a, ( α ) , y i, ( α ) ) and ( x a, ( α ) , ξ ( α ) i ) with ≤ α ≤ k ), on T ( k ) E and T ( k ) E ∗ (resp. T k E and T k E ∗ ), (2.3) D ( x a, ( ǫ ) , ξ ( ǫ ) i ) , ( x a, ( ǫ ) , y i, ( ǫ ) ) E T ( k ) σ = X i X ǫ ∈{ , } k ξ ( ǫ ) i y i, (1 ,..., − ( ǫ ) , (2.4) D ( x a, ( α ) , ξ ( α ) i ) , ( x a, ( α ) , y i, ( α ) ) E T k σ = X i X ≤ α ≤ k (cid:18) kα (cid:19) ξ ( α ) i y i, ( k − α ) . (b) h· , ·i T ( k ) σ and h· , ·i T k σ are non-degenerate pairings.(c) h· , ·i T k σ = h· , ·i ( k ) σ , i.e., the pairing h· , ·i T k σ is the complete lift of h· , ·i σ (up to the isomorphism T k ( E ∗ × M E ) ≃ T k E ∗ × T k M T k E ).Proof. We get (2.3) by iterated differentiation of the function h· , ·i σ : (cid:0) ( x a , ξ i ) , ( x a , y i ) (cid:1) P i ξ i y i .Taking into account (2.2) we find (2.4). Since the tangent lift of a non-degenerate pairing is non-degenerate, so it is for h· , ·i T ( k ) σ . The non-degeneracy of the pairings h· , ·i T ( k ) σ and h· , ·i T k σ can alsobe easily seen from the above local formulas. Finally, (c) follows immediately from the definition ofthe complete lift and the local expression (2.4) of h· , ·i T k σ .We stress that the above lifting procedure can be expressed in the framework of Weil functors andFrobenius algebras [26, 13].undle-theoretic methods for higher-order variational calculus 5 Canonical flip κ k and its dual ε k . It is well-known that the iterated tangent bundle TT M admitsan involutive double vector bundle isomorphism (called the canonical flip ) κ : TT M −→ TT M, which intertwines the projections τ T M : TT M → T M and T τ : TT M → T M (see, e.g., [16]).Such a notion of canonical flip can be generalized to a family of isomorphisms κ k : T k T M → TT k M, t kt =0 t s =0 γ ( t, s ) t s =0 t kt =0 γ ( t, s ) , which map the projection T k τ : T k T M → T k M to τ T k M : TT k M → T k M over id T k M and τ k T M : T k T M → T M to T τ k : TT k M → T M over id T M . Morphisms κ k can be also definedinductively as follows: κ := κ and κ k +1 := T κ k ◦ κ T k M (cid:12)(cid:12) T k +1 T M , i.e., as the unique morphismmaking the diagram(2.5) TT k T M T κ k / / TTT k M κ T kM / / TTT k M T k +1 T M κ k +1 / / ❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴ (cid:31) ? O O TT k +1 M (cid:31) ? O O commutative. The local description of κ k is very simple. If x a, ( α,ǫ ) are natural coordinates on T k T M and x a, ( ǫ,α ) are natural adapted coordinates on TT k M (with ≤ α ≤ k and ǫ = 0 , ), then x a, ( α,ǫ ) corresponds to x a, ( ǫ,α ) via κ k .Introduce now the dual ε k : T ∗ T k M → T k T ∗ M of the canonical flip κ k defined via the equality,(2.6) h Ψ , κ k ◦ V i τ T kM = h ε k ◦ Ψ , V i T k τ , where V ∈ T k T M and Ψ ∈ T ∗ T k M is a vector such that both pairings make sense (cf. [2]).Formula (2.6) shows that κ k and ε k are “adjoint” to each other with respect to canonical pairings, asschematically shown by the commutative diagram TT k M { { T k T M κ k o o R h· , ·i τ T kM h· , ·i T k τ M R T ∗ T k M c c ε k / / T k T ∗ M. ; ; In the coordinates ( x a, ( α ) , p a, ( α ) = ∂ x a, ( α ) ) on T ∗ T k M and ( x a, ( α ) , p ( α ) a ) on T k T ∗ M (adaptedfrom standard coordinates ( x a, ( α ) ) on T k M , and ( x a , p a ) on T ∗ M , respectively), we find from (2.4)that(2.7) ε k (cid:16) x a, ( α ) , p a, ( α ) (cid:17) = x a, ( α ) , p ( α ) a = (cid:18) kα (cid:19) − p a, ( k − α ) ! . In this section the construction of the vector bundle morphisms Υ k,σ and υ k,σ , associated with an in-teger k and a vector bundle σ : E → M , will be described. These morphisms are closely related withthe geometric integration-by-parts procedure and will play a crucial role in the geometric constructionof the Euler-Lagrange equations in the next Section 4. Bundles of semi-holonomic vectors.Definition 3.1.
For non-negative numbers n , . . . , n r ≥ let denote ¯ n := n + . . . + n r . The set e T n ,...,n r E = (cid:0) ι n ,...,n r M (cid:1) ∗ T n ,...,n r E = { X ∈ T n ,...,n r E : T n ,...,n r σ ( X ) ∈ T ¯ n M ⊂ T n ,...,n r M } , consisting of all ( n , . . . , n r ) –velocities in E projecting to ¯ n –velocities ( holonomic vectors ) in T n ,...,n r M ,is a vector subbundle of T n ,...,n r σ : T n ,...,n r E → T n ,...,n r M . The restriction e T n ,...,n r σ of T n ,...,n r σ to e T n ,...,n r E is the bundle of semi-holonomic vectors . Given a vector bundle σ ′ : E ′ → M ′ and a morphism φ : σ → σ ′ we define e T n ,...,n r φ : e T n ,...,n r σ → e T n ,...,n r σ ′ as the restriction of T n ,...,n r φ to e T n ,...,n r E . Thus e T n ,...,n r is a functor in the category of vector bundles.In agreement with our previous notation we shall denote by e T ( k ) E = e T ,..., E the subbundle ofsemi-holonomic velocities in T ( k ) E = T ,..., E .In future considerations the bundles e T k,k E and e T ( k ) E will be of our special interest. Let usremark that although, in general, there is no canonical projection T ( k ) E → T k E , there exists anatural projection P k : e T ( k ) E → T k E . It is defined as the left inverse to the canonical inclusion ι kE but considered as a map to e T k E , as is explained in the following proposition. Proposition 3.2.
Consider a semi-holonomic vector X ∈ e T ( k ) E ⊂ T ( k ) E lying over the k –velocity v k ∈ T k M ⊂ T ( k ) M . Then the formula h P k ( X ) , Ψ i T k σ = h X, Ψ i T ( k ) σ , where Ψ ∈ T k E ∗ ⊂ T ( k ) E ∗ lies over v k , defines a canonical projection P k : e T ( k ) E → T k E ,i.e., P k ◦ ι kE = id T k E . Moreover, P k is a vector bundle morphism and it can be expressed in localcoordinates as P k (cid:16) x a, ( ǫ ) , y i, ( ǫ ) (cid:17) = (cid:16) x a, ( α ) , y i, ( α ) (cid:17) , where y i, ( α ) = (cid:0) kα (cid:1) − X | ǫ | = α y i, ( ǫ ) is the arithmetic average of all the coordinates of total degree α .Proof. It follows immediately from the properties of the non-degenerate pairing h· , ·i T k σ (see Propo-sition 2.1). The map Υ k,σ . The result below describes the construction of a certain canonical and universalvector bundle morphism Υ k,σ from e T k,k σ to σ . The precise sense of the word universal will begiven later in Lemma 3.4. Informally speaking, any other morphism e T k,k σ → σ can be derived inan easy way from e T k,k . In the next Section 4 we show that this morphism is directly connected withintegration by parts in the procedure of deriving the Euler-Lagrange equations.To fix some notation denote an element Φ ∈ e T k,k E ⊂ T k T k E by Φ ( k,k ) and its projections tolower-order velocities by Φ ( m,n ) := τ ( k,k )( m,n ) ,E (Φ) ∈ T m T n E, undle-theoretic methods for higher-order variational calculus 7where m, n ≤ k . Observe that since Φ lies over some k –velocity, say v k ∈ T k M , then all theelements Φ ( m,n ) project under T m T n σ to a fixed m + n –velocity v m + n ∈ T m + n M ⊂ T m T n M independently on the numbers in the sum m + n . In particular, different elements Φ ( m,n ) with m + n fixed, belonging a priori to different bundles T m T n E , can be added together in the vector bundle T ( m + n ) σ : T ( m + n ) E → T ( m + n ) M , which contains all of them. Denote by e Φ the following elementof T ( k ) E :(3.1) e Φ := Φ (0 ,k ) − (cid:18) k (cid:19) Φ (1 ,k − + (cid:18) k (cid:19) Φ (2 ,k − + . . . + ( − k Φ ( k, . Similarly, define a morphism υ k,σ : e T k,k σ → T k σ by the formula(3.2) υ k,σ (cid:16) Φ ( k,k ) (cid:17) := P k (cid:20)(cid:18) k + 11 (cid:19) Φ (0 ,k ) − (cid:18) k + 12 (cid:19) Φ (1 ,k − + . . . + ( − k +1 (cid:18) k + 1 k + 1 (cid:19) Φ ( k, (cid:21) , where P k : e T ( k ) E → T k E was defined in Proposition 3.2.Now we are ready to state the main result of this section. Theorem 3.3 (Bundle-theoretic integration by part) . Let
Φ = Φ ( k,k ) ∈ e T k,k E be a semi-holonomicvector projecting to v k ∈ T k M and let v k := τ kk ( v k ) ∈ T k M . Let t k ξ be any element in T kξ E ∗ which projects to v k under T k σ ∗ . Then, if e Φ is given by (3.1) , the value of D e Φ , t k ξ E T ( k ) σ does notdepend on the choice of t k ξ . Hence it defines a canonical vector bundle morphism Υ k,σ : e T k,k σ → σ covering τ k : T k M → M given by (3.3) h Υ k,σ (Φ) , ξ i σ := De Φ , t k ξ E T ( k ) σ = * k X j =0 ( − j (cid:18) kj (cid:19) Φ ( j,k − j ) , t k ξ + T ( k ) σ . Moreover,(a) in coordinates, if Φ ∼ (cid:0) x a, ( α ) , y j, ( β,γ ) (cid:1) where ≤ α ≤ k and ≤ β, γ ≤ k , then (3.4) Υ k,σ (Φ) = x a , k X α =0 ( − α (cid:18) kα (cid:19) y i, ( α,k − α ) ! ; (b) Υ k,σ satisfies the recurrence formulas Υ ,σ (Φ) = ν E [ τ T E (Φ) − T τ E (Φ)] , (3.5) Υ k,σ = Υ ,σ ◦ Υ k − , TT σ (cid:12)(cid:12)(cid:12) e T k,k E , (3.6) where ν E denotes the projection T E (cid:12)(cid:12) M ∼ = E × M T M → E ;(c) υ k,σ satisfies the recurrence formulas υ ,σ = id σ , (3.7) D υ k,σ (cid:16) Φ ( k,k ) (cid:17) , t k ξ E T k σ = D Υ k,σ (cid:16) Φ ( k,k ) (cid:17) , ξ E σ + D υ k − , T σ (cid:16) Φ ( k − ,k ) (cid:17) , t k ξ E T ( k ) σ , (3.8) where in the last term we consider Φ ( k − ,k ) as a semi-holonomic vector in e T k − ,k − T E ⊃ e T k − ,k E ; (d) Υ k,σ and υ k − ,σ are related by the “bundle-theoretic integration by parts formula” (3.9) D Φ (0 ,k ) , t k ξ E T ( k ) σ = D Υ k,σ (cid:16) Φ ( k,k ) (cid:17) , ξ E σ + D T υ k − ,σ (cid:16) Φ ( k,k − (cid:17) , t k ξ E T ( k ) σ , where in the last term we consider Φ ( k,k − as a vector in T e T k − ,k − E ⊃ e T k,k − E ;(e) morphism Υ k, · commutes with the tangent functor T up to the canonical isomorphism e κ k,k,E : e T k,k T E → T e T k,k E , i.e., the diagram (3.10) e T k,k T E Υ k, T σ / / e κ k,k,E (cid:15) (cid:15) T E T e T k,k E TΥ k,σ ♥♥♥♥♥♥♥♥♥♥♥♥ is commutative;(f) morphism υ k, · commutes with the tangent functor T up to the canonical isomorphisms e κ k,k,E : e T k,k T E → T e T k,k E and κ k,E : T k T E → TT k E , i.e., the diagram (3.11) e T k,k T E υ k, T σ / / e κ k,k,E (cid:15) (cid:15) T k T E κ k,E (cid:15) (cid:15) T e T k,k E T υ k,σ / / TT k E is commutative. Lemma 3.4 is a natural continuation of Theorem3.3, but we decided to keep it separated since itsproof is rather technical (see Appendix).
Lemma 3.4 (Universality of Υ k,σ ) . Let M be a connected manifold. Then any functorial vectorbundle morphism ( F E , F E ) (3.12) e T k,k E e T k,k σ (cid:15) (cid:15) F E / / E σ (cid:15) (cid:15) T k M F E / / M is a linear combination of the morphisms Υ l,σ ◦ τ ( k )( l ) ,E , where τ ( k )( l ) ,E : e T k,k E → e T l,l E is the canonicalprojection induced by τ ( k,k )( l,l ) ,E : T k,k E → T l,l E for ≤ l ≤ k .Proof of Theorem 3.3. We shall prove first that Υ k,σ is a well-defined mapping and, simultaneouslypart (b). To this end, we proceed by induction with respect to k for arbitrary σ : E → M .Consider k = 1 . Take Φ = Φ (1 , ∈ TT E and let v := TT σ (Φ) ∈ T M . TT σ has the structureof a double vector bundle with the projections T τ E and τ T E onto T E . Vectors Φ (0 , = τ T E (Φ) and Φ (1 , = T τ E (Φ) project to the same vector v = τ T M ( v ) = T τ ( v ) in T M and to the same point τ E (Φ (0 , ) = τ E (Φ (1 , ) = Φ (0 , in E . It follows that their difference with respect to the vectorbundle structure T σ : T E → T M belongs to T E (cid:12)(cid:12) M ∼ = E × M T M . Hence, from the properties ofthe pairing h· , ·i T σ , D Φ (0 , − Φ (1 , , t ξ E T σ = D ν E (cid:16) Φ (0 , − Φ (1 , (cid:17) , ξ E σ . undle-theoretic methods for higher-order variational calculus 9In other words Υ ,σ is well-defined and satisfies (3.5).Assume now that the assertion holds for every l ≤ k − and every vector bundle σ . Observe that,for every l , e T l,l E ∋ Φ l X i =0 ( − i (cid:18) li (cid:19) Φ ( i,l − i ) ∈ T ( l ) E is a functorial vector bundle morphism over τ ll : T l M −→ T l M ⊂ T ( l ) M being the combination,with constant coefficients, of functorial vector bundle morphisms Φ Φ ( i,l − i ) . In other words, givena vector bundle σ ′ : E ′ → M ′ and a vector bundle morphism α : σ → σ ′ , it holds T ( l ) α l X i =0 ( − i (cid:18) li (cid:19) Φ ( i,l − i ) ! = l X i =0 ( − i (cid:18) li (cid:19) (cid:16) (T l T l α )Φ (cid:17) ( i,l − i ) . This fact, combined with functoriality of h· , ·i σ and the inductive assumption, guarantees that(3.13) α (Υ l,σ (Φ)) = Υ l,σ ′ ((T l T l α )Φ) for every l ≤ k − . In particular, we may take as α in (3.13), the following vector bundle morphisms T τ E : TT E → T E over T τ M , Ψ Ψ (1 , and τ T E : TT E → T E over τ T M , Ψ Ψ (0 , to get (cid:16) Υ k − , TT σ (Φ) (cid:17) (1 , = Υ k − , T σ (cid:16) Φ ( k,k − (cid:17) , (3.14) (cid:16) Υ k − , TT σ (Φ) (cid:17) (0 , = Υ k − , T σ (cid:16) Φ ( k − ,k ) (cid:17) , (3.15)where we treat Φ ( k,k ) as an element of T k − ,k − T , E while Φ ( k,k − and Φ ( k − ,k ) as elements of T k − ,k − T E . Now for the pairing (3.3) of our interest we can write: (cid:28) Φ (0 ,k ) − (cid:18) k (cid:19) Φ (1 ,k − + (cid:18) k (cid:19) Φ (2 ,k − + . . . + ( − k Φ ( k, , t k ξ (cid:29) T ( k ) σ = (cid:28) Φ (0 ,k ) − (cid:18) k − (cid:19) Φ (1 ,k − + (cid:18) k − (cid:19) Φ (2 ,k − + . . . + ( − k − Φ ( k − , , t k ξ (cid:29) T ( k ) σ + − (cid:28) Φ (1 ,k − − (cid:18) k − (cid:19) Φ (2 ,k − + (cid:18) k − (cid:19) Φ (3 ,k − + . . . + ( − k Φ ( k, , t k ξ (cid:29) T ( k ) σ Thanks to our inductive assumption the right-hand side later equals D Υ k − , T σ (cid:16) Φ ( k − ,k ) (cid:17) , t ξ E T σ − D Υ k − , T σ (cid:16) Φ ( k,k − (cid:17) , t ξ E T σ , Now we can use equalities (3.14) and (3.15) to bring the above expression to the form (cid:28)(cid:16) Υ k − , TT σ (Φ) (cid:17) (0 , − (cid:16) Υ k − , TT σ (Φ) (cid:17) (1 , , t ξ (cid:29) T σ (3.5) = (cid:10) Υ ,σ (cid:0) Υ k − , TT σ (Φ) (cid:1) , ξ (cid:11) σ . This sums up to formula (3.6) and assures that Υ k,σ is indeed correctly defined.The proof of (a) is simple. Locally, Φ and t k ξ (in canonical induced graded coordinates on e T k,k E and T k E ∗ ) are given by Φ ∼ (cid:16) x a, ( α ) , y j, ( β,β ′ ) (cid:17) , and t k ξ ∼ (cid:16) x a, ( α ) , ξ ( β ) i (cid:17) ; where α = 0 , , . . . , k and β, β ′ = 0 , , . . . , k .0 We have shown that De Φ , t k ξ E T ( k ) σ , which has a coordinate form of a polynomial in ξ ( β ) i and y j, ( β ′ ,β ′′ ) , actually does not depend on ξ ( β ) i for β ≥ . Hence from (3.3) we get that Υ k,σ (Φ) =( x a , y i ) , where y i is a ξ i = ξ (0) i -coefficient in the coordinate expression for De Φ , t k ξ E T ( k ) σ . By (2.3)we know that y i is the coordinate of total degree k in e Φ . We conclude that the formula (3.4) for Υ k,σ is true.To prove (c) we use the standard decomposition of binomial coefficients (cid:0) k +1 i +1 (cid:1) = (cid:0) ki (cid:1) + (cid:0) ki +1 (cid:1) toobtain (cid:18) k + 11 (cid:19) Φ (0 ,k ) − (cid:18) k + 12 (cid:19) Φ (1 ,k − + . . . + ( − k (cid:18) k + 1 k + 1 (cid:19) Φ ( k, = (cid:20)(cid:18) k (cid:19) Φ (0 ,k ) − (cid:18) k (cid:19) Φ (1 ,k − + . . . + ( − k (cid:18) kk (cid:19) Φ ( k, (cid:21) + (cid:20)(cid:18) k (cid:19) Φ (0 , k − − (cid:18) k (cid:19) Φ (1 , k − + . . . + ( − k − (cid:18) kk (cid:19) Φ ( k − , (cid:21) . In the first expression on the right-hand side of this equality we easily recognize formula (3.3),whereas, in the second, formula (3.2) k − .The proof of (d) is very similar. We decompose Φ (0 ,k ) = (cid:20)(cid:18) k (cid:19) Φ (0 ,k ) − (cid:18) k (cid:19) Φ (1 ,k − + . . . + ( − k (cid:18) kk (cid:19) Φ ( k, (cid:21) + (cid:20)(cid:18) k (cid:19) Φ (1+0 ,k − − (cid:18) k (cid:19) Φ (1+1 ,k − + . . . + ( − k (cid:18) kk (cid:19) Φ (1+ k − , (cid:21) . Now on the right-hand side of this equality we easily recognize formulas (3.3) and (3.2) k − .Finally, to prove (e) and (f), observe that the tangent functor T commutes (up to canonical isomor-phisms) with the projections τ k,ki,k − i and P k . Thus it commutes (up to canonical isomorphisms) with Υ k, · and υ k, · which are linear combinations of the compositions of the mentioned projections. Remark . Note that in formula (3.4) only the coordinates of the highest degree ( k ) matter but thecoordinates of lower degrees may be non-trivial. For example, if k = 2 , we have e Φ = Φ (0 , − (1 , + Φ (2 , ∈ TT E and Φ (0 , ∈ T E , interpreted as an element of TT E by means of ι , :T E → TT E , equals to (cid:0) x a, (0) , x a, (1) , x a, (1) , x a, (2) , y i, (0 , , y i, (0 , , y i, (0 , , y i, (0 , (cid:1) . Similarly for Φ (2 , . Therefore, in coordinates, e Φ looks like e Φ ∼ (cid:16) x a, (0) , x a, (1) , x a, (2) , y i, (0 , , y i, (1 , , y i, (0 , , y i, (1 , (cid:17) , where y i, (0 , = 0 , y i, (1 , = − y i, (0 , = y i, (1 , − y i, (0 , and y i, (1 , = y i, (2 , − y i, (1 , + y i, (0 , . Remark . Observe that, for any k and l , the bundle e T l,l e T k,k E is the pullback of T l,l T k,k E withrespect to the canonical inclusion T l,l ι k,k ◦ ι l,l T k M : T l, k M ⊂ T l,l T k,k M , i.e., e T l,l (cid:16)e T k,k σ (cid:17) = e T l,l (cid:16)(cid:16) ι k,k (cid:17) ∗ T k,k σ (cid:17) = (cid:16) ι l,l T k M (cid:17) ∗ T l,l (cid:16)(cid:16) ι k,k (cid:17) ∗ T k,k σ (cid:17) = (cid:16) ι l,l T k M (cid:17) ∗ (cid:16) T l,l ι k,k (cid:17) ∗ (cid:16) T l,l T k,k σ (cid:17) = (cid:16) T l,l ι k,k ◦ ι l,l T k M (cid:17) ∗ (cid:16) T l,l T k,k σ (cid:17) . Consider now the inclusion T k + l,k + l E ⊂ T l,l T k,k E . Any semi-holonomic vector X ∈ e T k + l,k + l E lying over v k +2 l ∈ T k +2 l M ⊂ T k + l,k + l M is mapped via this inclusion to an element lying over anundle-theoretic methods for higher-order variational calculus 11element in T l T k M ⊂ T l,l T k,k M . Thus we have the canonical inclusion e T k + l,k + l E ⊂ e T l,l e T k,k E .Therefore the inductive formula (3.6) can be described as follows: e T k,k E (cid:31) (cid:127) / / Υ k,σ (cid:15) (cid:15) e T k − ,k − e T , E Υ k − , e T1 , σ (cid:15) (cid:15) E e T , E. Υ ,σ o o Expressing Υ k,σ as a composition of morphisms Υ , · according to (3.6) we can obtain the moregeneral formula Υ k + l,σ = Υ k,σ ◦ Υ l, e T k,k σ (cid:12)(cid:12) e T k + l,k + l E , i.e.,(3.16) e T k + l,k + l E (cid:31) (cid:127) / / Υ k + l,σ (cid:15) (cid:15) e T l,l e T k,k E Υ l, e T k,kσ (cid:15) (cid:15) E e T k,k E. Υ k,σ o o Similarly, formula (3.8) generalizes to D υ k + l,σ (cid:16) Φ ( k + l,k + l ) (cid:17) , t k + l ξ E T k + l σ = D υ k,σ (cid:16) Υ l, e T k,k σ (cid:16) Φ ( k + l,k + l ) (cid:17)(cid:17) , t k ξ E T k σ + D υ l − , T k +1 σ (cid:16) Φ ( l − ,l + k ) (cid:17) , t k + l ξ E T k + l σ . (3.17)The proof is left to the reader.In light of (3.9), we can interpret formulas (3.16) and (3.17) as follows: integration by parts k + l times can be obtained as the composition of k times and l times integration by parts.2 In this section we use Theorem 3.3 to give a geometric construction of the force and momentumin higher-order variational calculus (Theorem 4.2). As a consequence, in Theorem 4.3, we obtainnecessary and sufficient conditions (Euler-Lagrange equations and transversality conditions) for acurve to be an extremal of the higher-order variational Problem 4.1.
Formulation of the problem.
Consider a
Lagrangian function L : T k M → R and the associatedaction(4.1) γ S L ( t k γ ) = Z t t L ( t k γ ( t )) d t, where γ : [ t , t ] → M is a path and t k γ ( t ) ∈ T kγ ( t ) M its k th prolongation. The set of admissiblepaths t k γ will be denoted by ADM ([ t , t ] , T k M ) .By an admissible variation of an admissible path t k γ ( t ) we will understand a curve δ t k γ ( t ) ∈ T t k γ ( t ) T k M . Observe that every admissible variation δ t k γ ( t ) can be obtained from a homotopy χ ( · , · ) : [ t , t ] × ( − ǫ, ǫ ) → M such that χ ( t,
0) = γ ( t ) and δ t k γ ( t ) = t s =0 t k χ ( t, s ) . We see that δ t k γ is generated by δγ ( t ) = t s =0 χ ( t, s ) ∈ T γ ( t ) M , i.e., a vector field along γ ( t ) , in the sense that(4.2) δ t k γ = κ k (cid:16) t k δγ (cid:17) . So, δγ can be called a generator of the variation δ t k γ . Given an admissible variation δ t k γ we maydefine the differential of the action S L in the direction of this variation: D d S L ( t k γ ) , δ t k γ E := Z t t D d L ( t k γ ( t )) , δ t k γ ( t ) E τ T kM d t. Define now a natural projection P : ADM ([ t , t ] , M ) ∋ t k γ (cid:16) t k − γ ( t ) , t k − γ ( t ) (cid:17) ∈ T k − M × T k − M, which sends an admissible path t k γ to the pair consisting of its initial and final ( k − –velocity.Its tangent map T P sends a variation δ t k γ ∈ T t k γ T k M to the pair ( δ t k − γ ( t ) , δ t k − γ ( t )) ∈ TT k − M × TT k − M .Now we are ready to formulate the following variational problem . Problem 4.1 (Variational problem) . For a given Lagrangian function L : T k M → R and a submani-fold S ⊂ T k − M × T k − M which represents the admissible boundary values of ( k − –velocities,find all curves γ : [ t , t ] → M such that their k th prolongation t k γ satisfies D d S L ( t k γ ) , δ t k γ E = 0 for every admissible variation δ t k γ such that T P ( δ t k γ ) ∈ T S. Let us comment the above formulation. In our approach to variational problems we study thebehavior of the differential of the action functional in the directions of admissible variations (differ-ential approach), rather to compare the values of the action on nearby trajectories (integral approach).Hence, solutions of Problem 4.1 are only the critical, not extremal, points of the action functional(4.1). The philosophy of understanding a variational problem as the study of the differential of theaction restricted to the sets of admissible trajectories and admissible variations allows one to treat theunconstrained and constrained cases in a unified way (see, e.g., [7, 9, 11]).undle-theoretic methods for higher-order variational calculus 13
Higher-order variational calculus.
Let S L be the action functional (4.1) and δ t k γ the variation(4.2). Define the force F L,γ ( t ) ∈ T ∗ M and the momentum M L,γ ( t ) ∈ T k − T ∗ M along γ ( t ) by F L,γ ( t ) = Υ k,τ ∗ (cid:16) t k Λ L ( t k γ ( t )) (cid:17) , (4.3) M L,γ ( t ) = υ k − ,τ ∗ (cid:16) t k − λ L ( t k γ ( t )) (cid:17) , (4.4)where Λ L := ε k ◦ d L : T k M → T k T ∗ M and λ L := τ kk − , T ∗ M ◦ Λ L : T k M → T k − T ∗ M . Theorem 4.2.
The differential of the action S L in the direction of the variation δ t k γ equals (4.5) D d S L ( t k γ ) , δ t k γ E = Z t t h F L,γ ( t ) , δγ ( t ) i τ d t + D M L,γ ( t ) , t k − δγ ( t ) E T k − τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t t . Theorem 4.3 below is an immediate consequence of formula (4.5).
Theorem 4.3.
A curve γ is a solution of Problem 4.1 if and only if it satisfies the following Euler-Lagrange (EL) equation F L,γ ( t ) = 0 (4.6) and the transversality conditions (cid:0) − ε − k − (M L,γ ( t )) , ε − k − (M L,γ ( t )) (cid:1) ∈ T ∗ T k − M × T ∗ T k − M annihilates T S. (4.7) Proof of Theorems 4.2 and 4.3.
Let us calculate the variation of the action S L in the direction δ t k γ : D d L ( t k γ ( t )) , δ t k γ ( t ) E τ T kM (4.2) = D d L ( t k γ ( t )) , κ k (cid:16) t k δγ ( t ) (cid:17)E τ T kM (2.6) = D ε k ◦ d L ( t k γ ( t )) , t k δγ ( t ) E T k τ = D Λ L ( t k γ ( t )) , t k δγ ( t ) E T k τ . Now we can use formula (3.9) with Φ ( k,k ) = t k Λ L ( t k γ ( t )) , Φ (0 ,k ) = Λ L ( t k γ ( t )) and Φ ( k,k − = t k λ L ( t k γ ( t )) , since the element Φ ( k,k ) := t k Λ L ( t k γ ( t )) ∈ T k T k T ∗ M is a semi-holonomic vectoras it projects to t k t k γ = t k γ ∈ T k M . We get D Λ L ( t k γ ( t )) , t k δγ ( t ) E T k τ (3.9) = D Υ k,τ ∗ (cid:16) t k Λ L ( t k γ ( t )) (cid:17) , δγ ( t ) E τ + D T υ k − ,τ ∗ (cid:16) t k λ L ( t k γ ( t )) (cid:17) , t k δγ ( t ) E T ( k ) τ . In the first summand we recognize the force F L,γ ( t ) defined by (4.3). To the second we can apply theequality T υ k − ,τ ∗ (cid:16) t k λ L ( t k γ ( t )) (cid:17) = T υ k − ,τ ∗ (cid:16) t t k − λ L ( t k γ ( t )) (cid:17) = t h υ k − ,τ ∗ (cid:16) t k − λ L ( t k γ ( t )) (cid:17)i . We conclude that D Λ L ( t k γ ( t )) , t k δγ ( t ) E T k τ = h F L,γ ( t ) , δγ ( t ) i τ + D t h υ k − ,τ ∗ (cid:16) t k − λ L ( t k γ ( t )) (cid:17)i , t t k − δγ ( t ) E TT k − τ = h F L,γ ( t ) , δγ ( t ) i τ + dd t D υ k − ,τ ∗ (cid:16) t k − λ L ( t k γ ( t )) (cid:17) , t k − δγ ( t ) E T k − τ = h F L,γ ( t ) , δγ ( t ) i τ + dd t D M L,γ ( t ) , t k − δγ ( t ) E T k − τ , M L,γ ( t ) is defined by (4.4). Thus the variation (4.5) reads(4.8) D d L ( t k γ ( t )) , δ t k γ ( t ) E τ T kM = h F L,γ ( t ) , δγ ( t ) i τ + dd t D M L,γ ( t ) , t k − δγ ( t ) E T k − τ . Integrating the above expression over [ t , t ] we get (4.5), concluding the proof of Theorem 4.2.Theorem 4.3 follows easily, as δ t k − γ ( t ) = κ k − ( t k − δγ ( t )) and ε k − is dual to κ k − , in light ofequation (2.6). Remark . The process of constructing the EL equations (4.6) starting from the Lagrangian function L can be followed on the diagram(4.9) ker Υ k,τ ∗ (cid:127) _ (cid:15) (cid:15) T ∗ T k M ǫ k / / ( τ T kM ) ∗ (cid:15) (cid:15) T k T ∗ M T k τ ∗ (cid:15) (cid:15) t k Λ L ( t k γ ) / / ❴❴❴❴❴❴ ∃ ? ❧❧❧❧❧❧❧ e T k,k T ∗ M Υ k,τ ∗ (cid:15) (cid:15) I t k γ / / T k M d L H H Λ L T k M T ∗ M. Similarly, the geometric construction of the momenta (4.4) corresponds to the diagram(4.10) T ∗ T k M ǫ k / / ( τ T kM ) ∗ (cid:15) (cid:15) T k T ∗ M τ kk − , T ∗ M (cid:15) (cid:15) I t k γ / / T k M λ L / / d L H H Λ L T k − T ∗ M t k − λ L ( t k γ ) / / ❴❴❴❴❴❴ e T k − ,k − T ∗ M υ k − ,τ ∗ / / T k − T ∗ M. Above we used dotted arrows to denote the objects associated with the Lagrangian function, whereasdashed arrows are maps defined only along the images of γ ( t ) .Note also that the map Υ k,τ ∗ allows us to define higher-order EL equations with external forces .Namely, given an external force , i.e., a map F : [ t , t ] → T ∗ M , we can consider equation Υ k,τ ∗ (cid:16) t k Λ L ( t k γ ( t )) (cid:17) = F ( t ) . When F ( t ) = 0 , the equation above reduces to the EL equation (4.6). Local form of the forces and momenta.
We shall now derive the local form of the force (4.3) andmomentum (4.4).Let us first calculate the force. Consider a trajectory γ ( t ) ∼ ( x a ( t )) . The differential d L ( t k γ ( t )) ∈ T ∗ T k M is given by p a, ( α ) = ∂L∂x a, ( α ) ( t k γ ( t )) , hence using (2.7) we calculate the coordinates of Λ L ( t k γ ( t )) ∈ T k T ∗ M , namely, p ( α ) a (Λ L ( t k γ ( t ))) = (cid:18) kα (cid:19) − ∂L∂x a, ( k − α ) ( t k γ ( t )) , x a, ( α ) (Λ L ( t k γ ( t ))) = d α x a ( t )d t α . Therefore, p ( α,β ) a ( t k Λ L ( t k γ ( t ))) = (cid:0) kβ (cid:1) − α d t α ∂L∂x a, ( k − β ) , and using (3.4) we find that our formula (4.3)of the force takes the following well-known form(4.11) F L,γ ( t ) a = k X α =0 ( − α d α d t α (cid:18) ∂L∂x a, ( α ) ( t k γ ( t )) (cid:19) . undle-theoretic methods for higher-order variational calculus 15Concerning momentum, a direct (i.e., by means of formulas (3.2) and (4.10)) derivation of itslocal form is a quite complicated computational task, so that we will obtain it by using formula (4.8),instead.To this end, consider a generator of the variation δγ ( t ) ∼ ( x a ( t ) , δx a ( t )) and its k th tangentlift t k γ ( t ) ∼ (cid:0) x a, ( α ) ( t ) , δx a, ( α ) ( t ) (cid:1) α =0 ,...,k , where x a, (0) ( t ) = x a ( t ) , δx a, (0) ( t ) = δx a ( t ) and x a, ( α +1) ( t ) = dd t x a, ( α ) ( t ) , δx a, ( α +1) ( t ) = dd t δx a, ( α ) ( t ) , for α = 0 , . . . , k − . Let the momen-tum M L,γ ( t ) along the trajectory γ ( t ) be locally given by (cid:16) x a, ( α ) ( t ) , (cid:0) k − α (cid:1) − p a, ( k − − α ) ( t ) (cid:17) . Thus,by (2.7) k − , ǫ − k − (M L,γ ( t )) ∼ (cid:0) x a, ( α ) ( t ) , p a, ( α ) ( t ) (cid:1) ∈ T ∗ T k M . Locally D M L,γ ( t ) , t k − δγ ( t ) E T k − τ = D ǫ − k − ( M L,γ ( t )) , δ t k − γ ( t ) E τ T k − M = X a k − X α =0 p a, ( α ) ( t ) δx a, ( α ) ( t ) . Thus dd t D M L,γ ( t ) , t k − δγ ( t ) E T k − τ = X a k X α =0 (cid:0) p a, ( α − ( t ) + ˙ p a, ( α ) ( t ) (cid:1) δx a, ( α ) ( t ) , where in the last formula we take p a, ( k ) ( t ) = p a, ( − ( t ) = 0 . The left-hand side of (4.8) equals P a P kα =0 ∂L∂x a, ( α ) ( t k γ ( t )) δx a, ( α ) , so from (4.11) we get X a k X α =0 δx a, ( α ) ( t ) (cid:0) p a, ( α − ( t ) + ˙ p a, ( α ) ( t ) (cid:1) = X a k X α =0 ∂L∂x a, ( α ) ( t k γ ( t )) δx a, ( α ) ( t ) − X a F L,γ ( t ) a δx a, (0) ( t ) . Since at a fixed time t the variation δx a, ( α ) ( t ) can be arbitrary, above equation splits into the followingset of linear equations p a, ( k − ( t ) = ∂L∂x a, ( k ) ( t k γ ( t )) , ˙ p a, ( k − ( t ) + p a, ( k − ( t ) = ∂L∂x a, ( k − ( t k γ ( t )) ,. . . ˙ p a, (1) ( t ) + p a, (0) ( t ) = ∂L∂x a, (1) ( t k γ ( t )) , ˙ p a, (0) ( t ) = ∂L∂x a, (0) ( t k γ ( t )) − F L,γ ( t ) a , whose unique solution is(4.12) p a, ( α ) ( t ) = k − − α X β =0 ( − β d β d t β (cid:18) ∂L∂x a, ( α + β +1) ( t k γ ( t )) (cid:19) , for α = 0 , . . . , k − ;i.e., the well-known formula for momenta.6 Tulczyjew’s approach to higher-order geometric mechanics.
The problem of geometric for-mulation of higher-order variational calculus on a manifold M has a few solutions. The first ap-proach is due to Tulczyjew [20, 21], who gave a geometric construction of a 0-order derivation E : Sec(T ∗ T ∞ M ) → Sec(T ∗ T ∞ M ) such that E (d L ) = 0 are the higher-order Euler-Lagrangeequations for any Lagrangian function L : T k M → R . Tulczyjew expressed his construction usingthe language of derivations and infinite jets. The latter allowed him to cover all orders k by a uniqueoperator. Later Tulczyjew’s theory was interestingly extended by Crampin, Sarlet and Cantrijn [4].Another approach was inspired by Tulczyjew’s papers [22, 23] on the first order mechanics. Theidea was to generate the EL equations from a Lagrangian submanifold. Two similar solutions wheregiven by Crampin [3] and de Leon and Lacomba [14]. They constructed the equations from a La-grangian submanifold in TT k − T ∗ M or TT ∗ T k − M generated by the Lagrangian L .All these solutions have, however, some drawbacks. First of all, they describe only a part of theLagrangian formalism, namely the EL equations, whereas the full structure of variational calculusshould contain also momenta (boundary terms). Secondly, the correctness of these constructionsis checked in coordinates. Note, however, that it is not the coordinate expression that defines theEL equations, but the opposite: we deduce the right local expression from the proper variationalprinciple. Therefore a fully satisfactory geometric construction should somehow explain the stepsperformed while deriving the known form of the equations (as it is in our approach described in theprevious Section 4), not give a black-box answer.In later years Tulczyjew [24] extended his work to give a full description of higher-order La-grangian formalism (i.e., including momenta) in the language of derivations. Another approach wascommunicated to us by Grabowska [6], who derived the k th order formalism as the first order formal-ism on T k − M . Her ideas are, to some extend, similar to these of [3, 14], where the canonical inclu-sion T k M ⊂ TT k − M was also used. Another quite general approach to the topic was presented byA.M. Vinogradov and his collaborators in the framework of secondary calculus (see Section 3 of [25]and the references therein).Below we relate Tulczyjew’s approach to our results from the previous Section 4. Comparison with Tulczyjew’s formulas.
In [21, 24] Tulczyjew introduced graded derivations ofdegree 0 E := k X n =0 ( − n n ! (cid:16) τ kk + n (cid:17) ∗ (d T ) n ι F n : Ω • (cid:16) T k M (cid:17) −→ Ω • (cid:16) T k M (cid:17) and P := k X n =1 ( − n n ! (cid:16) τ k − k + n − (cid:17) ∗ (d T ) n − ι F n : Ω • (cid:16) T k M (cid:17) −→ Ω • (cid:16) T k − M (cid:17) defined by means of two basic derivations, namely:(i) the total derivative d T : Ω • (T s M ) → Ω • (cid:0) T s +1 M (cid:1) being a d ∗ –derivation characterized by d T f ( α ) = f ( α +1) for ≤ α ≤ s , where f ( α ) denotes the ( α ) –lift of a smooth function f on M as defined in (2.1), and(ii) the i ∗ –derivations ι F n : Ω • (T k M ) → Ω • (T k M ) , associated with the canonical (nilpotent)endomorphism F n : TT k M → TT k M of the tangent bundle, F n (cid:16) t s =0 t kt =0 γ ( s, t ) (cid:17) := t s =0 (cid:16) t kt =0 γ ( st n , t ) (cid:17) . undle-theoretic methods for higher-order variational calculus 17Recall that a derivation a of the algebra of differential forms Ω • ( M ) is called a d ∗ -derivation (resp. i ∗ -derivation) if a commutes with de Rham differential (resp. if a vanishes on Ω ( M ) = C ∞ ( M ) )([21], see also [13], chapter ). As the algebra Ω • ( M ) is generated by Ω ( M ) and Ω ( M ) , hence a d ∗ -derivation (resp. i ∗ -derivation) is fully determined by its values on Ω ( M ) (resp. Ω ( M ) ). Thevalues of d T on Ω (T k M ) are given in the above characterization of d T , while for ι F n one defines h ι F n µ, v i := h µ, F n v i for a -form µ and a tangent vector v on T k M . Note that F n = F n and F isthe canonical higher almost tangent structure on T k M [15].It turns out that the form E (d L ) ∈ Ω (T k M ) is vertical with respect to the projection τ k :T k M → M and that the form P (d L ) ∈ Ω (T k − M ) is vertical with respect to τ k − k − : T k − M → T k − M . Therefore taking the appropriate vertical parts of these forms we can define operators EL : T k M −→ T ∗ M and PL : T k − M −→ T ∗ T k − M. Formulas EL ( t k γ ) = Υ k,τ ∗ (cid:0) t k Λ L ( t k γ ) (cid:1) and PL ( t k − γ ) = ε k − (cid:0) υ k − ,τ ∗ (cid:0) t k − λ L ( t k γ ) (cid:1)(cid:1) relateTulczyjew’s constructions to ours. Applications to mechanics on algebroids.
In [10] we showed that with every almost-Lie algebroidstructure on the bundle σ : E → M , one can canonically associate an infinite tower of graded bundles(5.1) . . . −→ E k −→ E k − −→ . . . −→ E = E equipped with a family of graded-bundle relations κ k : T k E → ⊲ T E k . The relation κ k is of special kind – it is dual of a vector bundle morphism ε k : T ∗ E k → T k E ∗ . Anatural example of such a structure is provided by the higher tangent bundles E k = T k M togetherwith the canonical flips κ k : T k T M → TT k M . Another example is E k := T ke G , the higher tangentspace at identity e ∈ G to a Lie group G . Both examples should be considered as the extreme cases ofwhat should we call a higher algebroid . Except for k = 1 there is no Lie bracket on sections of E k . Itis the relation κ k which is responsible for the algebraic structure on E k . More general examples canbe obtained by reducing higher tangent bundles of Lie groupoids.Given a smooth function L : E k → R one can naturally define a variational problem on E k .Such problems cover, on one hand, the standard variational problems like Problem 4.1 (in which case E k = T k M ), and, on the other hand, the reduction of invariant higher-order variational problems ona Lie groupoid. We showed in [10] that for such problems an analog of Theorem 4.2 holds, as well.Thus, we can characterize the variation of an action by means of the force Υ k,σ ∗ (cid:0) t k ε k (cid:0) d L ( t k γ ( t )) (cid:1)(cid:1) and momentum υ k − ,σ ∗ (cid:0) t k − τ kk − (cid:0) ε k (cid:0) d L ( t k γ ( t )) (cid:1)(cid:1)(cid:1) , where now ε k : T ∗ E k → T k E ∗ is the dualof the relation κ k , τ kk − : E k → E k − is the tower projection (5.1), σ ∗ : E ∗ → M is the dual of σ ,while Υ k,σ ∗ and υ k,σ ∗ are the same maps introduced in Section 3. Example: Riemannian cubic polynomials.
Let us consider one of the simplest, but interesting,second-order variational problem: given an integer n ≥ and points a = x < x < . . . < x n = b on the real line R , and values y , y , . . . , y n , v a , v b ∈ R , find an f ∈ C ([ a, b ]) such that f ( x i ) = y i for ≤ i ≤ n and f ′ ( a ) = v a , f ′ ( b ) = v b which minimizes the integral R ba f ′′ ( x ) dx . This problemhas a unique solution, called a complete cubic spline [1], which is a piece-wise cubic polynomial P determined uniquely by the following properties: it is a polynomial of degree ≤ on each interval [ x i , x i +1 ] , ≤ i ≤ n − , P ′ ( a ) = v a , P ′ ( b ) = v b , and it has continuous second derivatives ateach “slope” x , . . . , x n − . Note that the EL equations (4.11) read as f (4) = 0 , i.e., f is locally apolynomial of degree ≤ .8 Above example generalizes to a variational problem on any Riemannian manifold ( M, g ) . Indeed,let ∇ denote the Levi-Civita connection for the metric g and, for a smooth curve γ : R → M denoteby D t the covariant derivative ∇ ˙ γ ( t ) along γ . Following [18] we define(5.2) L ( t t =0 γ ( t )) := g γ (0) ( D t | t =0 ˙ γ ( t ) , D t | t =0 ˙ γ ( t )) . Locally, for γ ( t ) ∼ ( x a ( t )) , D t | t =0 ˙ γ ( t ) = (¨ x c + Γ cab ( x ) ˙ x a ˙ x b ) ∂ x c , where Γ cab ( x ) are the Christoffel symbols of the metric g and ˙ x a (resp. ¨ x a ) are (resp. second-order)derivatives of x a ( t ) at t = 0 . Therefore (5.2) is indeed a function on T M .We shall now compute the second-order EL equations associated with the Lagrangian L given by(5.2). To this end, recall two fundamental properties of the Levi-Civita connection: ∇ X Y − ∇ Y X = [ X, Y ] , (5.3) Xg ( Y, Z ) = g ( ∇ X Y, Z ) + g ( Y, ∇ X Z ) , (5.4)which hold for any vector fields X, Y, Z on the manifold M ..Note that although the Lie bracket [ X, Y ] is defined for vector fields, to calculate its value at a point p ∈ M it is enough to know vectors X ( Y )( p ) := T Y ◦ X ( p ) ∈ T Y ( p ) T M and Y ( X )( p ) ∈ T X ( p ) T M (see, e.g., [13]). According, we introduce the following notion: for a vector X ∈ T p M and a vector A ∈ T X T M lying over Y := T τ M ( A ) by an A –extension of X around p we will understand any(local) vector field e X on M such that e X ( p ) = X and Y ( e X )( p ) = A . This means that A is tangent tothe graph of e X at X .Consider now a curve γ ( t ) ∈ M and any generator δγ ( t ) ∈ T γ ( t ) M of an admissible variation δ t γ ( t ) = κ ( t δγ ( t )) . Let e ˙ γ be any δ ( t γ )( t ) –, i.e., κ ( t δγ ( t )) –extension, in the aforementionedsense, of t γ ( t ) = ˙ γ ( t ) and let f δγ be any t δγ ( t ) –extension of δγ ( t ) along γ ( t ) (in particular f δγ = δγ along γ ( t ) ). It follows immediately from [13] (or [10], Proposition 2.2) that(5.5) [ f δγ, e ˙ γ ] = 0 along γ ( t ) . We shall now compute the differential of the action S L in the direction of δ t γ ( t ) . Note that g ( ∇ e ˙ γ e ˙ γ, ∇ e ˙ γ e ˙ γ ) is a function on M coinciding with L ( t γ ) = g ( ∇ ˙ γ ˙ γ, ∇ ˙ γ ˙ γ ) along γ ( t ) , hence (cid:10) d S L ( t γ ) , δ t γ (cid:11) = Z t t δγg ( ∇ e ˙ γ e ˙ γ, ∇ e ˙ γ e ˙ γ ) d t (5.4) = 2 Z t t g ( ∇ δγ ∇ e ˙ γ e ˙ γ, ∇ ˙ γ ˙ γ ) d t =2 Z t t g ( R ( δγ, ˙ γ ) ˙ γ, ∇ ˙ γ ˙ γ ) + g (cid:16) ∇ ˙ γ ∇ f δγ e ˙ γ + ∇ [ f δγ, e ˙ γ ] e ˙ γ, ∇ ˙ γ ˙ γ (cid:17) d t, where R ( X, Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X,Y ] Z is the Riemann tensor of g . By (5.5) and bythe standard symmetry property g ( R ( X, Y ) Z, T ) = g ( R ( T, Z ) Y, X ) , the later equals Z t t g ( R ( ∇ ˙ γ ˙ γ, ˙ γ ) ˙ γ, δγ ) + g (cid:16) ∇ ˙ γ ∇ f δγ e ˙ γ, ∇ ˙ γ ˙ γ (cid:17) d t. Let us transform the last integrand as follows g (cid:16) ∇ ˙ γ ∇ f δγ e ˙ γ, ∇ ˙ γ ˙ γ (cid:17) (5.3), (5.5) = g (cid:16) ∇ ˙ γ ∇ ˙ γ f δγ, ∇ ˙ γ ˙ γ (cid:17) (5.4) =˙ γg ( ∇ ˙ γ δγ, ∇ ˙ γ ˙ γ ) − g ( ∇ ˙ γ δγ, ∇ ˙ γ ∇ ˙ γ ˙ γ ) (5.4) = ˙ γg ( ∇ ˙ γ δγ, ∇ ˙ γ ˙ γ ) − ˙ γg ( δγ, ∇ ˙ γ ∇ ˙ γ ˙ γ ) + g ( δγ, ∇ ˙ γ ∇ ˙ γ ∇ ˙ γ ˙ γ ) . undle-theoretic methods for higher-order variational calculus 19This gives us (cid:10) d S L ( t γ ) , δ t γ (cid:11) =2 Z t t g ( R ( ∇ ˙ γ ˙ γ, ˙ γ ) ˙ γ + ∇ ˙ γ ∇ ˙ γ ∇ ˙ γ ˙ γ, δγ ) d t +2 [ g ( ∇ ˙ γ δγ, ∇ ˙ γ ˙ γ ) − g ( δγ, ∇ ˙ γ ∇ ˙ γ ˙ γ )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t t . We see that the boundary term depends only on t δγ , hence comparing the last expression with (4.5)we get that Υ ,τ ∗ (cid:0) t Λ L ( t γ ) (cid:1) = g ( R ( ∇ ˙ γ ˙ γ, ˙ γ ) ˙ γ + ∇ ˙ γ ∇ ˙ γ ∇ ˙ γ ˙ γ, · ) . Consequently, the EL equationsfor L read(5.6) D t ˙ γ ( t ) + R ( D t ˙ γ ( t ) , ˙ γ ( t )) ˙ γ ( t ) = 0 , in agreement with [18]. Solutions of (5.6) are called Riemannian cubic polynomials .0 A Appendix
Proof of Lemma 3.4.
Let us explain that the functoriality of a vector bundle morphism e T k,k E → E means that actually we have a family F = { F E } of vector bundle morphisms as in (3.12), parame-terized by vector bundles σ : E → M , such that for any morphism f : E → E between vectorbundles σ i : E i → M i , i = 1 , , we have(A.1) F E ◦ e T k,k f = f ◦ F E . We shall derive the local coordinate form of F E . Observe first that the base map of F E , denoted by F E : T k M → M , has to be functorial as well. In other words F E is a natural transformation betweenWeil functors T k and Id . It follows from [12] that such a transformation corresponds to a uniquehomomorphism between the corresponding Weil algebras, namely h : D k = R [ ν ] / (cid:10) ν k +1 (cid:11) → R given by and ν . We conclude that F E = τ k is the canonical bundle projection.Every vector bundle can be locally described as E = M × V , where M is the base and V is the model fiber. In this case e T k,k E = T k M × T k T k V and hence F E must be of the form F E ( v k , X ) = ( τ k ( v k ) , L v k ( X )) , where L v k : T k T k V → V is a linear map, v k ∈ T k M and X ∈ T k T k V . Consider now a morphisms of the form f = f × id V : M × V → M × V , where f : M → M is a smooth function. It follows from (A.1) that L v k does not depend on v k andhence, locally, F E is of the form F E ( v k , X ) = ( τ k ( v k ) , L ( X )) , where L : T k T k V → V is a linear map.Now we shall find the coordinate form of L . Since every vector space V is a vector bundle overa one-point base, it follows that F induces a functorial morphism F V : e T k,k V = T k T k V → V . But T k T k V ≃ D k,k ⊗ V , canonically, where D k,k denotes the Weil algebra D k,k = R [ ν, ν ′ ] / D ν k +1 , ν ′ k +1 E .Any functorial linear map D k,k ⊗ V → V is of the form f ⊗ id V for some fixed linear map f : D k,k → R . If we denote c αβ := f ( ν α ν ′ β ) , then we conclude that general local form of F E : e T k,k E → E is F E (cid:16) x a, ( r ) , y i, ( α,β ) (cid:17) = x a = x a, (0) , y i = X ≤ α,β ≤ k c αβ y i, ( α,β ) , where ( x a, ( r ) , y i, ( α,β ) ) are the adapted coordinates on e T k,k E (as defined in Preliminaries) inducedfrom the standard coordinates ( x a , y i ) on E and we have underlined the coordinates in the co-domain.To get more information on the coefficients c αβ it will be enough to consider the case E = M × R with V = R . Consider the vector bundle morphism by e φ : M × R → M × R given by e φ ( x, y ) =( x, φ ( x ) · y ) , where φ ∈ C ∞ ( M ) . We are looking for possible F E such that the following diagram(A.2) e T k,k E e T k,k e φ (cid:15) (cid:15) F E / / E e φ (cid:15) (cid:15) e T k,k E F E / / E commutes. In our case, e T k,k E = T k R × T k T k R and the commutativity of (A.2) reads as(A.3) X ≤ α,β ≤ k c αβ y ( α,β ) = φ ( x ) · X ≤ α,β ≤ k c αβ y ( α,β ) , undle-theoretic methods for higher-order variational calculus 21where (cid:0) x ( r ) , y ( α,β ) (cid:1) (cid:0) x ( r ) , y ( α,β ) (cid:1) is the coordinate expression of the morphism e T k,k e φ . To obtainthe coordinate expression of T k T k e φ : T k T k E → T k T k E , for E = M × R and M = R , whichsends the class [ γ ] of a map γ : R × R → E to the class [ e φ ◦ γ ] in T k T k E , we write x ( e φ ( γ ( s, t ))) = x ( γ ( s, t )) = X ≤ α,β ≤ k x ( α,β ) ([ γ ]) s α t β α ! β ! + o ( s k , t k ) , (A.4) y ( e φ ( γ ( s, t ))) = φ ( x ( γ ( s, t ))) y ( γ ( s, t )) = φ ( x ) + r X r =1 φ ( r ) ( x ) r ! h r + o ( h k ) ! X ≤ α,β ≤ k y ( α,m ) ([ γ ]) s α t β α ! β ! + o ( s k , t k ) , (A.5)where x = x ( γ (0 , x (0 , ([ γ ]) and h = x ( γ ( s, t )) − x ( γ (0 , as in (A.4). Clearly, y ( α,β ) is thecoefficient of s α t β /α ! β ! in y ( e φ ( γ ( s, t ))) . For example, in case k = 2 we find that(A.6) y (2 , = y (2 , φ ( x ) + 2 y (1 , x (1 , φ ′ ( x ) + y (2 , x (0 , φ ′ ( x ) + . . . For any l, m ≥ , T l T m E is a three-fold graded bundle with bases E , T l E and T m E [8]. Its algebraof multi-homogeneous functions, which is a subalgebra of all smooth functions on T l T m E , is Z -graded. For example, y ( α,β ) is of degree with respect to the vector bundle structure over T l T m M and of degrees α and β with respect to the bundle structures over T m E and T l E , respectively. Ofcourse, T k T k e φ preserves this grading. Restricting T k T k e φ to e T k,k E means just replacing x ( α,β ) with x ( α + β ) . Thus we get y ( α,β ) = y ( α,β ) φ ( x ) + αy ( α − ,β ) x (1) φ ′ ( x ) + βy ( α,β − x (1) φ ′ ( x ) + . . . where, in case α = 0 or β = 0 , it is enough to put instead of y ( − ,β ) or y ( α, − . By comparing the φ ′ ( x ) coefficients in the above expression and separating the terms with respect to the gradation wefind that the necessary condition for (A.3) is X ≤ α,β ≤ k,α + β = s c αβ ( αy ( α − ,β ) + βy ( α,β − ) = 0 , for any ≤ s ≤ k , with the convention that y ( α,β ) = 0 whenever α or β is negative or greaterthan k . This gives a recurrence relation between the coefficients c αβ when the sum α + β = s fixed.Namely, ( α + 1) c ( α +1)( β − + βc αβ = 0 if ≤ α ≤ k − and ≤ β ≤ k . Moreover, c αk = 0 for ≤ α ≤ k . It follows that for ≤ s ≤ k the vector ( c s , c s − , . . . , c s ) is proportional to ( (cid:0) s (cid:1) , − (cid:0) s (cid:1) , (cid:0) s (cid:1) , . . . , ± (cid:0) ss (cid:1) ) and c αβ = 0 whenever α + β > k . Hence, X α,β c αβ y ( α,β ) = k X s =0 a s X α + β = s ( − α (cid:18) sα (cid:19) y ( α,β ) . for some coefficients a , . . . , a k ∈ R . Therefore, F E must have the desired form being a linearcombination of morphisms Υ s,σ given locally by (3.4).A slight modification of the proof above shows that Υ k,σ has no canonical extension to T k T k E . Corollary
A.1 . Any functorial vector bundle morphism(A.7) T k T k E (cid:15) (cid:15) F E / / E (cid:15) (cid:15) T k T k M F E / / M τ ( k,k ) E : T k T k E → E covering τ ( k,k ) : T k T k M → M . In particular, Υ k,σ has, ingeneral, no canonical extension to a functorial vector bundle morphism on T k T k E . Proof.
Just go over the same reasoning of the proof of Lemma 3.4. By considering a trivial vectorbundle E = M × V and endomorphisms of the form f = f × id V : E → E , where f ∈ C ∞ ( M ) ,we find that a general local form of F E : T k T k E → E has to be F E ( x a, ( α ′ ,β ′ ) , y i, ( α,β ) ) = ( x a = x a, (0 , , y i = X ≤ α,β ≤ k c αβ y i, ( α,β ) ) . In the same way we find that the coefficients c αβ have to satisfy (A.3), where ( x ( α ′ ,β ′ ) , y ( α,β ) ) ( x ( α ′ ,β ′ ) , y ( α,β ) ) is the coordinate expression of the morphism T k T k e φ , where e φ : R × R → R × R , ( x, y ) ( x, φ ( x ) · y ) for some function φ ∈ C ∞ ( R ) . In the transformation expression for y ( α,β ) weseparate terms with respect to the grading and collect the terms which contain φ ′ ( x ) . Then we findeasily that (A.3) implies that c α,β = 0 unless α = β = 0 . Therefore F E = c · τ ( k,k ) E , as claimed.undle-theoretic methods for higher-order variational calculus 23 Acknowledgments
This research was supported by Polish National Science Center grant under the contract numberDEC-2012/06/A/ST1/00256.The authors are grateful for professors Paweł Urba´nski and Janusz Grabowski for reference sug-gestions. We wish to thank especially the second of them for reading the manuscript and givinghelpful remarks. We are also grateful for the reviewers for many helpful suggestions.
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