Lepage forms, closed two-forms and second order ordinary differential equations

Transcription

1 Lepage forms, closed two-forms and second order ordinary differential equations Olga Krupková Department of Algebra and Geometry, Palacký University Tomkova 40, Olomouc, Czech Republic and Department of Mathematics, La Trobe University, Bundoora, Victoria 3086, Australia and G.E. Prince Department of Mathematics, La Trobe University, Bundoora, Victoria 3086, Australia Abstract: Lepage 2-forms appear in the variational sequence as representatives of the classes of 2-forms. In the theory of ordinary differential equations on jet bundles they are used to construct exterior differential systems associated with the equations and to study solutions, and help to solve the inverse problem of the calculus of variations: since variational equations are characterized by Lepage 2-forms that are closed. In this paper, a general setting for Lepage forms in the variational sequence is presented, and Lepage 2-forms in the theory of second-order differential equations in general and of variational equations in particular, are investigated in detail. Key words: Second order ordinary differential equations, Euler Lagrange equations, the variational sequence, dynamical forms, Lepage forms, Helmholtz form, the inverse problem of the calculus of variations. MSC 2000: 34A26, 58E30. PACS 2006: Xx, k, Vh. 1

2 1 Introduction Ordinary differential equations in jet bundles can be studied with help of Krupka variational sequence ([13, 14]), and of differential forms that appear as representatives of classes in the sequence. In this paper we are interested in second-order ordinary differential equations, and we investigate the role and properties of Lepage forms. We consider differential equations represented by dynamical forms on jet-prolongations of fibred manifolds ([18, 23, 25]). In Section 4 we study different representations of classes in the variational sequence; in particular, we present a generalisation to the concepts of source forms and Lepage forms, and we study Lepage equivalents of dynamical forms. In Section 5 Lepage forms are used to construct exterior differential systems associated with second-order differential equations, and to provide different classifications of the equations: (1) with respect to the properties of Lepage equivalents of dynamical forms that measure variationality (existence of Lagrangians), and (2) with respect to the properties of the solutions. In the classification, namely regular equations on one hand, and semivariational and variational equations on the other hand, are important distinguished classes of equations. The last two sections are devoted to regular equations. Following Krupková ([19, 23]) we present their geometric descriptions by means of semisprays and semispray connections, and study in detail Lepage forms corresponding to regular dynamical forms and semisprays. We focus on the so-called Kähler lift Ω of a regular symmetric (0,2)-tensor field g, discovered by Crampin, Prince and Thompson ([4]) and clarify the place of this 2-form in the variational sequence and in the general theory of Lepage forms. We also recall the role of this form in the solution of the inverse variational problem for semisprays. 2 Fibred calculus Throughout the paper, manifolds and mappings are smooth. We consider a fibred manifold π : Y X where dim X = 1 and dim Y = m + 1 (with m > 0), and its jet prolongations, π r : J r Y X; in what follows, most frequently we assume r = 1, 2. Natural fibred projections of J r Y onto J k Y, 0 k r 1, are denoted by π r,k (in this context, J 0 Y = Y ). Sections considered throughout are smooth 2

3 local sections, defined on open subsets of X. A section δ of π r is called holonomic if δ = J r γ for a section γ of π. We denote fibred coordinates on Y by (t, x i ), 1 i m, and associated coordinates on J r Y by (t, x i k ), 1 i m, 0 k r; usually we write x i 0 = x i, x i 1 = ẋ i, x i 2 = ẍ i, x i 3 =... x i. In coordinate expressions, summation over repeated indices is always assumed. In jet bundles we have distinguished vector fields, differential forms and operators, adapted to the fibred and prolongation structures. In this section we recall the main concepts according to Krupka [11, 12] (see also [16, 23, 35]. We denote by X(J r Y ) the module of vector fields on J r Y. A vector field ξ X(Y ) is called π-projectable if there exists a vector field ξ 0 X(X) such that T π.ξ = ξ 0 π, and π-vertical if ξ 0 = 0. Similarly a π r -projectable and π r -vertical (respectively, π r,k -projectable and π r,k -vertical) vector field on J r Y is defined. Let Λ q (J r Y ), r 0, q 0, denote the module of smooth q- forms on J r Y over the ring of functions (for q = 0 we have smooth functions on J r Y ). η Λ q (J r Y ) is called π r,k -projectable if there is η 0 Λ q (J k Y ) such that η = πr,k η 0. A form η Λ q (J r Y ) is called π r -horizontal if i ξ η = 0 for every π r -vertical vector field ξ on J r Y. A form η Λ q (J r Y ) is called π r,k -horizontal, 0 k < r, if i ξ η = 0 for every π r,k -vertical vector field ξ on J r Y. Let η Λ q (J r Y ). There is a unique horizonal form hη Λ q (J r+1 Y ) such that for every section γ of π, J r γ η = J r+1 γ hη. The mapping h : Λ q (J r Y ) Λ q (J r+1 Y ) is a homomorphism of the exterior algebras and is called horizontalization operator [11]. A form η Λ q (J r Y ) is called contact if J r γ η = 0 for every section γ of π. Obviously, η is contact iff hη = 0. On fibred manifolds where the base X is one-dimensional, every q-form for q 2 is contact. Contact forms on J r Y form a closed ideal in the exterior algebra locally generated by the 1-forms ω i = dx i ẋ i dt, ω i = dẋ i ẍ i dt,..., ω i r 1 = dx i r 1 x i rdt, (1) and their exterior derivatives, called the contact ideal. Let q 1, and let η Λ q (J r Y ) be a contact form. We say that η is one-contact if for every π r -vertical vector field ξ on J r Y 3

4 the (q 1)-form i ξ η is π r -horizontal; we say that η is k-contact, 2 k q, if i ξ η is (k 1)-contact. Every q-form η on J r Y admits the unique decomposition [12] π r+1,r η = p q 1 η + p q η into a sum of a (q 1)-contact and a q-contact form (above we have denoted hη = p 0 η). The form p i η is called the i-contact part of η. A contact q-form is called strongly contact [13] if π r+1,r η = p q η. Contact 1-forms on J r Y annihilate a distribution of constant corank mr, called contact or Cartan distribution of order r, and denoted by C πr. This distribution is not completely integrable. It is equivalently spanned by m + 1 vector fields t + ẋi x + + i xi r x i r 1, x j r, 1 j m. Vector fields belonging to the contact distribution that are everywhere non-vertical are called semisprays. Note that holonomic sections are characterized as integral sections of semisprays. Contact 1-forms (1) can be completed to a basis of linear forms that is well adapted to the fibred structure. In what follows, we shall often use for a local expression of forms on J 1 Y the adapted basis (dt, ω a, dẋ a ) instead of the canonical basis (dt, dx a, dẋ a ), and similarly, for forms on J 2 Y, the adapted basis (dt, ω a, ω a, dẍ a ) instead of (dt, dx a, dẋ a, dẍ a ). In the calculus of variations and the theory of differential equations on fibred manifolds, the basic objects are horizontal 1-forms on J r Y, called Lagrangians of order r, and 1-contact 2-forms, horizontal with respect to the projection onto Y, called dynamical forms. In a fibred chart a Lagrangian λ Λ 1 (J r Y ) and a dynamical form E Λ 2 (J r Y ) take the form and λ = Ldt, where L = L(t, x j, ẋ j,..., x j r), E = E i ω i dt, where E i = E i (t, x j, ẋ j,..., x j r), respectively. To a Lagrangian λ a distinguished dynamical form E λ is assigned, called the Euler Lagrange form of λ [11]. If λ is of order r then E λ 4

5 is of order 2r, and its components E i (L), called Euler Lagrange expressions, are defined by E i (L) = L x d L dr L + + i ( 1)r. dt ẋi dt r The mapping Λ 1 (J r Y ) λ E λ Λ 2 (J 2r Y ) is then called the Euler Lagrange mapping. A dynamical form E is called globally variational if there exists a Lagrangian λ such that (possibly up to a jet projection) E = E λ. E is called locally variational if every point in the domain of E has a neighbourhood U where E is variational. 3 The variational sequence A general framework for our exposition is the variational sequence, introduced by Krupka in 1990 [13, 14]. In the study of geometry of differential equations on fibred manifolds developed in this paper, two kinds of differential forms play a fundamental role: dynamical forms and Lepage 2-forms. Both they appear as distinct representatives of classes in the second column of the variational sequence on a fibred manifold π : Y X, where dim X = 1. Following Krupka, we denote Ω r q the sheaf of q-forms on J r Y, Ω r 0,c = {0}, and Ω r q,c the sheaf of strongly contact q-forms on J r Y. Set Θ r q = Ω r q,c + dω r q 1,c, x i r where dω r q 1,c is the image sheaf of Ω r q 1,c by the exterior derivative d. There arises a sequence 0 Θ r 1 Θ r 2 Θ r 3, (2) where the morphisms are the exterior derivative, i.e. a subsequence of the De Rham sequence 0 R Ω r 0 Ω r 1 Ω r 2 Ω r 3. The sequence (2) is an exact sequence of soft sheaves, so that the quotient sequence 0 R Ω r 0 Ω r 1/Θ r 1 Ω r 2/Θ r 2 Ω r 3/Θ r 3 (3) is also exact. It is called the variational sequence of order r on π. 5

6 As proved in [13], the variational sequence is an acyclic resolution of the constant sheaf R over Y. Hence, due to the abstract De Rham theorem, one gets the identification of the cohomology groups of the cochain complex of global sections of the variational sequence with the De Rham cohomology groups H q Y of the manifold Y. It should be stressed that the elements of the quotient sheaves Ω r q/θ r q, q 1, are not forms, but classes of (local rth-order) q-forms. We denote by [ρ] v an element of Ω r q/θ r q, that is the class of ρ Ω r q. The quotient mappings are denoted by E q : Ω r q/θ r q Ω r q+1/θ r q+1, hence E q ([ρ] v ) = [dρ] v. The name of the sequence relates to the fact that the quotient mapping E 1 : Ω r 1/Θ r 1 Ω r 2/Θ r 2 identifies with the Euler Lagrange mapping of the calculus of variations. The next quotient mapping, E 2 : Ω r 2/Θ r 2 Ω r 3/Θ r 3 is called Helmholtz mapping. The image of a class [ρ] v Ω r 2/Θ r 2 i.e. the class [dρ] v Ω r 3/Θ r 3 is called Helmholtz class. The condition E 1 ([ρ] v ) = 0 by exactness of the variational sequence means that there exists f Ω r 0 such that [ρ] v = [df] v. Hence, we get a (local) function f, such that the Euler Lagrange mapping maps [df] v to zero: in other words, the class [df] v has the meaning of a null Lagrangian. If in addition, H 1 Y = {0}, f may be chosen globally defined on J r Y. Similarly, the condition E 2 ([α] v ) = 0 gives us a class [ρ] v Ω r 1/Θ r 1 such that [α] v = [dρ] v = E 1 ([ρ] v ), i.e. [α] v is the image by the Euler Lagrange mapping of a class [ρ] v. Thus, condition E 2 ([α] v ) = [dα] v = 0 means that [α] v is locally variational (comes from a local Lagrangian, represented by the class [ρ] v ). If H 2 Y = {0}, the existence of a global Lagrangian is guaranteed. 4 Source forms and Lepage forms The quotient sheaves Ω r q/θ r q in the variational sequence are determined up to natural isomorphisms of Abelian groups. As a consequence, classes in Ω r q/θ r q admit various equivalent representations 6

7 by differential forms; these forms, however, may be of order greater than r. Let us turn to the two most important kinds of representations of the elements of the quotient sheaves Ω 1 q/θ 1 q: by source forms, and by Lepage forms. The first possibility is to represent classes in the variational sequence by source forms (we use the original terminology introduced by Takens in his seminal paper [37]). A canonical source forms representation is obtained by means of the interior Euler Lagrange operator, I, introduced to the variational bicomplex theory by Anderson [2, 3], and adapted to the finite order situation of the variational sequence theory in [10, 17]. This operator reflects in an intrinsic way the procedure of getting a distinguished representative of a class [ρ] v Ω r q/θ r q by applying to ρ the operator p q 1 and the factorization by Θ r q. In the first column, i.e. for classes in Ω r 1/Θ r 1 the situation is simple, since p 0 = h and Θ r 1 is the sheaf of contact 1-forms, which makes the factorization trivial, and the source form representation unique. Hence, for [ρ] v Ω r 1/Θ r 1 we simply put Iρ = hρ, and we obtain the class [ρ] v represented by the 1-form λ = hρ Ω r+1 1, i.e. a Lagrangian. For q 2 the factorization is no more trivial. If [ρ] v Ω r q/θ r q, we put [17] Iρ = Ip q 1 ρ, where the operator I is defined by a recurrence formula to be found in [17]. In fibred coordinates, where p q 1 ρ = r k,l=0 the formula for the operator I reads Iρ = Ip q 1 ρ = 1 2 r k,l=0 p=0 H kl ij ω i k ω j l dt, k ( ) k d ( 1) k k p p (Hlk dtk p ji Hij lk )ω j p+l ωi dt. The operator I provides a representation by ω i -generated (q 1)- contact q-forms. It is an R-linear mapping Ω r q Ω 2r+1 q such that 7

8 (i) Iρ belongs to the same class as π 2r+1,rρ, (ii) I 2 = I (up to a canonical projection), (iii) the kernel of I : Ω r q Ω 2r+1 q is Θ r q. Note that (iii) means that Iρ is the same for any ρ belonging to the class [ρ] v. Definition 4.1. By a source form representation of the sheaf Ω r q/θ r q, q 1, we understand an R-linear mapping I : Ω r q Ω s q such that I ρ is a ω i -generated (q 1)-contact q-form, equivalent with Iρ. The form I ρ is then called a source form for the sheaf Ω 1 q/θ 1 q. From now on, let us consider the first-order variational sequence 0 R Ω 1 0 Ω 1 1/Θ 1 1 Ω 1 2/Θ 1 2 Ω 1 3/Θ 1 3. In fibred coordinates, if ρ is a 2-form on J 1 Y, and we denote p 1 ρ = (E i ω i + E 1 i ω i ) dt, we obtain ( ) Iρ = Ip 1 ρ = E i de1 i ω i dt. dt Thus the obtained source forms for Ω 1 2/Θ 1 2 are certain dynamical forms. We can also see that for ρ of order 1, Iρ may be of order 3. Proposition 4.2. For the quotient sheaf Ω 1 2/Θ 1 2 the source form representation by means of the interior Euler Lagrange operator I is unique. Proof. If [ρ] v Ω 1 2/Θ 1 2 then by definition 4.1, any source form σ representing the class [ρ] v is a dynamical form equivalent with Iρ. Thus σ = Iρ + p 1 dη where η is a 1-form such that p 1 dη is a dynamical form, and Iσ = Iρ, i.e. Ip 1 dη = 0. However, since p 1 dη is a dynamical form, the above formulas give us Ip 1 dη = p 1 dη, hence p 1 dη = 0 and σ = Iρ. In the source form representation the Euler Lagrange mapping takes the expected form E 1 : Ω 2 1 Iρ = λ Idρ = Idλ = E λ Ω

9 The Helmholtz mapping in the canonical source form representation becomes E 2 : Ω 3 2 Iρ = E Idρ = IdE = H E Ω 4 3, where the source 3-form H E is called the Helmholtz form of E. In fibred coordinates where E = E i ω i dt is a second-order dynamical form we have (see [17]) H E = 1 Ei 2[( x E j 1 d ( Ei j x i 2 dt ẋ E ) j + 1 d 2 ( Ei j ẋ i 2 dt 2 ẍ E )) j j ẍ ( i Ei + ẋ + E j 2 d E ) ( j ω j Ei + j ẋ i dt ẍ i ẍ E ) ] j ω j ω i dt. j ẍ i The source 3-form representation by means of the interior Euler Lagrange operator I is no more unique: We note that for E of order 2, the obtained Helmholtz form is of order 4. As shown by Krupka in [15], a lower-order source form representation is possible by means of an equivalent Helmholtz form, that is of order 3: H E = 1 ( Ei 2 x E j 1 d ( Ei j x i 2 dt ẋ E )) j ω j ω i dt j ẋ i + 1 ( Ei 2 ẋ + E j d ( Ei j ẋ i dt ẍ + E )) j ω j ω i dt j ẍ i + 1 ( Ei 2 ẍ E ) j ω j ω i dt. j ẍ i Proposition 4.3. Let E be a second-order dynamical form. Then where H E = H E + p 2 dη, η = 1 d ( Ei 4 dt ẍ E ) j ω j ω i. j ẍ i Note that the conditions for the components of the Helmholtz form to vanish have the meaning of necessary and sufficient conditions for a dynamical form E be locally variational (that means that around every point there is a Lagrangian λ such that E = E λ ); they are called Helmholtz conditions in honour of Helmholtz ([7]). A different representation of classes in the variational sequence is realized by so-called Lepage forms [11, 17, 18]. ω j 9

10 Definition 4.4. A q-form ρ, q 1, is called Lepage form if p q dρ is a source form. If σ is a source q-form, we say that ρ is a Lepage equivalent of σ if ρ is a Lepage q-form and p q 1 ρ = σ. The above definition of a Lepage form is wider than that in [17] (where a form ρ is called Lepage if p q dρ = Idρ) admitting Lepage q-forms related with different equivalent source (q + 1)-forms, and consequently, Lepage equivalents of a given source form, that have different orders. From definition 4.4 we compute Lepage equivalents of Lagrangians and dynamical forms: A Lepage equivalent of a Lagrangian λ is defined to be a 1-form ρ such that hρ = λ and p 1 dρ is ω i -generated (=horizontal with respect to the projection onto Y ). Note that this is precisely the original definition due to Krupka [11]. A direct computation then gives that every Lagrangian has a unique Lepage equivalent; it is denoted by θ λ and called the Cartan form. In fibred coordinates, if λ Λ 1 (J 1 Y ), λ = Ldt, one gets θ λ = Ldt + L ẋ i ωi. Note that by definition, p 1 dθ λ = E λ. Let us now turn to Lepage equivalents of dynamical forms. Let E be a second-order dynamical form. By definition 4.4, a 2-form α is a Lepage equivalent of E if p 1 α = E and p 2 dα is a source 3-form. Due to the non-uniqueness of the source form representation for 3- forms, we expect to have Lepage equivalents of a dynamical form, associated with different corresponding source forms. By a direct computation we obtain: Theorem 4.5. (1) Every dynamical form E Λ 2 (J 2 Y ) has a unique global Lepage equivalent α E on J 3 Y, associated with the canonical source form IdE = H E. It is defined by p 2 d α E = H E, and in fibred coordinates where E = E i ω i dt, takes the form α E = E i ω i dt + 1 ( Ei 4 ẋ E j d ( Ei j ẋ i dt ẍ E )) j ω i ω j j ẍ i + 1 ( Ei 2 ẍ + E ) j ω i ω j. j ẍ i 10

11 (2) Every dynamical form E Λ 2 (J 2 Y ) has a unique global Lepage equivalent α E on J 2 Y, associated with the (minimal-order) source form H E. It is defined by p 2 dα E = H E, and in fibred coordinates where E = E i ω i dt, takes the form α E = E i ω i dt+ 1 ( Ei 4 ẋ E j )ω i ω j + 1 ( Ei j ẋ i 2 ẍ + E ) j ω i ω j. (4) j ẍ i Note that α E = α E + η, where η is defined in proposition 4.3. By an easy calculation we have: Theorem 4.6. Let E Λ 2 (J 2 Y ) be a dynamical form. The following conditions are equivalent: (1) The Lepage equivalent α E of E is projectable onto J 1 Y. (2) The Lepage equivalent α E of E is projectable onto J 1 Y. (3) In every fibred chart E = E i ω i dt, where the functions E i are affine in the second derivatives, E i = A i + g ij ẍ j, and g ij satisfy the integrability conditions g ij = g ji, g ij ẋ k = g ik ẋ j. (5) Note that (5) mean that there exists a function f(t, x i, ẋ i ) such that g ij = 2 f ẋ i ẋ. j A dynamical form E satisfying conditions of theorem 4.6 is called semivariational. Variational dynamical forms are characterized by the following theorem [18, 24]: Theorem 4.7. Let E Λ 2 (J 2 Y ) be a dynamical form. The following conditions are equivalent: 11

12 (1) E is locally variational. (2) α E = α E, and the Lepage equivalent of E is projectable onto J 1 Y and closed. (3) There is a unique 2-contact 2-form F such that α = E + F is projectable onto J 1 Y and closed. The concept of a Lepage equivalent of a dynamical form presented above is a generalization of a Lepage equivalent of a locally variational form, introduced in [18] (see also [4, 9, 18, 36] for the relation between locally variational forms and closed 2-forms). Note that in the Lepage form representation morphisms E q, q 0, in the variational sequence become simply the exterior derivatives, and on Lepage forms we have where p 0 = h and p 1 = id. E q p q 1 = p q d, 5 Second order differential equations and Lepage 2-forms Let E Λ 2 (J 2 Y ) be a dynamical form. A section γ of π is called path of a dynamical form E if E J 2 γ = 0. (6) In fibred coordinates equation (6) for paths of E becomes a system of m second-order differential equations for sections of π : Y X, E i (t, x j, ẋ j, ẍ j ) = 0. (7) Given a dynamical form E, we say that a 2-form α is an extension of E if E = p 1 α. We have seen that every dynamical form E Λ 2 (J 2 Y ) has a global second-order extension the Lepage equivalent α E of E. Usually we shall consider also local extensions (i.e. defined on open subsets of J 2 Y ); we denote by dom α the domain of definition of α. A dynamical form E Λ 2 (J 2 Y ) is called pertinent with respect to J 1 Y, if around every point in J 2 Y it has a local extension α that is projectable onto an open subset of J 1 Y. 12

13 Proposition 5.1. [24] A dynamical form E Λ 2 (J 2 Y ) is pertinent with respect to J 1 Y if an only if E i = A i (t, x k, ẋ k ) + g ij (t, x k, ẋ k )ẍ j. (8) Then every local projectable extension of E takes the form α = A i ω i dt + g ij ω i dẋ j + F, where F is a 2-contact 2-form on an open subset of J 1 Y. We can classify dynamical forms (i.e. differential equations) according to the properties of their Lepage equivalents: (1) A general dynamical form E Λ 2 (J 2 Y ); it has a unique global second-order Lepage equivalent α E (4). (2) J 1 Y -pertinent dynamical forms: they have the Lepage equivalent α E locally decomposed into the sum of a first-order extension α 0 E of E, and a second-order 2-contact form ϕ E as follows where and α E = α 0 E + ϕ E, (9) α 0 E = A i ω i dt + g ij ω i dẋ j ϕ E = 1 4 ( Ai ẋ A ) j ω i ω j, (10) j ẋ i (( gik ẋ g ) ) jk ẍ k ω i ω j + 1 ) (g j ẋ i ji g ij ω i ω j. (11) 2 Pertinent dynamical forms correspond to second-order differential equations g ij (t, x k, ẋ k ) ẍ j + A i (t, x k, ẋ k ) = 0. (12) (3) Pertinent dynamical forms such that ϕ E in (9) is π 2,0 -horizontal. This means that g ij = g ji, hence α E = A i ω i dt + g ij ω i dẋ j + 1 ( Ai 4 ẋ A ) j ω i ω j j ẋ i + 1 (( gik 4 ẋ g ) jk ẍ )ω k i ω j. j ẋ i 13

14 (4) Semivariational dynamical forms: the Lepage equivalent α E is projectable onto J 1 Y, and reads α E = A i ω i dt + 1 ( Ai 4 ẋ A ) j ω i ω j + g j ẋ i ij ω i dẋ j (i.e. α E = α 0 E, ϕ E = 0 in the formula for α E in (2) above). It can be shown that a semivariational dynamical form E splits canonically into a sum of a variational dynamical form E g, coming from a Lagrangian τ = T dt, uniquely determined by the g = (g ij ) (generalised kinetic energy ), and a first order dynamical form φ (a force ), so that equations for paths (12) take the form T x d T i dt ẋ = φ i. i (4) Locally variational dynamical forms: their Lepage equivalent α E is closed. (5) Globally variational dynamical forms: the Lepage equivalent α E is exact. Given a pertinent dynamical form E Λ 2 (J 2 Y ) there arise the following distributions ([21, 22, 25]): (1) Dynamical distribution: is associated with every extension α of E and is defined on dom α by α = annih{i ξ α ξ runs over vertical vector fields on dom α}. (2) Characteristic distribution of the 2-form α: is associated with every extension α of E and is defined on dom α by χ α = annih{i ξ α ξ X(dom α)} = span{ζ X(dom α) i ζ α = 0}. (3) Evolution distribution D E on J 1 Y : It is defined by D E : J 1 Y x ( α C π1 )(x) T x J 1 Y, where α is a local projectable extension of E, defined in a neighbourhood of x. It is easily seen that this definition is correct, since the vector space ( α C π1 )(x) does not depend upon the choice of α. Note that rank D E 1, and the rank may be nonconstant. Given a local projectable extension α of E, the above distributions are related as follows: D E χ α α 14

15 on dom α J 1 Y, so that χ α and α serve as local enveloping distributions for D E. Prolongations of paths of a pertinent dynamical form E (= solutions of equations (12)) coincide with integral sections of the evolution distribution D E. It is also important that for any extension α of E, the dynamical distribution α and the characteristic distribution χ α have the same holonomic integral sections, and these coincide with prolongations of paths of E in domα. This means, in particular, that equations (12) have the following equivalent coordinate free representations: For any projectable extension α of E, paths of E in dom α are (1) solutions of the equations J 1 γ i ξ α = 0 for every π 1 -vertical vector field ξ on dom α, (2) holonomic integral sections of the characteristic vector fields of the 2-form α, i.e. vector fields ζ that are solutions of the equation i ζ α = 0. Note that equation i ζ α = 0 may have solutions ζ that are not continuous and are not semisprays. The above exterior differential systems analysis of equations (12) leads to the following classification of pertinent dynamical forms with respect to the properties of their paths ([21]): A dynamical form E is called (1) regular if rank D E = 1, (2) weakly regular if the distribution D E is weakly horizontal and has a (locally) constant rank, (3) with no semispray constraints if D E is weakly horizontal, (4) semiregular if αe is weakly horizontal, of a (locally) constant rank, and completely integrable. Equipped with this classification, one can effectively study the Cauchy problem and develop integration methods for second-order ordinary differential equations. For this analysis, one uses local extensions of a dynamical form in general, and the Lepage equivalent α E in particular, namely if E is variational or semivariational. (For more details and techniques we refer to the work of Krupková [20, 15

16 21, 22, 23, 25, 26], and the survey paper by Krupková and Prince [27]). 6 Regular dynamical forms In the sequel we shall be interested in regular pertinent dynamical forms. Theorem 6.1. [19, 24] Let E be a pertinent dynamical form on J 2 Y. The following conditions are equivalent: (1) E is regular. (2) The matrix is regular. ( Ei ) g := (g ij ) = ẍ j (3) The evolution distribution D E is locally spanned by a semispray. It takes the form Γ = t + ẋi x i gij A j ẋ, (13) i where (g ij ) = g 1. (4) Equations for paths of E have an equivalent normal form ẍ i = g ij A j. (14) (5) Every projectable extension α of E has the maximal rank (equal to 2m). (6) For every projectable extension α of E, D E is the characteristic distribution of α. (7) For every projectable extension α of E, the equation i Γ α = 0 with the additional condition dt(γ) = 1 has a unique local solution Γ. It takes the form (13). 16

17 In view of the above theorem we refer to (14) as to equations of paths of E in a contravariant form. Let us turn to study extensions of regular pertinent dynamical forms. We have seen that E relates with a family of extensions defined on open subsets of J 1 Y. In fibred coordinates they read α = A i ω i dt + g ij ω i dẋ j + F ij ω i ω j, (15) where F ij are arbitrary functions, skewsymmetric in the indices i, j. Put f i = g ij A j, h i j = g ik F kj, and ω i Γ = dẋ i f i dt, ψ i h = h i jω j + ω i Γ. Given an α, there are two bases of one forms on J 1 Y that are adapted to α: (dt, ω i, ω Γ), i (dt, ω i, ψh). i With this notation, the family of local projectable extensions (15) of E consists of 2-forms α = g ij ω i ω j Γ + F = g ijω i ψ j h, (16) where F (resp. h) is arbitrary. Among these local 2-forms we have one that is canonically associated with the semispray Γ: it reads where Ω = g ij ω i ψ j, (17) ψ i = ω Γ i 1 f i 2 ẋ j ωj. (18) For the Lepage equivalent α E of E we get by (9)-(11) the expression α E = αe 0 + ϕ E, where αe 0 = Ω 1 ( gik 4 ẋ g ) jk f k ω i ω j. (19) j ẋ i Note that α E = Ω + 1 g ik 2 ẋ j (ẍk f k )ω i ω j + 1 ) (g ji g ij ω i ω j. 2 By an easy computation we get the following proposition, showing the place of the form Ω in the variational sequence: 17

18 Proposition 6.2. The 3-form p 2 dω is ω i -generated (i.e. is a source form) if and only if g is symmetric. The above proposition means that for g symmetric p 2 dω is equivalent with the Helmholz form H E = p 2 dα, i.e., the components of p 2 dω correspond to the Helmholtz conditions for local variationality of E. The 2-form Ω is important in the study of the inverse variational problem for semisprays. We shall return to it in the next section, where we also show that if moreover the underlying fibred manifold π is fibred over R and admits a global trivialisation, i.e. Y = R M, then Ω is global on J 1 Y = R T M (i.e., the splitting α E = Ω + 1 g ik 2 ẋ j (ẍk f k )ω i ω j is invariant with respect to the change of fibred coordinates adapted to the product structure of Y ), and can be obtained by an elegant intrinsic construction. Note also that if E is semivariational then α E = Ω, so that Ω is global on J 1 Y and p 2 dω = H E. E is locally variational iff Ω is closed, and globally variational iff Ω is exact. Returning now back to theorem 4.7, we obtain its version for regular dynamical forms as follows: Theorem 6.3. Let E Λ 2 (J 2 Y ) be a regular dynamical form. The following conditions are equivalent: (1) E is locally variational. (2) There is a unique 2-form α E = Ω such that (i) dω = 0, (ii) E = p 1 Ω, (iii) Ω has the maximal rank 2m. The evolution distribution D E = span{γ} of E is the characteristic distribution of Ω. This means that i Γ Ω = 0, and span{γ} is the unique solution of the above equation. Note that if E is variational then Ω = dθ λ. 18

19 7 Closed forms, Lepage forms and semisprays By a (second order) semispray connection we mean a section w : J 1 Y J 2 Y. The horizontal distribution H w of a semispray connection w is locally spanned by the semispray Γ = t + ẋi x + f i i ẋ, i where f i = ẍ i w are components of w, or equivalently, by the system of one-forms ω i, ω i Γ = w ω i = dẋ i f i dt. In this way we come to the basis (dt, ω i, ω Γ i ) of one-forms on J 1 Y, adapted to the semispray connection, considered in the previous section. Also the second kind of adapted bases to Γ considered in Sec. 6, namely (dt, ω i, ψ i ) with ψ i given by (20), can be justified by geometric constructions, provided that π is a trivial fibred manifold over R (see [4, 8]). Given a semispray connection w, respectively, a semispray Γ on J 1 Y there arise local adapted dual bases {Γ, H i, V j } and {dt, ω i, ψ j } constructed intrinsically as follows. Consider the 1-jet bundle J 1 Y fibred over Y, π 1,0 : J 1 Y Y ; in terms of the trivialisation Y = R M, the fibre of this vector bundle over (t, x) is just T x M. π 1,0 -vertical vectors on J 1 Y are tangent to these fibres, and the submodule of X(J 1 Y ) they generate is denoted V (J 1 Y ). Locally V (J 1 Y ) = span{v j := }. ẋ j The vertical endomorphism S is the unique (1, 1) tensor field on J 1 Y with the properties (1) S vanishes on π 1,0 -vertical vectors and semisprays, (2) S is vertical valued acting as a vector valued 1-form, (3) S( t ) =, the dilation field on the fibres of π 1,0 : J 1 Y Y. In coordinates S = ẋ i ω i. While S characterises the structure of J 1 Y, the eigenspaces of its deformation, L Γ S, create a direct sum decomposition X(J 1 Y ) = span{γ} V (J 1 Y ) H(J 1 Y ), with associated projectors I J 1 Y = P Γ + P V + P H. 19

20 H(J 1 Y ) is called the Γ-horizontal distribution and in coordinates { H(J 1 Y ) = span H i := x + 1 f i i 2 ẋ j }. ẋ i This completes the construction of the vector basis, the dual basis of 1-forms is {dt, ω i, ψ j } with ψ i := dẋ i f i dt 1 f i 2 ẋ j ωj. (20) With these bases the projectors have the coordinate expressions: P Γ = Γ dt, P H = H i ω i, P V = V j ψ j. There are a further two important intrinsically defined objects, the Jacobi endomorphism Φ := P V L Γ P H, and the Massa and Pagani connection (of Berwald type) ˆ (see [30, 8, 31]). This connection is defined by requiring that the covariant differentials ˆ dt, ˆ S and ˆ Γ are all zero and V (J 1 Y ) is flat. In coordinates, where Φ = Φ i j V i ω j = ( B i j Γ i kγ k j Γ(Γ i j) ) V i ω j, (21) Bj i := f i and Γ i x j j := 1 f i 2 ẋ. j The most useful components of ˆ are ˆ Γ Γ = 0, ˆ Γ H j = Γ i jh i, ˆ Γ V j = Γ i jv i. We will save the Lie bracket relations between the vector basis elements until later. Given a semispray connection w : J 1 Y J 2 Y, sections of π satisfying w J 1 γ = J 2 γ are called paths or geodesics of the connection w. Prolongations of paths of w coincide with integral curves of the vector field Γ, and are solutions of a system of m second-order ordinary differential equations in normal form ẍ i = f i (t, x j, ẋ j ). (22) Semispray connections are closely connected with regular dynamical forms: 20

21 Theorem 7.1. [19] Given a pertinent dynamical form E on J 2 Y there is a unique semispray connection w : J 1 Y J 2 Y such that equations for paths of E coincide with the geodesics of w. The connection w is defined by w E = 0. In fibred coordinates, where E = (A i + g ij ẍ j )ω i dt, the components of w take the form f i = g ij A j, where g 1 = (g ij ) is the inverse matrix to g. Conversely, given a semispray connection w : J 1 Y J 2 Y, there is a family of pertinent dynamical forms such that geodesics of w coincide with paths of any of the dynamical form. Each of the dynamical forms related with w is given as a solution of the equation w E = 0. In fibred coordinates where w has components f i, the components of the dynamical forms become E i = g ij (ẍ j f j ), where g = (g ij ) is an arbitrary regular matrix. If w and E are related then the horizontal distribution of the connection w is the evolution distribution D E, i.e. D E = H w = span{γ}, Γ = t + ẋi x i gij A j ẋ. i It is now natural to pose the so-called multiplier problem of Douglas [6]. We begin with m second-order ordinary differential equations in normal form (22) and seek a regular multiplier g ij with the property that the dynamical form E := (g ij ẍ j g ij f j )ω i dt is locally variational. Theorem 6.3 then gives necessary and sufficient conditions on the f i for the local variationality of E. Following (17) we see that these are that the form Ω = g ij ω i ψ j must be closed and of maximal rank 2m. Before proceeding to examine this closure condition in detail we quote a theorem due to Crampin, Prince and Thompson [4], which gives an alternative set of necessary and sufficient conditions in this case. 21

22 Theorem 7.2. Given a system (22) and corresponding semispray Γ, necessary and sufficient conditions for the existence of a locally variational dynamical form E, whose evolution distribution is spanned by Γ, are that there exists Ω Λ 2 (J 1 Y ) : (1) Ω has maximal rank, (2) Ω(ξ 1, ξ 2 ) = 0, for all π 1,0 -vertical ξ 1, ξ 2, (3) L Γ Ω = 0, (4) i H dω(ξ 1, ξ 2 ) = 0 for all Γ-horizontal H and π 1,0 -vertical ξ 1, ξ 2. This theorem represents a minimal set of conditions on the two form Ω, but ultimately this amounts to the requirement that Ω = g ij ω i ψ j must be closed and of maximal rank 2m for some g ij. It would be aesthetically more pleasing to deal with the multiplier g directly in an intrinsic way; this is achieved through the Kähler lift construction to be described below. Firstly we introduce vector fields and forms along the projection π 1,0 : J 1 Y Y. We follow [8, 28, 29, 35, 34]. Vector fields along π 1,0 are sections of the pull back bundle π 1,0 (T Y ) over J 1 Y. X(π 1,0 ) denotes the C (J 1 Y )-module of such vector fields. Similarly, Λ(π 1,0 ) denotes the graded algebra of scalar-valued forms along π 1,0 and V (π 1,0 ) denotes the Λ(π 1,0 )-module of vector-valued forms along π 1,0. Basic vector fields and 1-forms along π 1,0 are elements of X(Y ) and X (Y ) respectively identified with vector fields and forms along π 1,0 by composition with π 1,0. Using this device tensor fields along the projection can be expressed as tensor products of basic vector fields and 1-forms with coefficients in C (J 1 Y ). The canonical vector field along π 1,0 is T = t + ẋi x, i and the natural bases for X(π 1,0 ) and X (π 1,0 ) are then {T, x i } and {dt, ω i }. The set of equivalence classes of vector fields along π 1,0 modulo T is denoted X(π 1,0 ) so that ξ X(π 1,0 ) satisfies dt(ξ) = 0. Then the obvious bijection between X(π 1,0 ) and V (J 1 Y ) provides a vertical lift from X(π 1,0 ) to V (J 1 Y ), given in coordinates by: ξ V = ξ i ẋ = i (ξi ẋ i ξ 0 ) ẋ i 22

23 where ξ = ξ 0 + t ξi. x i The horizontal lift ξ H of ξ X(π 1,0 ) is given by ξ H = ξ i H i. Finally, we can lift along Γ by ξ Γ := dt(ξ)γ for any ξ X(π 1,0 ) (so that T Γ = Γ). Then any vector field ζ X(R T M) can be decomposed as ζ = (ζ Γ ) Γ + (ζ H ) H + (ζ V ) V for unique ζ Γ span{t}, ζ H X(π 1,0 ) with ζ H (t) = ζ(t), and ζ V X(π 1,0 ). This decomposition is the main aim of the lifting exercise. In coordinates, ζ Γ = dt(ζ)t, ζ H = dt(ζ) t + dxi (ζ) x i = dt(ζ)t + ωi (ζ) x i, ζ V = ψ i (ζ) x i. We can now give the fundamental Lie brackets in an intrinsic form: [Γ, ξ V ] = ˆ Γ ξ V ξ H, [Γ, ξ H ] = ˆ Γ ξ H + Φ(ξ) V. (23) The observation we made after theorem 7.2 about the simple structure of the Cartan 2-form dθ λ in the adapted co-frame means that the 2-form Ω on J 1 Y of theorem 7.2 which we seek is completely determined by a symmetric nondegenerate type (0,2) tensor along π 1,0, of the form g = g ij ω i ω j (i.e. g vanishes on T) (see [5, 33]). To be precise, Ω is the so-called Kähler lift of g, Ω = g K, which vanishes on Γ and satisfies g K (ξ V, ζ V ) = g K (ξ, ζ) = 0, g K (ξ V, ζ) = g(ξ, ζ). In this formulation the conditions (equivalent to those) in theorems 6.3 and 7.2 are g = 0, g(φξ, ζ) = g(ξ, Φζ), ˆDξ V g(ζ, η) = ˆD ζ V g(ξ, η), (24) where ˆD is the covariant derivative along π 1,0 defined by ˆ ζ ξ = ( ˆD ζ ξ Γ ) Γ + ( ˆD ζ ξ H ) H + ( ˆD ζ ξ V ) V for all ζ, ξ X(J 1 Y ). 23

24 While there is a certain elegant economy in this presentation of the multiplier g and the conditions for local variationality, there are certainly some drawbacks. For example, exterior calculus cannot be performed in this setting and there is a whole new hybrid calculus to be dealt with. (We refer the interested reader to the recent thesis of Aldridge [1] in which an intrinsic version of (24) is given on J 1 Y using the Massa and Pagani connection.) So we finish by examining in coordinate form the conceptually simpler condition, dg K = 0, which along with the maximal rank condition, are the necessary and sufficient conditions for the dynamical form E := (g ij ẍ j g ij f j )ω i dt to be locally variational. We set Ω := g ij ψ i ω j (this ordering is traditional) and compute dω: dω = (Γ(g ij ) g kj Γ k i g ik Γ k j )dt ψ i ω j + (H l (g ij ) g kj V i (Γ k l ))ψ i ω j ω l + V k (g ij )ψ k ψ i ω j + g ij ψ i ψ j dt + g ki Φ k j ω i ω j dt + g ki H j (Γ k l )ω i ω j ω l. This delivers the four Helmholtz conditions of Douglas in the form originally given by Sarlet ([32]) as dω(γ, V i, V j ) = 0, dω(γ, V i, H j ) = 0, dω(γ, H i, H j ) = 0, dω(h i, V j, V k ) = 0. The remaining conditions arising from dω = 0, namely dω(h i, H j, V k ) = 0 and dω(h i, H j, H k ) = 0, can be shown to be derivable from the first four: Indeed, they appear in the 3-contact part of dω, however, as we have seen in Sec. 6, necessary and sufficient conditions of local variationality arise from the requirement p 2 dω = 0. These conditions form the basis of a significant line of investigation of this inverse problem in the calculus of variations and the author is referred to the recent review article [27] by the present authors for more detail. Acknowledgements. The first author appreciates support of the Czech Science Foundation, grant GACR 201/06/0922, and of 24

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